*Chapter No.3***GASES**

__States of Matter__

Matter exists in four states solid , liquid , gas and plasma.

The gases are the simplest form of matter.

__Properties of Gases__ The general properties of gases are the following:

*Gases have no definite volume:*They occupy all the available space. The volume of the gas is the volume of the container.*Gasses have no definite shape.*They take the shape of the container.*Gasses have low densities.*Gases have low densities as compared with liquids and solids. The gases bubble through liquids and tend to rise up.*Gasses con diffuse and effuse.*They can diffuse and effuse rapidly through each other in all directions. The odour spreads in this way. The property of diffusions operates in liquids as well but is negligible in solids.*Gases are compressible.*They can easily be compressed by applying pressure because there are large empty spaces between the molecules.*Gases are expendables.*They can expand on heating or increasing the available volume . They expand to fill the entire volume of their containers.

Liquids and solids do not show an appreciable increase in volume when they are heated. On sudden expansion of gases, cooling effect is observed. this phenomenon is called

*Joule-Thomson effect*.*Gases exert pressure*. They exert pressure on the walls of the container.*Gases are miscible.*They can be mixed in all proportions forming a homogeneous mixture.*weak intermolecular forces.*The intermolecular forces in gases are very weak.*Gases are liquefiable.*They may be converted to the liquid state with sufficiently low temperature and high pressure.

__Properties of Liquids__ All liquids show the following general properties.

1.

*Liquids have definite volume but indefinite shape.*They adopt the shape of the container in which they are placed but their volume does not change.2.

*Liquids molecules are in constant random.*The evaporation and diffusion of liquid molecules are due to this motion.3.

*Liquids are denser than gases.*The densities of liquid are much greater than those of gases but are close to those of solids.4.

*Liquid molecules are very close together*. Molecules of liquid lie close together with very little space between them so they cannot be compressed.5.

*Intermolecular attractive forces.*The intermolecular attractive forces in liquids are intermediate between gases and solids. The melting and boiling points of gases, liquid and solids depend upon the strength of such forces. The strength of these forces is different in different liquids.6.

*Liquid molecules possess kinetic energy*. Molecules of liquids possess kinetic energy due to their motion. Liquid can be converted into solids on cooling, e I , by decreasing their kinetic energy. Molecules of liquid collide among them-selves and exchange energy but those of solid cannot do so.**Properties of Solids**

1.

*Solid particles are very close together.*The particles of solid substances are very close to each other and they are tightly packed. Due to tight packing of particles solids are non-compressible and non-diffusible.2.

*Strong attractive force .*There are strong attractive forces in solids which hold the particles together firmly. Due to this reason solids have definite shape and volume.3.

*The solid particles possess only vibration motion.*

__Units of Pressure__*“The pressure of air (Earth’s atmosphere) at sea level that will support a column of mercury 760 mm in height is called one atmosphere.”*

It is the force exerted by 76 cm long column of mercury on an area of 1 cm

^{2 }at O^{0}C .It is the average pressure of atmosphere at sea level. A second common pressure unit is the torr. One torr is the pressure exerted by a column of mercury 1 mm in height.**So,**

**1 atm=760 torr=760mm Hg**

The SI unit for pressure is the Pascal (pa). Remember that pressure is force per unit area, so Pascal can by expressed using the SI units for force and area. The SI unit of force is the Newton (N), and are area is measured in square (m

^{2}).Thus**, I pa =1Nm**^{-2 }, and IN =1 kg ms^{-2},**So**,

**1 pa =1 kg m**

^{-1 }S^{-2} Hence,

**1 atm =760 torr =760mm Hg=101325 pa =101.325 kpa**(kilopascal). The most commonly unit of pressure used in engineering work is pounds per square inch (psi).

**1 atm -760=760torr =14.7 pounds inch 2 .**

The unit of pressure mill bar is commonly used by meteorologists.

__Gas Laws__*“The relationships between the volume of a given amount of gas and the prevailing conditions of temperature and pressure are called the gas laws.”*

When the external conditions of temperature and pressure are changed, the volume of a given quantity of all gases in affected. This effect is nearly the same irrespective of the nature of the gas. Thus gases show a uniform behavior towards the external conditions. The gas laws describe this uniform behavior of gases.

__Boyle,s Law__*“At constant temperature, the volume of a given mass of a gas is inversely proportional to the pressure applied a the gas.”*

Mathematically:

(T and n constant)

PV=K ----------(1)

Where”K”is a proportionality constant. The value of k is different for different amounts of the same gas .

According to Eq (1), boyle’s law can be stated as follows:

*“At constant temperature, the product of pressure and volume of a fixed amount of a gas remains constant.”*

So, P

_{1}V_{1 }=k and P_{2}V_{2}=k Hence,

**P**_{1}V_{1 }=P_{2 }V_{2 } Where P

_{1 }and V_{1 }are the initial pressure and volume while P_{2 }and V_{2 }are the final pressure and volume.

__Experimental Verification of Boyles,s Law__ Consider a fixed amount of a gas in a cylinder at constant temperature say at 25

^{o}C. The cylinder is fitted with a moveable piston and a manometer to read the pressure of the gas directly .Let the initial volume of the gas is 1 dm^{3 }and its pressure is 2 atm when the piston has one weight on it. When the piston is loaded with a weight two times greater with the help of two equal weights, the pressure becomes 4 atm and the volume is reduced to dm^{3}. Similarly when the piston is loaded with a weight three times greater.Then the pressure becomes 6 atm and volume is reduced to dm

^{3 }.ThusP

_{1 }V_{1 }=2atmx1dm^{3 }=2 atm dm^{3 }=kP

_{2 }V_{2 }=4atm x dm^{3 }= 2atm dm^{3 }=kP

_{2 }V_{2 }=4atm x dm^{3 }= 2atm dm^{3 }=kHence the Bye’s law is verified.

The value of k will remain the same for the same quantity of a gas at the same temperature.

**Picture**

**Fig, Verification of Boyle,s Law**

**Example 1 :**A gas having a volume of 10 dm

^{3 }is enclosed in a vessel at 0

^{o}C and the pressure is 2.5 atmospheres . This gas is allowed to expand until the new pressure is 2 atmospheres. what will be the new volume of this gas, if the temperature is maintained at 273 k.

**Solution: Given:**P

_{1 }=2.5atm ; P

_{2 }=2atm

V

_{1 }=10dm^{3}; V_{2 }=? Since temperature is constant ( 0o C =273k )

Formula Used: V

_{2 }=P_{1 }V_{1 } V

_{2 }= V

_{2}=**V**

_{2}=12.5 dm^{3 }Answer

__Graphical Explanation of Boyles,s Law__ Take a given amount of gas and enclose it in a cylinder having a piston in it. The Boyle,s law can be represented graphically in three different ways as described below:

(i) Keeping the temperature constant at 0o C, when the pressure of the gas is varied, its volume changes. On plotting a graph between pressure on the x-axis and volume on the y-axis, a curve is obtained. Fig (a). This graph shows that volume is inversely proportional to pressure. Increase in pressure decreases the volume.

*“The pressure –volume curve obtained at constant temperature is called isotherm”*

Now increase the temperature of the same gas to 25o C .keeping the temperature constant graph between pressures of the gas is varied, its volume changes. Again on plotting a obtained .this curve goes away from both the axes, fig (b). The reason is that, at higher temperature, the volume of the gas has increased. Again, this graph shows that volume is inversely proportional to pressure. The increase in pressure decreases the volume.

Similarly, if we increase the temperature further, keeping it constant, and again plot another isotherm, It further goes away from the axes and produces the same result.

**Picture**

Fig (a) . Isotherm of a gas at 0

^{o}C Fig (b). Isotherm of a gas at different temperatures(ii) Keeping the temperature constant at T

_{1}, when the volume of the gas is increased, the gas spreads over a larger region of space. Consequently, it exerts less pressure on the container. It means increase in column decreases the pressure of the gas.On plotting a graph between the reciprocal of volume ( )n x-axis and pressure (P) on the y-axis. It shows that p is directly proportional ( )to . It indicates that the pressure goes down as the gas expands. This straight line will meet at the origin where both p and ( )are zero. Thus we can say that p goes to zero as V goes to infinity ( =0). Now increase the temperature of the same gas from T

_{1 }to T_{2 }. Keeping the temperature constant at T_{2 }. When volume is varied, its pressure changes. Again plot the graph between ( )on x-axis, and P on y-axis, again a straight line is obtained which meet at the origin . This straight line will be away from the x-axis, Fig(a) . Again the straight line straight line indicates that P is directly proportional to ( ).**Picture**

Fig.(a) A plot between P and Fig (b) A plot between pressure and product pf P V

(iii) Keeping the temperature constant at T

_{1}, when the pressure of the gas is varied, its volume changes. On plotting a graph between P on x-axis and the product PV on y-axis, a straight line parallel to x-axis is obtained, Fig (b). This straight line shows that PV remains constant even if we change pressure.Now increase the temperature of the same gas form T

_{1 }to T_{2}. Keeping the temperature constant at T_{2}, when the pressure of the agars is varied, its volume change. On plotting graph between P on x-axis and the product PV on y-axis, again a straight line parallel to x-axis is obtained, Fig (b) . This straight line shows that PV remains constant even if we change pressure. However, the value of PV increases with increase in temperature.

__Charles’s Law__*“At constant pressure, the volume of a given mass of a gas is directly proportional to the absolute temperature.”*

**Mathematically:**V = T (P and N are constant)

V = kT

= k

Where k is a proportionality constant.

If the temperature is changed from T

_{1 }to T_{2 }and the volume is changed from V_{1 }to V_{2 }then we have = k

and = k

so, = = k

**=**

__Experimental Verification of Charles, s Law__ Consider a fixed amount of a gas enclosed in a cylinder fitted with a moveable piston. Let the volume of the gas is V1 and its temperature is T1. If the gas in the cylinder is heated, the temperature and the volume of the gas will increase . The new temperature be T2 and volume will be V-2 . It can be shown that the ratio.

=

Hence Charles’ s law is verified.

**Picture.**

Fig. Verification of Charles’s Law

**Example 2 :**250cm

^{3 }of hydrogen is cooled from 127

^{o }C to -27o C by maintaining the pressure constant. Calculate the new volume of the gas at the low temperature.

**Solution: Given:**V1 =250cm

^{3}; V

_{2 }=?

T

_{1 }=273+127=400k ; T_{2 }=273-27=246k Formula Used: =

V

_{2}= xT_{2} V

_{2}= V

_{2}= 153.75 cm_{3 }Answer

__Derivation of Absolute Zero__ In order to derive absolute zero of temperature considers the following quantitative statement of Charles’s law.

Statement: “At constant pressure , the volume of a given mass of gas increases or decreases by of its original volume at 0

_{o }C for every 1_{o}C rise or fall in temperature respectively.” In order to understand the above statement .suppose V

_{o}is the volume of a givenMass of gas at 0

^{o}C and V_{1 }is the volume at T^{o}C .Then from the statement of Charles,s law. For 1

^{o}C rise in temperature, the increase in volume =Vo x=And, for to C rise in temperature ,the increase in volume =xt

=

The volume , V

_{t }of the gas at to C becomes , V

_{t }=vol of gas at 0^{o}C =increase in vol.at t^{o}C V

_{t}=V_{o }+ V

_{t}=V_{o }(1+)This general equation can be used to know the volume of the gas at various temperatures.

