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 N_{1} molecules of mass m_{1} each and r.m.s
velocity C_{1}. Let gas Y contain N_{2} molecules each of
mass m_{2} with r.m.s velocity C_{2},
Now, PV = m_{1}N_{1}C_{1}^{2}
and
PV = m_{2}N_{2}C_{2}^{2}
\ m_{1}N_{1}C_{1}^{2}
= m_{2}N_{2}C_{2}^{2}
\ m_{1}N_{1}C_{1}^{2}
= m_{2}N_{2}C_{2}^{2} …(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.
\ m_{1}C_{1}^{2}
= m_{2}C_{2}^{2}
\ m_{1}C_{1}^{2}
= m_{2}C_{2}^{2} …( 2 )
From (1) and (2),
N_{1} = N_{2}
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.
Consider n perfect gases of densities r_{1}, r_{2} …r_{n} and having r.m.s. velocities of their
molecules as c_{1}, c_{2} …c_{n} mixed in the same
container.
The resultant pressure of the mixture,
p = r_{1}C_{1}^{2}
+ r_{2}C_{2}^{2}
+ … r_{n}C_{n}^{2}.
If the same container was separately occupied by each of the gases, their
partial pressures would be,
p_{1} = r_{1}C_{1}^{2},
p_{2} = r_{2}C_{2}^{2},
…p_{n} = r_{n}C_{n}^{2}.
\ p_{1}
+ p_{2} + … p_{n} = r_{1}C_{1}^{2}
+ r_{2}C_{2}^{2}
+ … r_{n}C^{2}_{n} = 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 (
mv^{2}), kinetic energy of rotation (
Iw^{2}), kinetic energy of
vibration ( mn^{2}, m is the reduced mass)
and potential energy of vibration (
kx^{2}).
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.
