# Gaseous State

### Gaseous State MCQs

## Vander Waal Equation or Real Gas Equation

Where: P = Pressure, R = Universal gas constant, T = Absolute Temperature, a & b = Vander Waal Constant, V = Molar Volume and n = Number of moles

### Under what condition a real gas behaves as an ideal gas

a.*At high temperature and low pressure*

b. At low temperature and high pressure

c. At high temperature and high pressure

d. At low temperature and low pressure

## Unit of Vander Waal Constants

Unit of 'a': atm lit² mol⁻²Unit of 'b': liter mol⁻¹

## Physical Significance of Vander Waal Constants a & b

**Physical Significance of Vander Waal Constants a**

Vander Waal constant 'a' represents the magnitude of intermolecular forces of attraction.

**Physical Significance of Vander Waal Constants b**

Vander Waals constant 'b' represents the effective size of the molecules.( or average volume excluded from v by a particle).

## Derivation of Real Gas Equation or Vander Waal Equation

Van der Waals observed that the failure of PV = nRT equation for real gasses is due to the neglect of the following two factors-1. Volume of gas molecules is negligible in comparison with the total volume of the gas.

2. The gas molecules exert no attraction upon one another. Van der Waals also made necessary corrections in the ideal gas equation, keeping above facts in mind as-

### 1. Volume Correction

The inner volume (V) of the container in gas is placed, is taken as the volume of the gas in which gas molecules are free to move. Since gas molecules have actually some volume, a part of volume V called the excluded volume is not available for free movement of molecules. If the excluded volume for one mole of a gas be 'b'(called van der Waals constant), then actual volume of the gas will be (V-b) instead of V. Then the real gas equation for one mole becomes-P(V - b) = RT

and for n moles, it becomes-

P(V – nb) = nRT.

### 2. Pressure Correction

We know that the pressure of a gas is the force per unit area exerted when the random moving molecules collide with the inner walls of the container. The inter molecular attractive forces become more significant at higher pressure when the gas molecules come closer and closer to one another. Then the colliding molecules must differ in the the velocities. Therefore, molecules dont strike the inner walls with the same force consequently pressure decreases so observed pressure will be less than the ideal pressure.The attractive force was found to be proportional to the square of the number of moles (n) per unit volume i.e. n

^{2}V

^{2}.

Or the attractive force = an

^{2}V

^{2}where 'a' is the proportionality constant called vander waal constant.

Hence, for 'n' and one mole of real gas, the vander waal equation having necessary correction in ideal gas equation becomes-

This equation is called Vander Waal Equation.

## Deviation Of Real Gas From Ideal Gas Behavior

The deviation of real gas from ideal gas behaviour occurs due to the postulates of kinetic theory of gases. Two posulates of the kinetic theory do not hold good. These are-1. There is no force of attraction between the molecules of a gas. and

2. Volume of the molecules of a gas is negligibly small in comparison to the space occupied by the gas.

If postulate '1' is correct, the gas will never liquify. However, we know that gases do liquify when cooled and compressed. Also, liquids formed are very difficult to compress. This means that forces of repulsion are powerful enough and prevent squashing of molecules in tiny volume.

If postulate '2' is correct, the pressure vs volume graph of experimental data (real gas) and that theoritically calculated from Boyles law (ideal gas) should coincide.

Real gases show deviations from ideal gas law because molecules interact with each other. At high pressures molecules of gases are very close to each other. Molecular interactions start operating. At high pressure, molecules do not strike the walls of the container with full impact because these are dragged back by other molecules due to molecular attractive forces. This affects the pressure exerted by the molecules on the walls of the container. Thus, the pressure exerted by the gas is lower than the pressure exerted by the ideal gas.

P

_{ideal}= P

_{real}+ an

^{2}/V

^{2}

Where, a is a constant.

Repulsive forces also become significant. Repulsive interactions are short-range interactions and are significant when molecules are almost in contact. This is the situation at high pressure. The repulsive forces cause the molecules to behave as small but impenetrable spheres. The volume occupied by the molecules also becomes significant because instead of moving in volume V, these are now restricted to volume (V – nb) where nb is approximately the total volume occupied by the molecules themselves. Here, b is a constant. Having taken into account the corrections for pressure and volume is Ideal Gas Equation becomes-

(P + an

^{2}/V

^{2}) (V − nb) = nRT -----(equation-1)

(equation-1) is known as van der Waals equation. In this equation n is number of moles of the gas. Constants a and b are called van der Waals constants and their value depends on the characteristic of a gas. Value of ‘a’ is measure of magnitude of intermolecular attractive forces within the gas and is independent of temperature and pressure.

The deviation from ideal behaviour can be measured in terms of compressibility factor Z, which is the ratio of product PV and nRT.