Suppose at 0

^{o}C , the original volume , V_{o }of the gas is 546 cm^{3 }.Thus Vo =546 cm3

At 10o C, V10 =546(1+dm

^{3} At 100 oC V100 =

cm

^{3 } Thus , the increase in temperature from 10

^{o }C to 100^{o}C, increases the volume from 566 cm^{3 }to 746cm^{3 }. Applying the Charles’s Law,

=

The two ratios are not equal. So , Charles’s law is not being obeyed when temperature is measured on the Celsius scale. For this reason a new temperature scale has been developed.

The new temperature scale starts from-273

^{o}C (more correctly-273.16^{o}C) which is called zero Kelvin (0k) or zero absolute. The advantage of this scale is that all the temperatures on this scale are in positive figures .In order to develop the new temperature scale, the best way is to plot a graph between the variables of Charles,s law . That is, V and T.**Table: volume –Temperature data for a given amount of a gas at constant pressure**

Volume (cm^{3}) | Celsius Temperature(^{o}C) | Temperature (K) | =k=cm^{3 }k^{-1} |

1092 | 273 | 546 | 2 |

846 | 150 | 423 | 2 |

746 | 100 | 373 | 2 |

646 | 50 | 323 | 2 |

566 | 10 | 283 | 2 |

548 | 1 | 274 | 2 |

546 | 0 | 273 | 2 |

544 | -1 | 272 | 2 |

526 | -10 | 263 | 2 |

400 | -73 | 200 | 2 |

346 | -100 | 173 | 2 |

146 | -200 | 73 | 2 |

0 | -273 | 0 | 2 |

__Graphical Explanation of Charles,s Law__ Keeping the pressure of a given mass (say 1 mole) of gas constant, when the temperature of a gas is varied, its volume changes. The volumes at different temperatures are measured, Now if a graph is plotted between temperature (

^{o}C) on x-axis and volume on y-axis , a straight line is obtained ,this straight line cuts the temperature-axis and (x-axis) at -273. 16^{o}Cif it is extrapolated. The volume of a gas becomes zero at -273.16^{o}C. This temperature is the lowest temperature which is attainable if the substance remains in the gaseous stat. Actually, all real gases first converted into liquid and then into solids before reaching this temperature**Picture**

**Fig. the graph between volume and temperature for a gas according to table.**

If many plots of this type are examined. It is found that a given gas follows different straight lines for different masses of gas and for different pressures. Greater the mass of gas taken, greater will be the slope of the straight line. The reason is that greater the number of moles greater the volume occupied .All these straight lines when extrapolated meet at a common point of -273.16

^{o}C (0k). It is clear that this temperature of -273.16oC will be attained when the volume becomes zero. It is true for an ideal gas. But for a real gas. Thus-273.16oC represents the coldest temperature. This is the zero point (0K) for an absolute scale of temperature. Charles’s law is obeyed when the temperature is taken on the Kelvin scale. For example, at 283K (10

^{o }C) the volume is 566 cm^{3 }and at 373 k (100^{o }C) the volume is 746 cm^{3 }. According to Charlie’s law. ==k

Hence , Charles’s law is obeyed .

__Scales of Thermometry__

For measuring temperature, three scales of thermometry are used.

(

**a) Celsius or Centigrade Scale** One this scale, the temperature of ice at 1 atmospheric pressure is 0

^{o}C, and the temperature of boiling water at 1 atmospheric pressure is 100^{o}C. The space between these temperature marks is divided into 100 equal parts and each part is 1^{o}C**(b) Fahrenheit Scale**

On this scale, the temperature of ice at 1 atmospheric pressure is 32Fand the temperature of boiling water is 212

^{o }F .The space between these temperature marks is divided into 180equal parts and each part is 1^{o}F(c) Absolute or Kelvin scale

One the absolute or Kelvin scale, the temperature of ice at 1 atmospheric pressure is 273 K and that of boiling water is 373 K. On the Kelvin scale, absolute zero (0K) is the temperature at which the volume of a gas becomes zero. It is the lowest possible temperature that can be achieved. Thus, temperatures on the Kelvin scale are not divided into degrees .Temperatures on this scale are reported in units of Kelvin , not in degrees temperature on Kelvin Scale=273.16

^{o}C or 0K=273.16^{o}CTemperature on Kelvin scale=273+temperature

^{o}C Or

**T (k)=273+**

^{o}C The following relationship helps us to understand the introversions of various scales of temperatures.

**T (k)=273+**

^{o}C

^{o}**C=**

^{o}**f= (**

^{o}C)+32**Picture**

**Remember**that the Fahrenheit and Celsius scales are both relative temperature scales. They define two reference points (such as 0

^{o}C and 100

^{o}C). The Kelvin scale, however, is an absolute scale. Zero on this scale is the lowest temperature that can achieved.

**Standard Temperature and Pressure (STP)**

The temperature 273.16 k (0o C) and pressure 1 atmosphere (760 mm Hg) have been chosen as standard temperature and standard pressure.

STP 273.16K (0o C) and 1 atm (760 mm Hg)

__General Gas Equation [Ideal Gas Equation]__*“ A gas equation which is obtained by combining Boyle ‘s law , Charles ,s law and Avogadro ,s law is called general gas equation.”*

It relates pressure, volume temperature and number of moles of gas.

According to Boyle’s Law V (at constant T and n)

According to Charles’s Law V T (at constant P and n)

According to Avogadro’s Law VT (at constant P and T)

If the variables P,T and n are not Kept constant then all the above three relationships can be combined together.

V

V=R

Where ‘R’ is a general gas constant.

**pV=nRT(First Form)**

*The above equation is called ideal or general gas equation.*

The general gas equation shows that if we have any quantity of an ideal gas then the product of its pressure and volume is equal to the product of number of moles,

__Variation on the Gas Equation__ During chemical and physical processes, any of the four variables (P,V,n ,T) in the ideal gas equation ,PV =nRT may be fixed and any of them may change.

*For examples:*1. For a fixed amount of gas at constant temperature.

PV= nRT =constant [ n and T constant]

PV=k (Boyle’s Law)

2. For a fixed amount of gas at constant pressure,

=[ n and P constant]

V=kT

3. For a fixed amount of gas at constant volume,

=[ n and V constant]

P=kT

4. For a fixed pressure and temperature,

[ P and T constant ]

V=kn (Avogadro’s Law)

5. For a fixed volume and temperature,

[ T and V constant]

P=kn

__General Gas Equation For one mole of gas__ For one mole of gas , the general gas equation is

PV==RT [n=1]

For two different pressure, volumes and absolute temperatures, the above equation can be written as,

Hence, we can write,

= (Second Form)

__Numerical Value of Ideal Gas Constant , R__ The numerical value of R depends upon the units chosen for P,V and T.

*(i) When pressure is expressed in atmosphere and volume in dm*^{3}. For 1 mole of a gas at STP, we have

N= 1 mole, P=1 atm, T =273.16k, V=22.414dm

^{3 } Putting these value in general gas equation,

R=

R=

**R=0.0821dm**

^{3 }atm k^{-1 }mol^{-1}

*(ii)*

*when pressure is expressed in mm of mercury or torr and volume of gas in cm*^{3 }.**R= 0.0821dm**

^{3 }atm k^{-1 }mol^{-1}Since 1 dm

^{3 }=1000cm^{3 }: 1 atm =760mm of HgR=0.0821x1000 cm

^{3 }x760mm Hg K^{-1 }mol^{-1}**R=62400 cm**

^{3 }mm Hg K^{-1 }mol^{-1}Since 1 mm=1 torr

**R=62400 cm**

^{3 }torr K^{-1 }mol^{-1}

*(iii)*

*When pressure is expressed in Nm*^{-2 }and volume in m^{3 }(SI Units)

*For 1 mole of gas at STP,*N= 1 mol, P=101325 Nm

^{-2}, T=273.16 K, V=0.022414m^{3}Substituting the values in general gas equation,

R=

R=

R=8.3143 NmK

^{-1 }mol^{-1}Since 1 Nm=1 j

**R=8.3143jk**

^{-1 }mol^{-1}**whenever the pressure is given in Nm**

*Remember that,*^{-2 }and the volume in m

^{3 },then the value of used must be 8.3143 jk

^{-1 }mol

^{-1 }.

**when the pressure is given in Nm**

*Remember that*^{-2 }and the volume in m

^{3}, then the value of R used must be 8.3143JK

^{-1 }mol

^{-1 }.joule (j) is unit of work and energy in the system.

(iv)

**.***When energy is expressed in ergs*R=8.3143jk

^{-1 }mol^{-1}^{ }Since 1 J =10

^{7 }erg

**R=8.3143x10**

^{7 }erg K^{-1 }mol^{-1}(

*v) When energyis expressed in calories.*R=8.3143jk -1 mol-1

Since 1 cal=4.184j

Or 1 j=

R=

**R = 1.987 cal K**

^{-1 }mol^{-1}

*(vi) R can be expressed in units of work or energy per Kelvin per mole .*

*Form ideal gas equation, we can write*R =

If the pressure is written as force per units area and volume as area x length,

R====

Hence R can be expressed in units of work or energy per Kelvin per mole.

__Physical Meaning of Value of R__ The physical meaning of the value 0.0821 dm3 atm K

^{-1 }mol^{-1 }of R is that if we have one mole of an ideal gas at 273.16 K and one atmospheric pressure and its temperature is increased by 1 K then it will absorb 0.0821 dm^{3 }atm of energy , dm3 atm being the unit of energy .Hence the value of R is a universal parameter for all the gases . It tells us that the Avogadro’s number of molecules of all the ideal gases have the same demand of energy.

__Density of an Ideal Gas__ We know that, n=

Where a is the number of moles of gas .m is the mass of the substance in grams and M is the molar mass of the substance.

On substituting the above expression into the ideal gas equation,

We obtained

PV=RT (Third From)-------(6)

PM==RT

PM=dRT (d=)

d= (Fourth From)---------(7)

Hence the density of an ideal gas is directly proportional to its molar mass.

*Greater the pressure on the gas , the closer will be the molecules and greater the density temperature of the gas ,the lower will be the density of the gas.*

With the help of equation (7) ,we can calculate the relative molar mass, M of an ideal gas if its pressure ,temperature and density are know.

