State the Law of Malus.

Explanation

According to this, law, in a classical system of particales, which is inequilibrium at absolute temperature T, the average internal (kinetic)
energy per particle associated with each degree of freedom is `(1)/(2) K _(B) T _(2)` where `k _(B)` is Boltzamann.s constant. If the particle has f degrees of freedom its average kinetic energy would be `1/2 f k _(B) T.`
Let us consider 1 mole of an ideal gas at absolute temperature T. It has N molecules. Since in an ideal gas there are no intermolecular forces, there is no internal potential energy. The internal energy of an ideal gas is entirely the kinetic energy of its molecules. As we know that the average kinetic energy per molecule of an ideal gas is `(1)/(2) fk _(B) T,` where fis the number of degrees of freedom. Hence the internal energy of 1 mole of an ideal gas would be
`U = N xx (1)/(2) fk _(B) T = (1)/(2)f RT (k _(B) = (R)/(N))`
Differentiating it we get
`(d)/(dT) = (1)/(2) fR`
Suppose the gas is heated at constant volume until its temperature rises through dT. The heat given would be dQ`= C _(v) aT,` and no external work would be done (as volume remains constant). Therefore, by the first law of thermodynamics.
`dU = dQ - dW` we have
`d _(v) = C _(v) dT`
`C _(v) = (d _(v))/( d T)`
`C _(v) = (d _(v))/(dT)`
substituting the volume of `d _(u) //dt` from above, we get
`C _(V) = (1)/(2) fR`
This is the molar specific heat of the gas at constant volume. The molar specific heat at constant pressure would be
`C _(p) = C _(V) + R = (1)/(2) f R + R`
`C _(P) = ((f )/(2) + 1) R`
The ratio of the specific heats of a gas would be
` gamma = (C _(P))/( C _(v)) = (((f)/(2) +1)R)/( (f)/(2) R)`
`gamma = 1 + (2)/(f)`