Z = PV/nRT

when-

Z = 1 (Ideal Gas at all Temperature)

Z = >1 (Positive deviation from Ideal Gas Above their Boyle Temperature)

Z = < 1 (negative deviation from Ideal Gas below their Boyle Temperature)

## Boyle's Temperature (T_{B}):

The temperature at which normal gases start to behave like ideal gases(due to the absence of both attractive and repulsive forces at that particular temperature).T

_{B}= a/Rb

## Critical Phenomenon and Critical Constants

Gasses can be liquefied if the temperature decreases and the pressure increases continuously, But for every gas there is a characteristic temperature above which it cannot be converted to the liquid state no matter how high the pressure is. This limiting value of temperature is not the same for all gases but is different for different gases.This limiting value of temperature is called the critical temperature and is denoted by Tc. Hence critical temperature is defined as

*the temperature below which the gasses can converted into liquid on continuously increase in pressure and above which no liquefication(i.e. a process of converting gas into liquid) is possible, no matter how high the pressure is*.

At the critical temperature, a certain minimum pressure has to be applied to the gas to liquefy it. This pressure is called critical pressure and is denoted by Pc may be defined as

*the minimum pressure which must be applied to a gas at its critical temperature to liquefy it*.

The volume occupied by one mole of a gas at its critical temperature and critical pressure is called the critical volume and is denoted by Vc.

At this stage or point, both gas and its corresponding liquid would occupy the same volume and therefore their densities are equal so, at this stage, it is not possible to distinguish between the liquid and gaseous states as the two forms are existing in equilibrium. The phenomenon of a smooth merging of a gas into its liquid state under a critical state or critical point is referred to as

**Critical Phenomenon**’. The density of the gas at the critical point is called the critical density.

Tc, Pc and Vc are known as

**Critical Constants**of the gas.

## Derivation of Critical Constants:

We know that-(P + a/V

^{2})(V − b) = RT (for one mole)

or, PV − bP + a/V − ab/V

^{2}= RT

multiplying the above equation by V

^{2}/P we get-

V

^{3}− V

^{2}b + aV/P −ab/P −RTV

^{2}/p = 0

or, V

^{3}− (b + RT/P)V

^{2}+ aV/P − ab/P = 0

when, T = T

_{c}and P = P

_{c}then-

V

^{3}− (b + RT

_{c}/P

_{c})V

^{2}+ aV/P

_{c}− ab/P

_{c}= 0 --------(equation 1)

at critical points,

V = V

_{c}

or, V − V

_{c}= 0

or, (V − V

_{c})

^{3}= 0

or, V

^{3}− 3V

^{2}V

_{c}+ 3V

_{c}

^{2}V − V

_{c}

^{3}= 0 --------(equation 2)

on equation the powers of V in equation (1) and equation (2), we get-

3V

_{c}= RT

_{c}/P

_{c}+ b --------(equation 3)

3V

_{c}

^{2}= a/P

_{c}--------(equation 4)

V

_{c}

^{3}= ab/P

_{c}--------(equation 5)

dividing equation (5) by equation (4) we get-

V

_{c}/3 = b

or,

**V**

_{c}= 3bputting the value of V

_{c}in equation (4) we get-

27b

^{2}= a/P

_{c}

or,

**P**

_{c}= a/27b^{2}Now putting the value of V

_{c}and P

_{c}in equation (3) we get-

9b − b = 27b

^{2}RT

_{c}/a

or, 8a = 27bRT

_{c}

or,

**T**

_{c}= 8a/27Rb## Law of Corresponding State or Reduced Equation of State:

According to this law,two different gases behave similarly, if their reduced properties (i.e. P, V and T) are same.

The ratio of P to P

_{c}, V to V

_{c}and T to T

_{c}are called reduced pressure(P

_{r}), reduced volume(V

_{r}) and reduced temperature(T

_{r}) respectively.

so, P

_{r}= P/P

_{c}

or, P = P

_{r}P

_{c}--------(equation 1)

V

_{r}= V/V

_{c}

or, V = V

_{r}V

_{c}--------(equation 2)

similarly,

T

_{r}= T/T

_{c}

or, T = T

_{r}T

_{c}--------(equation 3)

Now replacing the value P, V and T in Vander Waal equation, we get-

(P

_{r}P

_{c}+ a/V

_{r}

^{2}V

_{c}

^{2})(V

_{r}V

_{c}− b) = RT

_{r}T

_{c}--------(equation 4)

Now putting the values of critical constants in the equation (4) we get-

(P

_{r}a/27b

^{2}+ a/V

_{r}

^{2}9b

^{2})(V

_{r}3b − b) = 8a.RT

_{r}/27Rb

or, (P

_{r}a/27b + a/V

_{r}

^{2}9b)(3V

_{r}− 1) = 8a.T

_{r}/27b

multiply the above equation by 27b/a we get-

**(P**

_{r}+ 3/V_{r}^{2}) (3V_{r}− 1) = 8T_{r}This equation is completely free from constants such as R, a, b hence this is applicable to all substances n fluid state.This is law of corresponding states.