**Example3:**

A sample of nitrogen gas is enclosed in a vessel of volume 380 cm

^{3 }at 120^{o }C and pressure of 101325 Nm^{-2 }.This gas is transferred to a 10dm^{3}. Flask and cooled to 27

^{o }C.Calculate the pressure in Nm^{-2 }exerted by the gas at27^{o }C.**Solution**:

V

_{1 }=380cm^{3 }=0.38 dm^{3 }; V_{2 }=10dm^{3}P

_{1 }=101325 Nm^{-2 }; P_{2 }=?T

_{1 }=273+120=393k ; T_{2 }=273+27=300k**:**

*Formula used*=

P

_{2 }= P

_{2 }= P

_{2 }=0.028atm P

_{2 }=0.029x101325**=2938.4 Nm**^{-2 }Answer**Example 4:**

Calculate the density of CH

_{4 }at 0^{o }C and 1 atmospheric pressure, what will happen to the density if(a) temperature is increase to 27

^{o }C.(b)

**the pressure is increased to 2 atmospheres at 0**.^{o }C**Solution:**

M

_{CH4}=12+4=16g mol^{-1 } P=1 atm

R= 0.0821dm

^{3 }atm K^{-1 }mol^{-1}^{ }T = 273+0=273K

*Formula Used:* d =

d =

**d=0.7138 g dm3 Answer**

**(a)**

**Density at 27**

^{o }CM

_{CH4=}16g mol^{-1}P= 1 atm

R=0.0821 dm

^{3 }atm K^{-1 }mol^{-1}T=273+27=300K

^{ }**d=**

*Formula Used:* d =

**d = 0.6496g dm**

^{3}AnswerThus by increasing the temperature from 0

^{o }C , the density of gas has decreased form 0.7143 to 0.6496 g dm^{3 }.**(b) Density at 2 atm pressure and 0**

^{o }C. M

_{CH4}=16g mol-1 P = 2 atm

R =0.0821 dm3 atm K-1 mol-1

T = 273+0=273k

**d =**

*Formula Used:* d =

**d=1.428 g dm-3 Answer**

Thus the increasing of pressure has increased the density . the density becomes double by doubling the pressure .

**Example 5:**

Calculate the mass of 1 dm3 of NH

_{3 }gas at 30^{o }C and 1000 mm Hg pressure, considering that NH_{4 }is behaving ideally.**Solution: Given:**P=1000 mm Hg = 1.316 atm ; V=1 dm

^{3 }

MNH4 =14+3=17g mol-1 : R=0.0821dm3 atm K-1 mol-1

T=273+30-303l ; m=?

**PV =**

*Formula Used:* M=

M =

**M =0.0908 g Answer**

__Avogadro ‘s Law__ The law may be stated in a number of ways as follows;

*1.*

*“Equal volumes of ideal gases at the same temperature and pressure contain equal number of molecules.”*

*2.*

*“Equal number of molecules of ideal gases at the same temperature and pressure occupy equal volumes.”*

*3.*

*“At constant temperature and pressure, the volume of an ideal gas is directly proportional to the number of moles or molecules of gas.”*

**Mathematically:**Vn (at P and T constant)

V=kn

**Explanation**:

Since one mole of an ideal gas at STP has a volume of 22.414 dm

^{3 },So 22.414dm^{3 }of an ideal gas at STP will have Avogadro’s number of molecules , i.e, 6.02x10^{23 }. Mathematically , it can be written as,

**22.414 dm**

^{3 }of an ideal gas at STP has number of molecules =6.02x10^{23} 1 dm

^{3 }of an ideal gas at STP has number of molecules =molecules =2.68x10

^{22 }moleculesIn other words if we have one dm

^{3 }of each of H^{2 },O^{2 },N^{2 }and CH^{4 }in separate vessels at STP, then the number of molecules in each will be 2.68x 10^{22 }.Similarly when the temperature or pressure are equally changed for these four gases, then the new equal volumes will have the same number of molecules , 2.68x10

^{22 }. One dm

^{3 }of H_{2 }at STP weight approximately 0.0899 grams and one dm^{3 }of O_{2 }at STP weight 1.4384 grams, but their number of molecules are the same .Although , oxygen molecule is 16 times heavier than hydrogen but this does not disturb the volume occupied by the molecules because molecules , of the gases are widely separated from each other at STP .One molecule of gas is approximately at a distance of 300 times its own diameter from its neighbors at room temperature .

__Daltan’s Law of Partial Pressure__

*“The total pressure exerted by a mixture of non-reacting gases is equal to the sum of their individual partial pressure.”*

**Mathematically**: P

_{t }=P

_{1 }+P

_{2 }+P

_{3 }

Where Pt is the total pressure of the mixture of gases and P

_{1 },P_{2 }and P_{3 }are the partial pressure of gas_{1 }. gas_{2 }and gas_{ 3 }respectively in the mixture. The partial pressure of a gas in a mixture of gases in the pressure a gas would exert if it were the only gas in the container. Partial pressures are commonly represented by small p‘s.

Explanation: Suppose we have four 10 dm

^{3}cylinders. The first one contains H_{2 }gas at a pressure of 400 torr; the second one contains CH^{4 }gas at a pressure of 100 torr at the same temperature. Now transfer these gases to the fourth cylinder of capacity 10dm^{3 }at the same temperature. According to Dalton‘s law, the pressure in the fourth cylinder is found to be equal to the sum of the pressures that each gas exerted by itself. P

_{t }=P_{H2 }+PC_{H 4 }+P_{o2 } P

_{t}=400+500+100=1000 torrThere three non-reacting gases are behaving independently under the normal conditions. The rapidly moving molecules of each gas in a mixture have equal opportunities to collide with the walls of the container. Hence, each gas in the container exerts a pressure independent of the pressure of other gases in the container. The total pressure is the result of total number of collisions per unit area in a given time. Since the molecules of each gas move independently, so the general gas equation can be applied to the individual gases in the gaseous mixture.

(i)P

_{H2 }V=n_{H2 }RT P_{H2 }=n_{H2}P_{H2 }n_{H2 }(ii)P

_{H2 }V=n_{H4}RTP_{H2 }=n_{H2}P_{ch2}n_{Ch4}(iii)P

_{O2 }V=n_{o2 }RT P_{o2 }=no2 P_{o2 }n_{o2 }_{ }is a constant factor

H2 , CH4 and O2 have their own partial pressures . Since volume and temperature are the same so their number of moles will be different and will be directly proportional to their partial pressures .

Adding Equations (i),(ii) and (iii),we get ,

(P

_{H2 }+P_{CH4 }+P_{o2 })V= (n_{H2 }+ n_{CH4 }+n_{o2 })RT P

_{t }= n_{t }RT Where Pt =P

_{H2 }+P_{CH4 }+P_{o2 }, n_{t }=n_{H2 }+n_{CH4 }+ n_{o2 }Hence, the total pressure of the mixture of gases depends upon the total number of .moles of the gases.

__Calculation of Partial Pressure of a Gas__

The partial pressure of any gas in a mixture of gases can be calculated provided we know the mass or number of moles of the gas, the total pressure and the total number of moles present in the mixture of gases.

Suppose we have a mixture of gas A and gas B. The mixture is enclosed in a container having volume (V). The total pressure of the mixture is P

_{t }. The number of moles of the gases A and B are N_{A }and n_{B }respectively .If the temperature of the mixture is T, then we can write general gas equations For the mixture of gases: P

_{t }V=n_{t }RT --------(i) For gas A: P

_{A}V=n_{A}RT --------(ii) For gas B: P

_{B}V=n_{B}RT ---------(iii)Divide Eq (ii) by Eq (i) ,we get,

=

=

PA=

Similarly, by dividing Eq (iii) by Eq (i), we get

P

_{A}=In mole –fraction form, the above equations can be written as,

P

_{A}=X_{A }P_{t }[ X_{A }=] P

_{B }= X_{B }P_{t }[ X_{B }=]**Example 6:**There is a mixture of H2, He and CH4 occupying a vessel of volume 13 dm

_{3 }at 37

^{o}C and pressure of 1 atmosphere. The masses of H

_{2 }and He are 0.8g and 0.12g respectively. Calculate the partial pressure in torr Hg of each gas in the mixture.

**Solution: Given:**

P=1 atm : V=13dm3

n=? ; T=273+37=310K

R=0.0821 dm

^{3 }atm K^{-1 }mol^{-1 }

*Formula Used:* PV=nRT

n =

n=

n

_{H2 }= n

_{He}= n

_{CH4 }= 0.51-(0.396+0.03)=0.51-0.426=0.084mol P

_{H2 }= P

_{He }= =0.776x760=589.76torr Answer

P

_{He }= P

_{H2 }=**=0.058x760=44.08 torr Answer**

P

_{CH4 }= P

_{CH4 }=**=0.164x760=124.64 torr Answer**

__Applications of__Dalton ’s Law of Partial pressure The four important applications of Dalton ’s law of partial pressures are the following:

**The**

*1. Determination of the pressure of dry gas:**.The partial pressure exerted by the water vapour is called aqueous tension*.Thus

**Picture**.

*Remember that :* While solving the numerical, the aqueous tension is subtracted from the total pressure (P

_{moist gas})**In living beings, the process of respiration depends upon differences in partial pressures. When animals inhale air then oxygen moves into lungs as the partial pressure of oxygen in the air is 159 torr while the partial pressure of oxygen in the lungs is 116 torr . On the other hand, CO**

*2. Process of Respiration:*_{2 }produced during respiration moves out in the opposite direction as its partial pressure is low in the air relative to its partial pressure in the lungs .

**: At higher altitudes ,the pilot feel uncomfortable breathing because the partial pressure of oxygen in the unperssurized cabin is low as compared to torr in air where one feels comfortable breathing.**

*3 Pilots feel uncomfortable breathing at higher altitudes***deep Sea divers take oxygen mixed with an inert gas (He) and adjust the partial pressure of oxygen according to the requirement. In sea after every 100feet depth, the diver experiences about 3 atm pressure .Therefore ,normal air cannot be breathed in depth of sea . Moreover, the pressure of N**

*4 Deep sea divers feel uncomfortable breathing:*_{2 }increases in depth of sea and it diffuses in the blood.

__Diffusion and Effusion__

__Diffusion__

*“The spontaneous intermingling (intermixing) of molecules of one gas with another at a given temperature and pressure is called diffusion.”*

__Explanation of Diffusion from Kinetic Theory__ According to the Kinetic molecular theory of gases, the molecules of the gases move haphazardly. They collide among themselves, collide with the walls of the container and change their directions. In other words the molecules of gases are scattered or intermixed due to collisions and random motion to from a homogeneous mixture.

**Example:**(1) The spreading of fragrance of a rose or a scent is due to diffusion.

(2) Suppose NO

_{2 }(a brown gas an d O_{2 }( a colorless gas )are separated from each other by a partition . When the partition is removed, both diffused into each other due to collisions and random motion. They generate a homogeneous mixture. The partial pressure ofBoth are uniform throughout the mixture.

**Picture**

__Effusion:__ “The escape of gas molecules through an extremely small opening into a region of low pressure is called effusion.”

The spreading of molecules in effusion is not due to collisions but to their escape one by one .Actually the molecules of a gas are habitual in colliding with the walls of the vessel. When a molecule approaches just in front of the opening it enters the other portion of the vessel. This type of escape of gas molecules through a small hole into a region of low pressure or vacuum is called effusion.

**Picture**

__Graham’s Law of Diffusion__

*“The rate of diffusion or effusion of a gas is inversely proportional to the square root of it’ s density at constant temperature and pressure.”* Mathematically. Rate of diffusion (at constant T and P)

Rate of diffusion =

R =

Rx ==k

The constant k is same for all gases, when they are all studied at the same temperature and pressure .

