ideal gas equation of state derivation derive in class 11 physics chemistry using gas laws

find the ideal gas equation of state derivation derive in class 11 physics chemistry using gas laws ?

Assumptions Regarding the Model of an Ideal Gas

(i) A gas consists of very tiny particles called molecules which are identical in all respects.

(ii) The molecules are in random motion and obey Newton’s laws of motion.

(iii) The total number of molecules is very large.

(iv) The volume of molecules is negligibly small compared to the volume occupied by the gas.

(v) The molecules do not exert any appreciable force on each other except during collision.

(vi) The collisions of molecules between themselves and with the walls of the container are perfectly elastic. The collisions are of negligible duration.

Pressure Exerted by an Ideal Gas

The pressure of a gas in a container is a result of the continuous bombardment of the gas molecules against the walls of the container and is given by

where m = mass of each molecule, n = number of molecules in the container, v_rms = root mean square speed of molecules, V = volume of container, M = mass of gas in the container, r = density of the gas and U = internal energy of the gas.

Root Mean Square Speed

The root mean square (rms) speed is defined as

where v1, v2, v3, ……… vn are the speeds of the molecules 1, 2, 3, …….. n respectively. In terms of P and r, v_rms is given by

where m is the mass of each molecule, T is the absolute temperature of the gas and k is a constant called Boltzmann’s constant. Its value is

k = 1.38 x 10^–23 J K^–1 per molecule

R is the universal gas constant and its value is

R = 8.315 J K^–1 mol^–1

 Mean Translational Kinetic Energy

The mean translational kinetic energy of a molecule of a gas is given by

In terms of E, the pressure of the gas is given by

where U = nE is the total translational kinetic energy of all the n molecules of the gas. It is also called the internal energy of the gas.

Avogadro’s Law

Equal volumes of all gases at the same temperature and pressure contain an equal number of molecules. This is Avogadro’s law. The number of molecules in one mole of any gas is 6.0255 x 10²³ which is called the Avogadro number. The amount of a substance which contains as many elementary units as there are in 0.012 kg (or 12 g) or carbon-12 is called a mole. In other words, a mole of substance is the number of grams equal to its molecular mass in grams.

Thus, the mass of 1 mole of carbon is 12 g or 0.012 kg and that of 1 mole of oxygen is 32 g(0.032 kg).

Equation of State of an Ideal Gas

The relationship between pressure P, volume V and absolute temperature T of an ideal gas is called the equation of state. For n moles of a gas, this relation is PV = nRT where R is the molar gas constant. From Avogadro’s law, it follows that one mole of all gases, at the same temperature and pressure, occupies equal volume. Experiments confirm that one mole of any gas occupies 22.4 litres at STP. Consequently, for one mole the ratio PV /T is constant for all gases. This constant is the molar gas constant and can be evaluated as follows:

Van der Waal’s Equation of State

According to Van der Waal, the true pressure exerted by a gas is greater than P by an amount a/V² (where a is a constant) due to attractive forces between molecules and the true volume of the gas is less than V by an amount b (where b is another constant) because molecules themselves occupy a finite space. The Van der Waal’s equation of state is

At high pressures, when the molecules are too many and too close together, the correction factors a and b both become important. But at low pressures, when they are not too many and not too close together, a gas behaves like an ideal gas and obeys the equation PV = nRT.

 Degrees of Freedom and Equipartition of Energy

The total number of coordinates or independent quantities required to completely specify the position or configuration of a dynamical system is called the degrees of freedom of the system. The molecules of a monoatomic gas consist of single atoms. Therefore, the molecules of a monoatomic gas have

three degrees of freedom corresponding to translational motion. The molecules of a diatomic gas have five degrees of freedom-three corresponding to translational motion and two for rotational motion. A polyatomic molecule has six degrees of freedom including one of vibrational motion. The law of equipartition of energy is stated as follows. In any dynamical system with a uniform absolute temperature T, the total energy is distributed equally among all the degrees of freedom and the average energy per degree of freedom per molecule equals kT/2,

where k = 1.38 x 10^–23 J ^K–1. If the molecules of a gas have f degrees of freedom, then kinetic energy per molecule = f x  kT/2. Therefore, kinetic energy per mole is

Thus y = 1 + 2/3 = 5 /3 for a monoatomic gas = 1 + 2/ 5 = 7/ 5 for a diatomic gas = 1 + 2 /6 = 4/ 3 for a triatomic or polyatomic gas

Kinetic Theory of Gases

The molecules of a gas move in all directions with various speeds. The speeds of the molecules of a gas increase with rise in temperature. During its random motion, a fast molecule often strikes against the walls of the container of the gas. The collisions are assumed to be perfectly elastic, i.e. the molecule bounces back with the same speed with which it strikes the wall. Since the number of molecules is very large, billions of molecules strike against the walls of the container every second. These molecules exert a sizeable force on the wall. The force exerted per unit area is the pressure exerted by the gas on the walls. According to the kinetic theory, the pressure of a gas of density r at absolute temperature T is given by

Van der Waals Equation of State

The equation of state PV = nRT holds for an ideal gas. The behaviour of real gases shows departures from an ideal gas behaviour especially at high pressures. The model of an ideal gas is based on a number of assumptions. Van der Waals modified the ideal gas equation PV = nRT by taking into account two of those assumptions which may not be valid. He argued that (i) the volume of the molecules may not be negligible compared to the volume V occupied by the gas and (ii) the attractive forces between the molecules may not be negligible. He said that pressure P in equation PV = nRT is less than the true pressure by an amount p because of attractive forces between the molecules. According to him, the pressure ‘defect’ p is inversely proportional to the square of volume, i.e

where a is constant depending on the nature of the gas. Thus the true pressure of the gas is P’ = P + p = P + a/V ² . He further argued that V is not the true volume of the gas because the molecules themselves occupy a finite volume. According to him, the true volume of the gas is V’ = (V – b) where b is a factor depending on the actual volume of the molecules themselves. Thus Van der Waals’ equation for real gases is P’ V’ = nRT, i.e.

At high pressures, when the molecules are too many and too close together, the correction factors a and b become important.

Mean Free Path

During their random motion, the molecules of a gas often come close to each other. When the distance between two molecules is comparable with the diameter of a molecule, the forces between them become very strong. As a result, their individual momenta before and after the encounter are different. When this happens a ‘collision’ is said to have occurred. The average distance a molecule travels before it suffers a collision with another molecule is called the mean free path (lc), which can be estimated as follows. Suppose the average speed of a molecule of diameter d is v  . In one second, this molecule sweeps out a volume πd²v  . If it finds any other molecules in this volume, it will suffer collisions with them. If n is the number density (i.e. number per unit volume) of the molecules, then the number of molecules in this volume = πd²vn. The number of collisions per second = nc = πd²vn. Therefore, the average time between two collisions (called collision period Tc) is

Hence the mean free path (i.e. the average distance the molecule travels between two successive collisions) is