## Compressibility Factor (z)

The compressibility factor is a correction factor which describes the deviation of real gas from ideal gas behaviour. It is useful thermodynamic property for modifying the ideal gas law to account for real gas behaviour.Z = PV / nRT

If the gas behaves ideally, the value of Z is one. The extent to which Z differs from one is a measure of the extent to which the gas is behaving nonideally.

The term compressibility factor can also be defined as the ratio of the actual volume of a gas to the ideal volume of gas for a given temperature and pressure.

Z = V

_{actual}/ V

_{ideal}

### The value of compression factor of a van der Waals gas at critical point is

a. 3.735b. 0.735

c. 3.375

d.

*0.375*

## Liquefaction of Gas

The liquefaction of a gas is a phenomena which takes place when the intermolecular forces of attraction increases to such an extent that they combine the gas molecules together forming a liquid state. Liquefaction of gas can be increased by increasing the intermolecular forces of attraction, which in turn can be increased either by increasing the pressure, which will reduce the distance between the molecules or decreasing the kinetic energy by cooling the gas making them slower. Hence, a gas can be liquefied by cooling or by application of pressure or the combined effect of both.The temperature at which a gas can be liquified is called liquifaction temperature. Above this temperature, the gas can not be liquidfied no matter how high is the pressure. Thus liquifaction temperature of a gas is the critical temperature (T

_{c}). Above this temperature, gaseous state exists, at this temperature liquifaction occurs and below this temperature liquid state exists. Critical temperature depends on the attractive forces present in the gaseous molecules.

The volume and pressure corresponding to critical temperature is called critical volume(V

_{c}) and critical pressure(P

_{c})respectively.

The value of T

_{c}, P

_{c}and V

_{c}are-

T

_{c}= 8a/27Rb

P

_{c}= a/27b

^{2}

V

_{c}= 3b

### Which of the following gas can be most readily liquified

a. NH_{3}

b. Cl

_{2}

c.

*SO*

_{2}d. CO

_{2}

Hints: value of 'a' for NH

_{3}= 4.17, Cl

_{2}= 6.49, SO

_{2}= 6.71, CO

_{2}=3.59)

## Gas Constant in Different Units

8.314 J⋅K^{−1}⋅mol

^{−1}

8.314 × 10

^{7}erg K

^{−1}mol

^{−1}

(As 1J = 10

^{7}ergs)

5.189 × 10

^{19}eV K

^{−1}mol

^{−1}

(As 1 J = 6.2415×10

^{18}eV)

0.082 L atm K

^{−1}mol

^{−1}

1.985 cal K

^{−1}mol

^{−1}

1.985 × 10

^{−3}kcal K

^{−1}mol

^{−1}

### The value of the universal gas constant 'R' depends upon the

a. Nature of the gasb. Mass of the gas

c. Temperature of the gas

d.

*Units of measurement*

## Postulates of Kinetic Theory of Gases

The kinetic theory of gases explained the behavior of the gases. On the basis of extensive study on the behaviour of gases, Bernouli, Boltzman, Maxwell, Clausious and ohers gave the following main postulates of kinetic theory of gases-1. Every gas consists of large number of tiny particles called point masses i.e. the actual volume of molecules is negligible when compared to the total volume of the gas.

2. The gas molecules are always in a state of rapid zig-zag motion in all directions. These molecules collide with each other and with the walls of the container.

3. A molecule moves in a straight line with uniform velocity between two collisions.

4. The molecular collisions are perfectly elastic so that there is no net loss of energy when the gas molecules collide with one another or against the walls of the container.

5. There are no attractive forces operating between molecules or between molecules and the walls of the container in which the gas has been contained. The molecule move independently of one another.

6. The pressure of the gas is force per unit area exerted by the moving molecules collide with the walls of the container.

8. Gaseous molecules does not have potential energy i.e. all the energy of the gaseous molecules is kinetic.

7. The average kinetic energy of gas molecules is directly proportional to absolute temperature. This means that the average kinetic energy of molecules is the same at a given temperature.

## Relation between Kinetic Energy and Temperature

Relationship between kinetic energy and temperature can be established from the kinetic gas equation.We know that-

PV = 1/3(mn)C

^{2}

or, PV = 2/3 X 1/2 (mn)C

^{2}

or, PV = 2/3 KE

or, KE = 3/2 RT

For n moles-

KE = 3/2 nRT

or,

**KE ∝ T**

From above relation we can say that the kinetic energy of an ideal gas is dependent only on the temperature of the gas and not on the nature of the gas or its pressure.