Suppose we have two gases 1 and 2 whose rates of diffusion are r

_{1 }and r_{2 }and densities are D1 and d2 respectively. According to Graham’s law, r

_{1 }x_{1}=k r

_{2 }x_{2}=k Divide the two equations and rearrange

Since the density of a given gas is directly proportional to its molecular mass.

dM

Therefore, Graham’s law of diffusion can also be written as,

Where M

_{1 }and M_{2 }are the molecular masses of gases.Experimental Verification of Graham’s Law of Diffusion

This law can be verified in the laboratory by noting the rates of diffusion of two gases in a glass tube when they are allowed to move from opposite ends. Two cotton plugs soaked in NH

_{4 }OH and HCI solution are introduced in the open ends of 100cm long tube at the same time .NH3 molecules travel a distance of 59.5 cm when HCI molecules cover 40.5 cm in the same time. Dense white fumes of NH_{4}CI are produced at the junction of two gases. =

=

Form Graham’s law the rates of diffusion of NH

_{3 }and HCI can be calculated by using their molecular masses as follows: =

This result is the same as obtained form the experiment.

Hence the law is verified

Remember that light gases diffuse more rapidly than heavy gases.

**Picture**

Fig: Verification of Graham’s Law

Example 7: 250 cm

^{3 }of the sample of hydrogen effuses four times as rapidly as 250 cm^{3 }of an unknown gas . Calculate the molar mass of unknown gas .Solution: Let the symbol for the unknown gas is x.

Rate of effusion of unknown gas r

_{x }=1 Rate of effusion of hydrogen gas r

_{H2}=1x4=4 Molar of effusion of hydrogen gas M

_{H2 }=2g mol^{-1 } Molar mass of unknown gas, M

_{x}=? Formula used:

Taking square of both the sides,

M

_{x }=16x2g mol^{-1} M

_{x}=32g mol^{ -1 }Answer

__Kinetic Molecular Theory of Gases__

*“A gas model which explains the physical behavior of gases and the gas laws is called the kinetic molecular theory of gases.* The kinetic molecular theory of gases was first suggested by Bernoulli (1738).R.J .Clausisus (1857) derived the Kinetic gas equation and deduced all the gas laws from it .Maxwell (1859) gave the law of distribution of velocities .Bolt Mann (1870) gave the concept of distribution of energies among the gas molecules. Vander walls (1873) modify the Kinetic gas equation in order to apply for real gases.

__Postulates of Kinetic Molecular Theory of Gases__ The basic assumptions of the Kinetic molecular theory of an ideal gas are the following:

*All gases consist of a large number of very small particles called molecules.*Noble gases (He,Ne, Ar,etc )have mono=atomic molecules .*The molecules of a gas mve randomly in straight lines in all directions with various velocities*. They collide with one another and with the walls of the container and change their direction.*The pressure exerted by a gas is due to the collections of the gas molecules with the walls fo the container.*The collisions among the molecules and with the walls of the container are perfectly elastic. There is no loss energy during collision or mutual friction.*The molecules of a gas are widely separated from one another .There is large empty spaces between the gas molecules.**There are no attractive forces between the gas molecules*.There is no force of gravity on gas molecules.*The actual volume of the gas molecules is negligible as compared to the volume of the gas*.*The motion of the gas molecules due to gravity is negligible as compared to the effect of continued collisions between them*.*The average kinetic energy of the gas molecules is directly proportional to absolute temperature of the gas*.

**Average K.E T**

**Average K.E=**

Note : The average kinetic energy is considered because the gas molecules are moving with different velocities . The velocity of each molecules change with each collision .Some of the molecules has very low velocities, while others move very rapidly.

__Kinetic Gas Equation__ Keeping in view the basic assumptions of kinetic molecular theory of gases, Clausius derived an expression for the pressure of an ideal gas . This equation relates the pressure of the gas with mass of a molecule of the gas, the number of molecules of the gas in the volume of the gas and the mean square velocities of gas molecules .The kinetic gas equation has the following form.

**PV=**

Where P=pressure of the gas

V=Volume of the gas

M=mass of one molecule of gas

N=number of gas molecules in volume

=mean square velocity (average of the square of the

Velocities of the gas molecules)

The idea of the mean square velocity is important because all the molecules of a gas at a particular temperature have different velocities .These velocities are distributed among the molecules.

__Mean velocity ,Mean Square Velocity ,Root Mean Square Velocity__**Mean(average ) velocity,**

*“The average value of the different velocities of all the molecules in a sample of gas at a particular temperature is called the mean velocit .”*

If there are n1 molecules with velocity c

_{1 }, n_{2 }molecules with velocity c_{2 }and n_{3 }molecules with velocity c_{3},then ,Where is known as the mean of average velocity .The bar (-) over the velocity (c ) indicates average or mean value.

In this reference n

_{1 }+ n_{2 }+n_{3 }=N

__mean square velocity ,c__^{2}

*“The average value of the square of the different velocities of all the molecules in a sample of gas at a particular temperatures is called the square velocity.”*Where is known as the mean square velocity.

__Root mean square Velocity ,c___{rms}

*“The square root of the mean of the squares of the different velocities of all the molecules in a sample of gas at a particular temperature is called the root mean square velocity.”*C

_{rms }=“The square root of the mean square velocity.

The expression for the root mean square velocity deduced from the kinetic gas equation is written as:

Where , C

_{mns }root mean square velocity M =molecular mass of the gas

T = absolute temperature

This equation is a quantitative relationship between the absolute temperature and the velocities of the gas molecules .According to this equation ,the higher the temperature of a gas ,the greater the velocities .

__Explanation of Gas Law From kinetic theory of Gases__**1.**

**Bo lye’s Law**

According to kinetic gas equation,

PV=

Multiplying and dividing by 2 on right hand side.

PV=

But =KE

PV=(K.E)

Now, according to kinetic theory of gases, kinetic is directly proportional to absolute temperature

K.E T

K.E =kT

PV =

If temperature is constant, then is constant

PV=constant (which is boyle’s law)

Thus boyle’s law is derived.

2. Charles,s Law

According to kinetic gas equation,

PV=

Multiplying and dividing by 2 on right hand side.

PV=

=KE

PV=(K.E)

Now ,according to kinetic theory of gases , kinetic energy is directly proportional to absolute temperature.

K.E T

K.E =kT

PV =

V =

If pressure is constant, then is constant

V=constant x T

V T (which is Charles’s law)

**3 Avogadro’s Law**

Consider two gases 1 and 2 at the same pressure P and the same volume V.Suppose their number of molecules are N

_{1 }and N_{2 },masses are m_{1 }and m_{2 }and mean square velocities are respectively. For gas 1: PV- …………..(1)

For gas 2: PV- ………….(2)

From Eq(1) and Eq (2) ,we get,

=……..(3)

If the temperature of the gas 1 and gas 2 are the same, their kinetic energy will also be the same.

Kinetic energy of gas

_{1 }=kinetic energy of gas_{2}=

Dividing eq(3) by eq(4)

N

_{1 }= N_{2}Thus ,number of molecules of gas1 =number of molecules of gas2

**4. Graham’s law of Diffusion**

According to kinetic gas equation,

PV =

For 1 mole gas , N=N

_{A}PV=

PV = (mN

_{A}=M)Where M is the molecular mass of gas.

P=

P= [ =d(density of gas)]

=

Taking the square root of both the sides

=

c

_{nms }=Since the root mean square velocity of the gas is equal to the rate of diffusion of the gas,

C

_{nms }=rSo, r=

r=

At constant pressure,

r=

r=constant x

r

Thus , Graham ‘s law of diffusion of gases is derived

__Kinetic Interpretation of Temperature__ Consider one mole of the gas containing N molecules. Suppose m is the mass of one molecule of the gas and is the mean square velocity of the molecule.

According to kinetic gas equation,

PV= …………..(1)

The Eq(1) can be rewritten as,

PV= ……………(2)

Now , the kinetic energy of one molecule of a gas due to its translational motion is given by the following equation ,

E

_{k}= …………….(3)Where Ek is the average translational kinetic energy of a gas.

On putting eq(3) in Eq (2) , we get,

PV =

Since the number of gas molecules in one mole of gas in N

_{A }(Avogadro’s-number)So,

N=N

_{A}PV =

For I mole of the ideal gas,

PV=RT

Therefore,

=RT

E

_{k}=E

_{k}We can draw the following conclusions from the above equation:

**1. Concept of Absolute Zero of Temperature**

According to the equation, T=, absolute temperature of a gas is directly

Proportional to the average translational kinetic energy of the gas molecules. It means that a change in temperature brings about a change in the intensity of molecular motion .Thus, when T=0k (-273.16

^{o }C), the translational kinetic theory, absolute zero of temperature, both velocity and kinetic energy of gas molecules become zero. Also at this temperature, the gas exerts no pressure on the walls of the container.The absolute zero temperature cannot be attained .However, a temperature as low as 10

_{-5}K is obtained.**2.**

**Concept of Average Kinetic Energy of a gas**

The average translational kinetic energy of a gas depends only on its absolute temperature and is independent of the pressure, volume, or type of gas. The mean kinetic energy of each gas molecule is the same at the same temperature. The mean kinetic energy of gas molecule does not depend upon its mass. Therefore, a small molecule such as H

_{2 }will have , at the same temperature , the same average kinetic energy as a much heavier molecule such as CO_{2 }.*Hence, for different gases average kinetic energy of molecules at the same temperature is the same. Thus at the same temperature.*K.E of gas 1 = K.E of gas2

**3. Concept of Heat Flow**

When heat flows from one body to another, the molecules in the hotter body give up some of their kinetic energy through collisions to molecules in the colder body. This process of flow of heat continues until the average translational kinetic energies of all the molecules become equal. Consequently, the temperature of both the bodies becomes equal.

**4. Concept of Temperature of Gases, Liquids and solids**

Temperature is the measure of average translational kinetic energies of molecules. In gases and liquids, temperature is the measure of average translational of kinetic energies of their molecules , In solids , temperature is the measure of vibrational kinetic energy of molecules because there is only vibrational motion about mean position .

__Liquefaction of Gases__**General Principle of Liquefaction of Gases**

Under suitable conditions of temperature and pressure all gases may be liquefied. Liquefaction occurs only when the attractive forces between the molecules are greater than the kinetic energy of the molecules.

Two conditions favour this situation.

1. High pressure. High pressure is applied on gases. This brings the molecules of a gas close to each other. Thus the forces between the molecules increase.

2. Low temperature. Low temperature is created in gases .This decreases the kinetic energy of the molecules. Thus the attractive forces between the molecules increase.

In general, the liquefaction of a gas requires high pressure and low temperature.

Liquefaction means the conversion of a gas into a liquid.

For each gas, there is a certain temperature above which the gas cannot be liquefied even if a very high pressure is applied. This temperature is known as critical temperature. The critical temperature of a gas may be defined as, “the highest temperature which a gas con be liquefied by increasing the pressure is called critical temperature.’ The critical temperature is represented by Tc. Below the critical temperature a gas can be liquefied however great the pressure may be. The value of the critical temperature of a gas depends upon its size, shape and intermolecular forces present in it.

“The pressure which is required to liquefy the gas at its critical temperature is called critical pressure.”

The critical pressure is represented by Pc.

The critical temperature and the critical pressure of the gases are very important because they provide us the information about the condition under which gases liquefy.

