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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Kinetic Theory of Gases Know More

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I. Just as Boyle's law and Charles' law can be deduced from the kinetic theory, Avogadro's law and
Dalton's law of partial pressures can be deduced.

Avogadro's Law: Equal volumes of all gases at the same temperature and pressure contain equal number of molecules.

Consider two gases X and Y at the same temperature and pressure and having the same volume. Let gas X have N1 molecules of mass
m1 each and r.m.s velocity C1. Let gas Y contain N2 molecules each of mass m2 with r.m.s velocity C2,

Now, PV =
m1N1C12 and
PV =

\ m1N1C12 = m2N2C22

\ m1N1C12 = m2N2C22 (1)

If the two gases are mixed, since they are at the same temperature, the average K.E. of any molecule of gas X is equal to the average K.E. of any molecule of gas Y.

\ m1C12 = m2C22

\ m1C12 = m2C22 ( 2 )

From (1) and (2),

N1 = N2

Thus, the number of molecules in each gas is the same, which proves Avogadro's law.

Dalton's Law of Partial Pressures
The resultant pressure exerted by a mixture of perfect gases is equal to the sum of the pressures exerted separately by its several components.

n perfect gases of densities r1, r2 rn and having r.m.s. velocities of their molecules as c1, c2 cn mixed in the same container.

The resultant pressure of the mixture,

p = r1C12 + r2C22 + rnCn2.

If the same container was separately occupied by each of the gases, their partial pressures would be,

p1 = r1C12, p2 = r2C22, pn = rnCn2.

\ p1 + p2 + pn = r1C12 + r2C22 + rnC2n = p.

Which proves
Dalton's Law.

II. Principle of Equipartition of Energy
In the kinetic theory, it is assumed that molecules behave like hard elastic spheres. Hence, the kinetic energy of a molecule is purely translational in nature. In the case of monoatomic molecules, the prediction of specific heat based on this model is satisfactory.

However, in the determination of specific heats all possible ways of absorbing energy should be considered. If we picture a molecule as an object with an internal structure, it can rotate and vibrate as well as possess translational motion. Thus, during collisions, the rotational and vibrational modes of motion would contribute to the internal energy of the molecule.

A molecule would then possess kinetic energy of translation ( mv2), kinetic energy of rotation ( I
w2), kinetic energy of vibration ( mn2, m is the reduced mass) and potential energy of vibration ( kx2).

From statistical mechanics it can be shown that if the number of particles is very large, all these terms all have the same average value which depend only on the temperature. Thus, the energy depends only on the temperature and distributes itself in equal parts to each of the independent ways in which the molecule can absorb energy. This theorem is called the equipartition of energy. Each independent mode of energy absorption is called a degree of freedom.

First Law of Thermodynamics
Consider a system in an initial equilibrium state
i. Let it absorb an amount of energy Q and change to a new equilibrium state f. Let w be the amount of work done during this process. Then, Q - w will give the change in the internal energy of the system. It is found that this quantity (Q - w) only depends, the initial and final states of equilibrium but not on the actual process or path. The quantity Q - w is called the internal energy function.

If the system undergoes an infinitesimal change in state, only an amount of heat
dQ is absorbed. The amount of work done is dw. The change in internal energy du is infinitesimal. Then, the change in internal energy
du = dQ - dw.

This expression is called the first law of thermodynamics.

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