**Examples:**

(1) CO2 is a nonpolar gas its critical temperature is 31.1

^{o}C. It must be cooled below this temperature before it can be liquefied by applying high pressure. However, if temperature is maintained below 31.1^{o}C. Then lower pressure than critical pressure is required to liquefy it.(2) NH3 is a polar gas. Its critical temperature is 132.44

^{o}C. It must be cooled below this temperature before it can be liquefy by applying sufficient high pressure.**Table : Critical Temperature and Critical Pressure of Some Gases**

Gas | Critical Temperature T _{c }(^{o}C) | Critical Pressure P _{c9 }(atm) |

Carbon dioxide , CO _{2} | 31.142(304.3k) | 73.0 |

Freon -12 , CC1 _{2}F_{2} | 111.54(384.7k) | 39.6 |

Ammonia , NH _{3} | 132.44(405.6k) | 111.5 |

Water Vapours, H _{2}O | 374.44(647.6k) | 217 |

Oxygen , O _{2} | -118.75(154.4k) | 49.7 |

Argon , Ar | -112.26(150.9k) | 48 |

Nitrogen, N _{2} | -147.06(126.1k) | 33.5 |

Joule-Thomson Effect (Joule-Kelvin Effect)

“The cooling effect produced when a compressed gas is allowed to undergo sudden expansion through the nozzle of a jet is called joule-Thomson effect.”

The effect is due to energy used in overcoming the intermolecular attractive forces of the gas. Joule-Thomson effect is used in the liquefaction of gases by cooling.

Explanation: In a compressed gas, the molecules are very close to each other. Therefore, the intermolecular attraction is fairly large. When the compressed gas is allowed to undergo sudden expansion in passing through a small orifice (hole) into a region of low pressure, the molecules move apart. In doing so, the intermolecular attractive forces must be overcome. For this purpose energy is required. This energy is taken from the gas itself which is thereby cooled.

__Method from the liquefaction of Gases__**Lind’s method of liquefaction of gases**

The basic principle of the Linde’s method of liquefaction of gases is the joule-Thosmson effect. The apparatus used for this purpose is shown in the fig.

For liquefaction of air, pure dry air is compression .The compressed air is then passed through a spiral pipe having a jet at the end. The air expands at the jet into the chamber where the pressure of the air falls down from 200 atm to 1atm. As a result of joule-Thomson effect a considerable drop of temperature occurs .The cooled air goes up and cools the incoming compressed air of the spiral tube .the cooled air goes again to the compression pump .The process of compression and expansion is repeated again and again till the air is cooled to such an extent that it liquefies. The liquefied air collects at the bottom of the expansion chamber from where it is drawn off. All gases except H

_{2 }and He can be liquefied by this method.Picture

**Fig. linde’s method for the liquefaction of air**

__Non-Ideal Behavior of gases__ An ideal gas obeys the gas laws and general gas equation for all valyes of temperature and pressure. All real gases, however, show marked deviations from ideal behavior. These deviations from ideal behavior depends on:

(i) Pressure, and (ii) temperature

A convenient way of showing the deviations of a real gas from ideal behavior is to plot a graph between on x-axis and on y-axis. The ratio is called the compressibility factor. Its value is equal to 1 for 1 mole of an ideal gas for all values of temperature and pressure. For a real gas, the compressibility factor deviate from 1.The extent of deviation from 1 depends upon the pressure and temperature. it may be either greater than 1 or lesser than 1 . A value greater than one shows that thee gas is less compressible than an ideal gas. On the other hand, a value less than one indicate that the gas is more compressible than an ideal gas.

For an ideal gas, since the increase of pressure decreases the volume in such a way that remains constant at constant temperature, so a straight line is obtained parallel to the pressure axis.

In order to understand the deviations of a real gas from ideal behavior, let us study the plot of against p of a few real gases like He, H

_{2}, N_{2 }and CO_{2 }first at 0^{o}C and then at 100^{o}c.**(a)**

__Effect of Pressure Variation On Deviations at 0__^{o}C In the fig, the curves between compressibility factor and pressure for the four real gases, He, H

_{2}, N_{2 }and CO_{2 }along with the behavior of an ideal gas, are shown.**Picture**

**Fig Non-ideal behavior of gases at 0**

^{o }C(i) At low pressures, all the curves for these four gases approach to unity as the pressure approaches zero. Thus all real gases behave as ideal gases at low pressures (upto10 atm).

(ii) At high pressure, the behavior of He and H

_{2 }differs from that of N_{2 }and CO_{2 }at 0^{o}C.For both he and H_{2 }at 0^{o}C, the compressibility factor is always greater than one. This curves show a continuous increase for both He. It means that these gases are less compressible than expected from ideal behavior. For both N_{2 }and CO_{2 }at 0^{o}C, the curve first decreases and then increases again as the pressure increases. This decrease in the compressibility factor below unity is more pronounced in CO2_{ }than in N_{2. }For CO_{2, the}dip in the curve is greatest.The extent of deviation of these four gases show that these gases have their own limitations for obeying general gas equation. It depends upon the nature of the gas that at which value of pressure, it will start disobeying .

__(b) Effect of Pressure Variation on Deviation at 100__^{o}CIn the fig, the curves between compressibility factor and pressure for the four real gases, He, H

_{2}, and CO_{2}at 100^{o}C along with the behavior of an ideal gas, are shown. It is clear from the shape of the curves, of these four gases that the deviations from the ideal behavior become less at 100^{o}C than at 0^{o}C. The graphs come closer to the expected straight line and the deviations are shifted towards higher pressure. It means that the increase in temperature makes the gases ideal.**Picture**

**Fig. Non-ideal behaviour of gases at 100**

^{o}C**Conclusions:**

On the basis of experimental observations, we draw the following two conclusions.

1. Gases are ideal at low pressure and non-ideal at high pressure.

2. Gases are ideal at high temperature and non-ideal at low temperature.

__Causes for Deviation From Ideality__

In the derivation of ideal gas equation, PV=nRT, it was assumed that all gases behaved exactly alike under all conditions of pressures and temperatures. In 1873, van der Waals pointed out that all real gases deviate from ideal behavior at high pressures and low temperatures. It was found that the following two incorrect postulates in the kinetic theory of gases are the cause of deviation.

**1. The molecules exert no forces of attraction on each other in a gas.**

When the pressure on a gas is high and the temperature is low then the intermolecular attractive forces become significant. Therefore, the ideal gas equation does not hold good .Actually, under these conditions, the gas does not remain ideal .

**2. The actual volume of gas molecules is negligible as compared to the volume of the container .**

The actual volume of gas molecules is very small as compared to the volume of the container, so it can be neglected. This volume, however, does not remain negligible when there is high pressure on the gas. This can be understood from the following figures.

**Picture**

**Fig. A gas at low pressure when Fig . A gas at high pressure when**

**Actual volume is negligible Actual volume is negligible**

__Van der Waals equation For Real Gases__Keeping in view the molecular volume and the intermolecular attractive forces,

Van derWallls pointed out that the ideal gas equation does not hold good for real gases

He suggested that both volume and pressure factors in ideal gas equation needed

Correction in order to make it applicable to the real gases.

**Volume Correction**

The volume of a gas is the free space in the container available for the movement

of molecules . For an ideal gas, the actual volume of the molecules is considered zero. So

the volume of an ideal gas is the same as the volume of the container . It also represents the volume in which the molecules are free to move. For a real gas, the molecules are rigid spherical particles which have a definite volume. Consequently, they take up part of the space in the container. Thus the volume in which the molecules of a real gas are actually free to move is less than the volume of the container. Thus, van der Walls suggested that a suitable volume correction term may be subtracted from the volume of the vessel containing the gas . The volume correction term for one mole of the gas is ‘b’. If v is the volume of the vessel containing the gas, then the volume available to gas molecules for their free movement is written as.

V

_{free}= V_{vessel}=bWhere b is constant depending upon the nature of the gas . b is also called the excluded volume or co-volume . It has been found that b is not the actual volume of molecules but it is four times the actual volume of the molecules. B =4 V

_{m}.Where Vm is the actual volume of one mole of gas molecules.

Pressure Correction

Consider a molecule in the interior of a gas. Since it is completely surrounded by other molecules equally in all directions so the resultant attractive force acting on the molecule is zero .However, when the molecule approaches the wall of the container, the equal distribution of molecules about it is upset. There is now a greater number of molecules away from the wall with the result that these molecules exert net inward pull on the molecule. Thus, when a molecule is about to strike the wall of the container, it experiences a force of attraction towards the other molecules in the gas .So it will strike the wall with a lower velocity and will exert a lower pressure. Hence the observed pressure, P Exerted by the molecules is less than the ideal pressure, Pi

_{ }by an amount P^{i }.It is, therefore, necessary to add a pressure correction term in order to obtain the ideal pressure.P

*=P+ P*_{i }^{i}P

*is the true kinetic pressure, if the forces of attraction would have been absent. P*_{i }*is the amount of pressure lessened due to attractive forces.*_{i }**Picture**

Fig : Force of attraction and Pressure correction

It is suggested that a part of the pressure P for one mole of a gas used up against intermolecular attractions should decrease as volume increase. Consequently, the value of p

*in terms of a constant ‘a’ which accounts for the attractive forces and the volume V of vessel can be written as*^{i }P

*=*^{i }

__How to prove it?__P

*es determined by the forces of attraction between molecules of type A which are striking the wall of the container and molecules of type B which are pulling inward. The net force of attraction is proportional to the concentrations of A type and B type molecules.*^{i}Let ‘n’ is the number of moles of A and B separately and total volume of both gases is ‘v’.Then concentration, C= in moles dm-3 of A and B separately. Therefore,

P

*x*^{i }P

*x*^{i }P

*=*^{i }Where’ a’ is a constant of proportionality.

**If n=1 , then P**

*=*^{i }

Greater the attractive forces among the gas molecules, smaller the volume of container, greater the value of lessened pressure P

*.*^{i} This ‘a’ is called coefficient of attraction or attraction per unit volume . It has a constant value for a particular real gas .Thus effective kinetics pressure of a gas is given by P

*, which is the pressure if the gas would have been ideal .*_{i }P

*+ P=*^{i}The kinetic gas equation for one mole of a gas may be written as,

(

**P+****)(V-b) =RT**For ‘n’ moles of a gas, the kinetic gas equation may be written as

(P+) (V-b) =RT

This is called van der Waals equation. ‘a’ and ‘b’ are known as van der waals constants.

**Units of ‘a’**

Since P

*=*^{i } a=

a=

a=atm dm

^{6 }mol^{-2}In sI units, pressure is in Nm

^{-2 }and volume in m^{3.}So, a=

a= Nm

^{4 }mol^{-2 }**Units ‘b’:**‘b’ is excluded or incompressible volume /mole of gas .So , its units should be dm3 mol-1 or m3 mol-1 .

The values of ‘a’ and ‘b’ can be determined by knowing the values of P,V and T of a gaseous system under two different conditions . The values of ‘a’ and ’b’ for some common gases are given in the following table.

Gas ‘a’ (atm dm6 mol-2) ‘b’(dm3 mol-1) |

Hydrogen 0.245 0.0266 Oxygen 1.360 0.0318 Nitrogen 1.390 0.0391 Carbon dioxide 3.590 0.0428 Ammonia 4.170 0.0371 Chlorine 6.493 0.0562 |

The presence of intermolecular forces in gases like CI

_{2 }and SO_{2 }increase their ‘a’ factor. The least valur of ‘a’ for H^{2 }is taken, then its small size and non-palar character. The ‘b’ value of H^{2 }is 0.0266 g (1 mol) of H^{2 }is taken, then it will occupy 0.0266 dm^{3 }mol^{-1 }. It means that if 2.0266 dm^{3 }or 266 cm^{3 }of volume at closet approach in the gaseous stat.**Example 8:**One mole of methane gas is maintained at 300 k. Its volume is 250 cm

^{3 }.calculate the pressure exerted by the gas under the following conditions.

(i) When the gas is ideal (ii) When the gas is non-ideal

a=2.253 atm dm6 mol-1 b=0.0428dm3 mol-1

**Solution:**

(i) when methane gas is behaving as ideal, general gas equation is applied , that is , PV=nRT

v=250 cm =0.25 dm

^{3}n=1 molT=300k R=0.0821atm dm

^{3 }k^{-1 }mol^{-1}P=?

P=

P=

P=98.52 atm Answer

(ii) When methane gas is behaving as non-ideal, van der Waals equation is applied,

(P)(V-nb)=nRT

One rearranging the above equation for P,

P=-

P=

P=

P=117.286-36.048

P=81.238atm Answer

In the non-ideal situation the pressure has lessened up to

=98.52-81.238=17.282atm Answer

__Plasma Stat (or Plasma)__**What is Plasma?**

*“A high temperature ionized gas mixture consisting of free electrons, positive ions and natural atom sis called plasma.”* It is the fourth state of matter, in addition to solid, liquid and gas. It was first identified by William Crookes in 1879. It is a gaseous state. It is a gaseous state. It is electrically neutral because it consists of almost equal numbers of free electrons and positive ions. It conducts electricity because it contains free electrons and positive ions. It constitutes more than 99% of the visible universe. It is everywhere in our space environment. Naturally occurring plasma is rare on earth. Plasma is used to conduct electricity in neon signs and fluorescent bulbs. It occurs only in lightning discharges and in artificial devices like fluorescent lights, neon signs, etc. Scientists have constructed special chambers to experiment with plasma in laboratories.

**How is Plasma formed?**

Plasma is formed by the interaction of energy with atoms or molecules. Atoms or molecules may be ionized on heating. An atom on losing an electron changes to a positive ion. In a sufficiently heated gas, ionization occurs many times, creating clouds of free electrons and positive ions. However, all the atoms are not necessarily ionized and some of them may remain neutral atoms.

*“A high temperature ionized gas mixture consisting of free electrons, positive ions and natural atom sis called plasma.”*Plasma contains a significant number of electrically charged particles which are free electrons and positive ions. It is due to this reason plasma shows electrical properties and behavior.

**What plasma can potentially do?**

Plasma can generate explosions, carry electrical currents and support magnetic fields within themselves. Space plasmas can contain enough heat to melt the earth thousands of times over. Crystal plasmas can freeze the earth at least a hundred times, one after the other.

**Artificial and Natural plasma:**

Artificial Plasma can be created by using charges on a gas, e.g,. in neon signs .Plasma at low temperature is not exist. This is because outside a vacuum, low temperature plasma reacts rapidly with any molecule it meets. This aspect makes it both very useful and not easy to use.

Natural plasma exists only at very high temperatures or low temperature vacuums. It does not break down or react rapidly. It is extremely hot having a temperature over 20.000

^{o}C. Plasmas possess very high energy. They vaporize any material they touch.**Characteristics of Plasma:**

1. Plasma shows a collective response to electric and magnetic fields. This is because it has sufficient numbers of charged particles. The motions of the particles in the plasma generate fields and electric currents from within plasma density. It refers to the density of the charged particles.

- Although plasma contains free electrons and positive ions and conducts electricity but as whole it is electrically neutral. It contains equal number of free electrons and positive ions.

**Where is plasma found?**

Plasma is the most abundant from of matter. The entire universe is almost of plasma. It existed before any other forms of matter came into being .Plasmas are found from the sun to qarks. Quarks are the smallest particles in the universe. Most of the matter in inner-stellar space is plasma. The sun is a 1.5 million kilometer ball of plasma which is heated by nuclear fusion.

On the earth, plasma only occurs in a few limited places such as lightning bolts, flames, auroras and fluorescent lights. On passing electric current through neon gas, both plasma and light are produced.

**Applications of plasma:**

The important applications of plasma are the following.

(i) A fluorescent light bulbs and tube is not like an ordinary light bulb. Inside the tube is a gas. When light is turned on, electricity flows through the tube. This electricity acts as a source of energy and chares up the gas. This charging and exciting of the atoms creates a glowing plasma inside the tube or bulb.

(ii) Neon signs are glass tubes filled with gas. When they are turned on, electricity flows through the tube. The electricity charges the gas and creates plasma inside the tube. The plasma glows a special colour depending on the nature of the gas inside the tube.

(iii) They are used for processing go semiconductors, sterilization of some medical products, lamps ,lasers, diamond coated films , high power micro wave sources, and pulsed power switches.

(iv) They provide foundation for the generation of electrical energy from fusion pollution control, and removal of hazardous chemicals.

(v) Plasmas are used to light up offices and homes, make computers and electronic equipment to work.

(vi) They drive lasers and particle accelerators, help to clean up the environment, pasteurize foods, and make tools corrosion-resistant.

**Future Horizons:**

Scientists are working on putting plasma to effective use. Low energy plasma should by able to survive without instantly reacting and degeneration. The application of magnetic fields involves the use of plasma. The magnetic fields create low energy plasma which relate molecules which are in a metastable state. The metastable molecules do not react until they collide with another molecule with just the right energy. This enables the met stable molecules to survive long enough to react with a designated molecule. The met stable molecules are selectivity. It makes a potentially unique solution to problems like radioactive contamination. Scientists are experimenting with mixtures of gases to work as met stable agents on plutonium and uranium.

**EXERCISE**

**Q1: Select the correct answer out of the following alternative suggestions:**

(i) Pressure remaining constant, at which temperature the volume of a gas will become twice of what it is at 0

^{o}C .a. 546

^{o}C b.200^{o}C c. 546 k d. 273kHind: VT.

(ii) Number of molecules in one dm3 of water is close to

a. b. c. d.

Hint: 1 dm

^{3 }of H_{2}O =1000cm^{3 }of H_{2}O; 1000cm^{3 }of H_{2}O =1000g of H_{2}ONo of moles of H

_{2}O =moles ( 1 ml of H_{2}O =1 f of H_{2}O)(iii) Which of the following will have the same number of molecules at STP?

a. 280cm

^{3 }of CO_{2 }and 280 cm^{3 }of N_{2 }Ob. 11.2dm

_{3 }of O_{2 }and 32 g of O_{2}c. 44g of CO2 and 11.2 dm3 of CO

d. 28 g of N

_{2 }and 5.6 dm^{3 }of oxygen(iv) If absolute temperature of a gas is doubled and the pressure is reduced to one half , the volume of the gas will

a. remain unchanged b. Increase four times

c. reduce to d. be doubled

Hint: PV = RT, V=: V

(v) How should the conditions be changed to prevent the volume of a given gas from expanding when its mass is increased?

a. Temperature is lowered and pressure is increased.

b. Temperature is increased and pressure is lowers.

c. Temperature and pressure both are lowered

d. Temperature and pressure both are increased.

Hint: PV = RT, V=: V( M and R being constant)

(vi) The molar volume of CO2 is maximum at

a. STP b. 127

^{o}C and 1 atmc. 0

^{o}C and 2 atm d. 273^{o}C 1 atm Hint:For1 mole of CO

_{2 }PV = RT, V=: V or V_{m}(vii) The order of the rate of diffusion of gases NH

_{3}, SO_{2 }C1_{2 }and CO_{2 }si :_{a. }NH

_{3}

_{ }>SO

_{2 }> C1

_{2 }> CO

_{2}

b. NH

_{3 }> CO_{2 }> SO_{2 }> C1_{2 }c. C1

_{2 }>SO_{2 }> CO_{2 }>NH_{3}Hint: r

(viii) Equal masses of methane and oxygen are mixed in an empty container at 25

^{o}C . The fraction of total pressure exerted by oxygen is a. b.

c. d.

Hint: In an empty container, the partial pressure of gas is directly proportional to the mole-fraction of the gas. Partial pressure mole fraction, at STP.

(viii) Gases deviate from ideal behavior at high pressure. Which of the following is correct for non-ideality?

a. At high pressure, the ga molecules move in one direction only.

b. At high pressure, the collision between the gas molecules are increased manifold.

c. At high pressure, the intermolecular attractions become significant.

d. At high pressure, the intermolecular attractions become significant.

(ix) The deviation of a gas from ideal behavior is maximum at

a. -10

^{o}C and 5.0atm b. -10^{o}C and 2.0atmc. 100 oC and 2.0 atm d. 0.

^{o}C and 2.0atm (xi) A real gas obeying van der Waals equation will resemble ideal gas if

a. both ‘a’ and ‘b’ are larger b. both ‘a’ and ‘b’ are small

c. ‘a’ is small and ‘b’ is larger d. ‘a’ is larger and ‘b’ is small

**Ans:**(i)c (ii)d (iii)a (iv)b (v)a (iv)b (vii)b (viii)a (ix)d (x)a (xi)b

**Q2. Fill in the blanks.**

(i) The product of PV has the S.I. unit of _______.

(ii) Eight grams each of O

_{2 }and H_{2 }at 27^{o}C will have total K.E in the ratio of ________.(iii) Smell of the cooking gas during leakage from a gas cylinder is due to of the property of _______of gases.

(iv) Equal _________of ideal gases at the same temperature and pressure contain ________number of molecules.

(v) The temperature above which a substance exists only as gas is called _____.

**Ans.**(i) atm dm3 (ii) :16 (iii) diffusion (iv) Volumes: equal (v) Critical temperature

**Q3. label the following sentences as true or false.**

(i) Kinetic energy of molecules is zero at 0

^{o}C.(ii) A gas in a closed container will exert much higher pressure at the bottom due to gravity than at the top.

(iii) Real gases show ideal gas behavior at low pressure and high temperature.

(iv) Liquefaction of gases involves decrease in intermolecular spaces.

(v) An ideal gas on expansion will show Joule-Thomson effect.

**Ans**. (i) False (ii) False (iii) True (iv) False (v) False

**Q4:**(a) What is Boyle’s law of gases? Give its experimental verification.

(b) What are isotherms? What happen to the positions of isotherms when they plotted at high temperature for a particular gas.

(c) Why do we get a straight line when pressures exerted on a gas are plotted against inverse of volumes. This straight line changes its position in the graph by varying the temperature. Justify it.

(d) How will you explain that the value of the constant k in equation PV=k depends upon

(i) The temperature of the gas (ii) the quantity of the gas.

Ans: (a) Isotherms: The P-V curves obtained at constant temperature are called isotherms. These curves are obtained by plotting a graph between pressure on the x-axis and volume on the y-axis. Similar curves are obtained at fixed temperatures.

When the isotherms (P-V curves are plotted at high temperature, they go away from the axes. The reason is that, at higher temperature, the volume of the gas increase.

Ans.(c) A plot of P versus gives a straight line at constant temperature. This shows that p is directly proportional to . This straight line will meet at the origin where both P and are zero. The P goes down as the gas expands, falling away to zero as the volume approaches infinity (==0)

This straight line changes its position because both pressure and volume varies on varying the temperature. When temperature is increased both pressure and volume will increase. Keeping T constant and plotting P versus another straight line is obtained. This straight line goes away from x-axis. However , when temperature is decreased both the values of P and V will decrease. Again a straight line is obtained. This straight line will be closer to the x-axis

Ans.(d) General equation: PV=nRT _______(1)

The general gas equation contains the Boyle’s law, for which nRT are constant at fixed T and n.

Bolyle’s law: PV=k _______(2)

Form Eq (1) and Eq (2), we get

K=nRT

K=

K=constant x mT (at fixed R and M )

K mT _______(3)

This relation indicates that

(i) k T; it means k depends upon the temperature of the gas

(ii) k m; it means k depends upon the quantity of the gas.

Q5. (a) What is the Charles’s law? Which scale of temperature is used to verify that = k (pressure and number of moles are constant)

(b) A sample of carbon monoxide gas occupies 150.0mL at 25.0o C. it is then cooled at constant pressure until it occupies 100.0mL. What is the new temperature?

(c) Do you think that the volume of any quantity of a gas becomes zero at -273

^{o}C. Is it not against the law of conservation of mass? How do you deduce the idea of absolute zero from this information?Ans.(a) The relation, =k can be verified only when T is taken on the Kelvin scale.

Ans.(b) V

_{1 }=150 mL T_{1 }=273+25=298K V

_{2 }=100 mL T_{2 }=?Formula Used:

=

or

T

_{2 }= T

_{2 }= T

_{2 }=198.67k T

_{2 }=273+t^{o}C t

^{o}C=T_{2 }– 273 t

^{o}C=198.67-273 =-74.33

^{o}C AnswerAns.(c) We know that : V

_{1 }=V_{o }(1+)At -273

_{o}C, V_{-273 }= V_{o }(1-)= V_{o }(1-1) =V_{o }x0=0 The volume of the gas become zero at- 273

^{o}C. But it is impossible to imagine that a Gas which is matter occupies no space. It goes against the law of conservation of mass.

Hence, it follows that -273

^{o}C is the lowest temperature which a body can ever have. So -273^{o }C is called the absolute zero of temperature. The scale of temperature on which -273^{o}C is taken as zero is called absolute scale of temperature.**Q6.**(a) What is Kelvin scale of temperature ? Plot a graph for one mole of an ideal gas to prove that a gas become liquid, carlier than -273o C .

(b) Throw some light on the factor in Charles’s law.

**Ans.(b)**In Charles’s law, the factor (0.00366 ) is the coefficient of expansion of given mass of gas at constant pressure. It shows that a gas expands by parts of its volume at 0

^{o }C for a rise of temperature of 1

^{o}C.

**Statement of Charles’s Law:**

*“At constant pressure, the volume of a given mass of gas increases or decreases by*

*of its volume at0*

^{o}C for every 1^{o}C rise or fall in temperature .”**Mathematically.**

V

*-=V*_{t }_{o }+*It means that if we have 273 cm*

^{3 }of gas at 0^{o}C, its volume will increase by 1 cm^{3 }for every 1^{o}C rise in temperature if it is heated at constant pressure Thus,*At 1*

^{o}C , the volume will become 274 cm

^{3}.

At 2

^{o}C , the volume will become 271 cm^{3}. At 273

^{o}C , the volume will become 0 cm^{3}.Similarly, if the gas is cooled by 1

^{o}C at constant pressure, its volume will decrease by 1 cm^{3 }. Thus, At -1

^{o}C , the volume of the gas becomes 272 cm^{3}. At -2

^{o}C , the volume of the gas becomes 275 cm^{3}. At -273

^{o}C , the volume of the gas becomes 546 cm^{3}.**Q7.**(a) What is the general gas equation? Derive it in various forms.

(b) Can we determine the molecular mass of an unknown gas if we know the pressure, temperature and volume along with the mass of that gas?

(c) How do you justify from general gas equation that increase in temperature or decrease of pressure decreases the density of gas?

(d) Why do we feel comfortable in expressing the densities of gases in the units of g dm

^{-3 }rather than g cm^{-3}, a which is used to express the densities of liquids and solids.**Ans.(b)**Yes, we can determine the molecular mass of an unknown gas if we know its, P,T,V and m by applying the following formula:

M=

**Ans.(c)**We know form general gas equation,

d=

d=constant x (M and R being constant)

d=

Density is directly proportional to pressure and inversely proportional to temperature or decrease of pressure, decreases the density of the gas.

**Ans.(d)**The densities of gases are very low. They are about 1000 times smaller than the densities of liquids and solids. So, if gas densities are expressed in g cm

^{-3}, then the values will be very small. They may go to fourth place of decimal for some gases. When we express the densities in g dm

^{-3}, then the values of the densities become reasonable to be expressed. For example, the density of CH

_{4 }gas is 0.7168 g dm

^{-3 }at STP, but if it is expressed in g cm

^{-3 }, then it is 0.0007168 g cm

^{-3 }at STP. Therefore, we feel comfortable in expressing the densities of gases in the units of g dm

^{-3 }rather than in g cm

^{-3}.

**Q8.**Derive the units for gas constant R in general gas equation:

(a) When the pressure is in atmosphere and volume in dm

^{3.} (b) when the pressure is in Nm

^{-2 }and volume in m^{3 }. ( c) when energy Is expressed in ergs.

**Q9.**(a) what is Avogadro’s law of gases?

(b) Do you think that 1 mole of H

_{2 }and 1 mole NH_{3 }at 0^{o}C and 1 atm pressure will have equal Avogadro’s number of particles? If not, why?(c) Justify that 1 cm

^{3 }of H_{2 }and 1 cm^{3 }of CH_{4 }at STP will have the same number of molecules, when one molecules of CH_{4 }is 8 times heavier than that of hydrogen.**Ans**(b) 1 mole of H

_{2 }and 1 mole of NH

_{3 }at 0

^{o}C and 1 atm pressure will have equal number of molecules under the same conditions of temperature and pressure. Hence, 1 cm3 of H2 and 1 cm3 of CH4 at STP will have the same number of molecules.

**Q10.**(a)

(c) Derive an equation to find out the partial pressure of a gas knowing the individual moles of component gases and the total pressure of the mixture.

(d) Explain that the process of respiration obeys the Dalton ’s law of partial pressure.

(e) How do you differentiate between diffusion and effusion? Explain Graham’s law of diffusion.

**Ans.**(a) For

**Q11.**(a) What is critical temperature of a gas? What is its importance for

Liquefaction of gases? Discuss Linde’s method of liquefaction of gases.

(b) What is Joule-Thomson Effect? Explain its importance in linde’s method of liquefaction of gases.

**Ans**. (a) Importance of Critical temperature for liquefaction of gases.

The critical temperature of the gases provides us the information about the

Condition under which gases liquefy. For example,O2 ,has a critical temperature 154.4k(-118,75

^{o}C). It must be cooled below this temperature before it can be liquefied by applying high pressure.**Q12**. (a) What is Kinetic molecular theory of gases? Give its postulates.

(b) How dose Kinetic molecular theory of gases explain the following gas laws:

(i) Boyle’s law (ii) Charles’s law

(iii) Avogadro’s law (iv) Graham’s law of diffusion.

**Q13. (a**) Gases show non-ideal behavior at low temperature and high pressure.

Explain this with the help of a graph.

(b) Do you think that some of the postulates of Kinetic molecular theory of gases are faulty? Point out these postulates.

(c) Hydrogen and helium are ideal at room temperature, but SO

_{2 }and C1_{2 }are non ideal. How will you explain this?**Ans.**(c) Hydrogen (b.p-253

^{o}C) and helium b.p-269

^{o}C)have a very low boiling points .They are far away from their boiling points at room temperature. Also, they have smaller number of electrons in their molecules and smaller molecular sizes, i.e., molecular weight. So, intermolecular forces are negligible at room temperature. Hence, they behave as an ideal gases at room temperature.

On the other hand, SO

_{2 }(b.p-10^{o}C) and C1_{2 }(b.p-34^{o}C) have boiling points near to room temperature. They are not far away from their boiling points at room temperature. Also, they have larger number of electrons in their molecules and larger molecular sizes. So, sufficient intermolecular attractive forces are present at room temperature. Hence, they behave as non-ideal at room temperature.**Q14.**(a) Derive van der Waals equation for real gases.

(b) What is the physical significance of van der Waals constant, ‘a’ and ‘b’.

Give their units.

**Ans. (b)**Physical Significance of van der Waals constant ‘a’ and ‘b’

(i)

**Significance of’ a’:**The value of constant ‘a’ is a measure of the intermolecular attractive forces and greater will be the ease of its liquefaction.**Units of ‘a’ :**

The units of ‘a’ are related to the units of pressure, volume and number of moles.

P= or

a= =

a= atm dm

^{6 }mol^{-2} In SI units: a= ==Nm

^{4 }mole^{-2}(iii) Significance of ‘b’: The value of constant ‘b’ us related to the size of the molecule. Larger the size of the molecule, lager is the value of ‘b’. It is effective volume of the gas molecules.

**Units of ‘b’:**

^{ }‘b’ is the compressible volume per mole of gas. So the units of ‘b’ are related to the units of volume and moles.

V=nb or b =

b==dm

^{3 }mol^{-1} In SI units: b===dm

^{3 }mol^{-1 }**Q15.**Explain the following facts:

(a) The plot of PV versus P is a straight line at constant temperature and with a fixed number of moles of an ideal gas.

(b) The straight line in (a) is parallel to pressure-axis and goes away from the pressure axis at higher pressure for many gases.

(c) The van der walls constant ‘b’ of a gas is four times the molar volume of that gas

(d) Pressure of NH

_{3 }gas at given conditions (say 1 atm pressure and room temperature) is less as calculated by van der Waals equation than that calculated by general gas equation.(e) Water vapors do not behave ideally at 273 k.

(f) SO

_{2 }is comparatively non-ideal at 273 k but behaves ideally at 327 K.**Ans. (a)**At constant temperature and with a fixed number of moles of an ideal gas, when the pressure of the gas is varied, its volume changes, but the product PV remains constant. Thus,

P

_{1 }V_{1 }= P_{2 }V_{2 }= P_{3 }V_{3 }= Hence, for any fixed temperature, the product PV when plotted against P.a straight line parallel to P-axis is obtained. This straight line indicates that PV remains constant quantity.

**Ans. (b)**Now, increase the temperature of the same from T

_{1 }to T

_{2 }.At constant temperature T

_{2 }and with the same fixed number of moles of an ideal gas, when the constant. However, the value of PV increase with increase in temperature. On plotting graph between P on x-axis is obtained. This straight line at T

_{2 }will be away from the x-axis. This straight line also shows that PV is a constant quantity.

**Ans. (c) Excluded volume, ‘b’ is four times the molar volume fo gas.**The excluding with each other as shown in Fig. The spheres are considered to be non-compressible. So the molecules cannot approach each other more closely than the distance, 2r . Therefore, the space indicated by the dotted sphere having radius, 2r will not be available to all other molecules of the gas. In other words, the dotted spherical space is excluded volume per pair of molecules.

**Let each**molecules be a sphere with radius =r

**Volume**of one molecules (volume of sphere)=

**The distance**of the closest approach of 2 molecules =2 r

**The excluded volume for 2 molecules=**

The excluded volume for 1 molecule=

=

=4V

_{m }=b The excluded volume for ‘n’ molecules=n b

Where V

_{m }is the actual volume of a molecule. Hence, the excluded volume or co-volume or non-compressible volume is equal to 4 times the actual volume of the molecules of the gas.

**Ans**. (d) The pressure of NH

_{3 }calculated by general gas equation is high because it is considered as an ideal gas. In an ideal gas, the molecules do not exert any force of attraction on one another. On the other hand, when the pressure of NH

_{3 }is considered as a real gas. Actually, NH

_{3 }is a real gas. It consists of polar NH3 molecules approaches the walls of the container, it experiences an inward pull. Clearly, the molecule strikes the wall with a lesser force than it would have done it these are no attractive forces. As a result of this , the real gas pressure is less than the ideal pressure.

**Ans.**(e) Water vapors present at 273K do not behave ideally because polar water molecules exert force of attraction on one another.

**Ans.**(f) At low temperature, the molecules of SO2 possess low kinetic energy. They come close to each other. The e intermolecular attractive forces become very high. So, it behave non-ideally at 273K. At high temperature, the molecules of SO2 have high kinetic energy. The molecules are at larger distances from one other another. The intermolecular attractive forces become very weak. So, it behaves ideally at 327K.

**Q16.**Helium gas in a 100 cm

^{3 }container at a pressure of 500 torr is transferred to a container with a volume of 250cm

^{3}.What will be the new pressure

(a) if no change in temperature occurs

(b) if its temperature changes from 20

^{o}C to 15^{o}C?**Solution:**

(a) Given: P

_{1}=500 torr P_{2 }=? V

_{1 }=100cm_{3}V_{2 }=250cm_{3 }Formula used: P

_{2 }V_{2 }= P_{1}V_{1} P

_{2 }= P

_{2 }=**=250torr Answer**

(b) Given: P

_{1 }=500torr ; P_{2 }=? V

_{1 }=100cm3 ; V_{2 }=250 cm_{3}_{ }T

_{1 }=273 +20=293K : T

_{2}=273+15=288K

Formula Used:

**=** P

_{2 }=x =

**=196.58 torr Answer**

**Q17.**

**(a)**What are the densities in kg/m

^{3 }of the following gases at STP

(i) Methance, (ii) oxygen (iii) hydrogen

(P=101325Nm

^{-2}, T=273k, molecular masses are in kg mol^{-1 })**(b)**Compare the values of densities in proportion to their mole masses.

**(c)**How do you justify that increase of volume upto 100 dm

^{3 }at 27

^{o}C of 2 moles of NH

_{3 }will allow the gas behave ideally.

**Solution:**

(a) (i)Given P=101325Nm

^{-2}^{ }T=273K

Molar mass of CH

_{4}=12+4=16g mol^{-1 }=16x10^{-3 }kg mol^{-1}^{ }R=8.3143NmK

^{-1 }mol

^{-1}

d=?

Formula Used: d=

d=

d= 0.714kgm

^{-3}P=101325Nm

^{-2 }T=273k

Molar mass of O

_{2 }=32g mol^{-1} d= ?

Formula used: d=

d=

d=1.428kgm

^{-3} (iii) P=101325Nm

^{-3}; T=273kMolar mass of H

_{2}=2x10^{-3}kg mol^{-1}:R=8.3143 NmK^{-1 }mol^{-1} d=?

Formula Used: d=

d=

**d=0.089kg m**

^{-3}**Q18.**A sample of krypton with a volume of 6.25 dm

^{3 }, a pressure of 765 torr and a temperature of 20

^{o}C is expanded to a volume of 9.55 dm

^{3 }and a pressure of 375 torr. What will be its final temperature in

^{o}C?

**Solution:**

P

_{1}= 765torr ; P_{2 }=375torr V

_{1 }=6.25dm^{3 }; V2 =9.55 dm^{3} T1 =273+20 =273k ;T

_{2 }=? Formula used: x

T

_{2}= T

_{2 =} T2 =

T1 =273+

^{o}C^{o}C =T-273=219.46-273

**=-53.54**

^{o}C Answer**Q19.**Working at a vacuum line, a chemist isolated a gas in a weighing bulb with a volume of 255 cm

^{3 }, at a temperature of 25

^{o}C and under a pressure in the bulb of 10.0torr. The gas weight 12.1 mg. what is the molecular mass of this gas?

**Solution:**

V=255 cm

^{3 }=0.255 dm^{3 }^{ }P=10.0torr =

T

_{1 }=273+25=298K m=12.1mg=0.0121g

R=0.0821dm

^{3 }atom K^{-1 }mol^{-1}^{ }Formula Used: PV=RT

M=

M=

**M=87.93 g Mol**

^{-1 }Answer**Q20.**What pressure is exerted by a mixture of 2.00g of H

_{2 }and 8.00g of N

_{2 }at 273 K in a dm

^{3 }vessel?

**Solution:**

Given: V=10dm

^{3 }; T=273k R=0.0821 dm

^{3 }atm K^{-1 }mol^{-1} P

_{H2 }=? P_{N2 }=? Mass of

_{ H2 }=2.00g Mass of_{N2 }=8.00g Molar mass of H

_{2 }=2g Molar mass of N_{2 }=28g mol^{-1} n

_{H2 }==1 mole ;n_{N2 }==0.286 mole n=m

_{H2 }+n_{N2 }=1+0.286=1.286 moles PV=nRT

P x10 dm

^{3 }=1.286 molx 0.0821dm^{3 }atm k^{-1 }mol^{-1 }x 273K P=

**P=2.88 atm Answer**

**Q21. (a)**The relative densities of two gases A and B are 1:1.5 find out the volume of B which will diffused in the same time in which 150 dm

^{3 }of A will diffuse?

**(b)**Hydrogen (H

_{2 }) diffuses through a porpous plate at a rate of 500cm3 per minute at 0

^{o}C. What is the rate of diffusion of oxygen through the same porpous plate 0

^{o}C?

**(c)**The rate of effusion of an unknown gas A through a pinhole is found to be 0.279 times the rate of effusion of H

_{2 }gas through the same pinhole.

Calculate the molecular mass of the unknown gas at STP.

**Solution:**

Given:

d

_{A }=1 r_{A}=150dm_{3 } rB=

**(b)**rH

_{2=500 }cm

^{3 }per minute M

_{H2 }= 2 g mol

^{-1 }

r

_{o2}=? M_{o2 }=32 g mol^{-1}^{ }

=

^{} =4

ro

_{2}=**=125cm**^{3 }Answer**(c)**Given: rH

_{2}=1 r

_{A}=0.279

M

_{A }=? MH_{2}=g mol^{-1} =

=

=

0.078=

M

_{A }= M

_{A }=25.64 g mol^{-1}Answer**Q22.**Calculate the number of molecules and the number of atoms in the given amounts of each gas

(a) 20cm

^{3 }of cH_{4 }at 0^{o}C and pressure of 700 mm of mercury(b) 1 cm

^{3 }of NH_{4 }at 100^{o}C and pressure of 1.5 atm**Solution:**

**(a)**Given: P=700 mm =0921atm

V=20cm

^{3 }=0.02dm^{3 } R=0.0821dm

^{3 }atm K^{-1 }mol^{-1 } T=273+0=273K

N=?

**Formula Used:**PV=nRT

n=

n=

No of molecules =No of moles x N

_{A }**=8.22x10**

^{20 }molecules Answer Since 1 molecule of CH

_{4 }contains =5 atoms Therefore, No of atoms =5x4.948x10

^{20}**=24.74x10**

^{20 }atoms Answer**(b)**Given P=105 atm ; R=0.0821dm

^{3 }atm K

^{-1 }mol

^{-1}

V=1 cm

^{3 }=0.001dm^{3}; T=273+100=373K n=?

**Formula Used:**n=

n=

No of molecules =4.89x10

^{-5 }x6.02x10^{23 } =29.44x10

^{18 }=2.944x10^{19 }molecules Answer No of atoms =2.944x10

^{19 }x4=1.18x10^{20 }atoms AnswerQ23. Calculate the masses of 10

^{20 }molecules of each of H_{2 },O_{2 }and CO_{2 }at STP. What will happen to the masses of these gases, when the temperature of these gases are increased by 100^{o}C and pressure is decreased by 100 torr.**Solution:**

(a) Given: Molecules of H

_{2 }=10^{20}Now , 6.02x10

^{23 }molecules of H_{2 }at STP =1mole =2g 10

^{20 }molecules of H_{2 }at STP = =0.332x10

^{-5 }g**=3.32x10**

^{-4 }g Answer (b) Given: Molecules of O

_{2}=10^{20}Now, 6.02x10

^{23 }molecules of O_{2 }at STP =32g 10

^{20 }molecules of CO_{2 }STP =**=5.32x 10**

^{-3 }g Answer (c) Given: Molecules of CO

_{2 }=10^{20}Now, 6.02x10

^{23 }molecules of CO_{2 }at STP =44g 10

^{20 }molecules of CO_{2 }at STP =**=7.30x10**

^{-3 }g Answer**Q24.**(a) Two moles of NH

_{3 }are enclosed in a 5dm

^{3 }flask at 27

^{o}C. calculate the pressure exerted by the gas assuming that

(i) it behaves like an ideal gas

(ii) it behaves like a real gas

a=1.17 atm dm

^{6 }mol^{-2 }b=0.0371 dm

^{3 }mol^{-1}(b) Also calculate the amount of pressure lessened due to forces of attractions at these conditions of volume and temperature.

(c) Do you expect the same decreased in the pressure of 2 moles of NH

_{3 }having a volume of 40dm_{3 }and at temperature of 27^{o}C.**Solution:**

(a.i) Given: V=5dm

^{3}; T=273+27=300K n=2mole ; R=0.821dm

^{3 }atm K^{-1 }mol^{-1 } P=?

**Formula Used:**PV=nRT

P=

P==

P=9.852 atm Answer

(a.ii) Given: n=2mol

a=4.17atm dm

^{3 }mol^{-1} b=0.0371dm

^{3 }mol^{-1} V=5dm

^{3} R=0.0821atm dm

^{3 }K^{-1 }mol^{-1} T=273+27=300K

P=?

Formula Used: (V-nb)=nRT

On rearranging the equation

P=-

On substituting the values

P=-

P=-

P=-

P=9.99-0.667

P=9.32atm Answer

(b) Difference of pressure , P=9.852-9.32=053atm Answer

(c) Given: n = 2 mol ; V=40dm

^{3};T=273+27=300K p =?

Formula Used: PV=nRT

P=

P=

P=1.232 atm

The decrease in pressure is not the same