Degree of Freedom, Laws of Equipartition of Energy and Atomicity of Gases

Degree of Freedom

The degree of freedom indicates the number of independent motion which a particle can undergo. A monoatomic gas molecule has three degree of freedom. A diatomic gas molecule has three degree of freedom of translation and two degrees of freedom of rotation. So, it has five degrees of freedom.

Laws of Equipartition of Energy

It states that the total kinetic energy of a dynamical system consisting of a large number of particles in thermal equilibrium is equally divided among its all the degrees of freedom and the energy associated with each degree of freedom is \(\frac 12kT\), where k is Boltzmann’s constant and T is absolute temperature of the system.

According to kinetic theory of gases, the mean kinetic energy of a molecule at a temperature T is given by,

\begin{align*} \frac 12 mc^2 &= \frac 32 kT\dots (1) \\ \text {But,}\: c^2 &= u^2 + v^2 + w^2 \end{align*}

As, x, y, and z are all equivalent, mean square velocities along the three axes are equal.
$$ u^2 = v^2 = w^2 \\ \text {or,}\: \frac 12mu^2 = \frac 12v^2 = \frac 12 w^2 $$\begin{align*} \therefore \frac 12mc^2 = 3\left (\frac 12mu^2\right ) = 3\left (\frac 12mv^2\right ) \\ = 3\left (\frac 12mw^2\right ) = \frac 32kT \\ \therefore \frac 12mu^2 = \frac 12mv^2 = \frac 12mw^2 = \frac 12 kT \end{align*}

The energy associated with each degree of freedom whether translator or rotator is \(\frac 12kT\)

Atomicity of Gases

Let us consider a gram molecule of a perfect gas to absolute temperature T with internal energy U which is equal to total kinetic energy of its molecules. As one molecules contains N (Avogadro’s number) molecules, we have internal energy,

$$ U = N \times \text {average K.E. of a molecule} $$

Now if each molecule of the gas possesses ‘f’ degrees of freedom, the average total K.E. of a molecule is

\begin{align*} U = N.f\frac 12 kT = \frac f2RT \dots (i) \\ \text {Where} R = Nk\end{align*}

Now the molar specific heat of a gas at constant volume is defined as the increase in internal energy of 1 gram mole of a gas per unit degree rise of temperature, i.e.,

Molar specific heat,

\begin{align*} C_v = \frac {dU}{dT} = - \frac {d}{dt} \left ( \frac f2 RT\right ) = \frac f2R \\ \text {As,}\: C_p – C_V &= R \\ \text {or,}\: C_p &= C_v + R = \frac f2R + R =\left (\frac f2 + 1\right )R \end{align*}

Hence the ratio of specific heats of a gas,

\begin{align*} r = \frac {\left (\frac f2 + 1\right )R }{\frac f2 + R} = 1 + \frac1f \end{align*}

Special cases

1. Mono-atomic Gas
   Each molecule has three degrees of freedom because it has one translator motion. i.e f=3
   \begin{align*} C_v &= \frac f2R = \frac 32R \\ C_p &= \left (\frac f2 + 1\right ) R = \frac 52R \\ \therefore r &= 1 + \frac 2f =1 + \frac 53 = 1.66 \end{align*}
   The value of r is found to be true experimentally for monoatomic gas like argon and helium.
2. Diatomic Gas
   Diatomic has three degrees of freedom of translation and two degree of freedom of rotation. Thus total number of degree associated with each molecule of a diatomic gas is 5 i.e. f = 5.
   \begin{align*} C_v &= \frac f2R = \frac 52R \\ C_p &= \left (\frac f2 + 1\right ) R = \frac 72R \\ \therefore r &= 1 + \frac 2f =1 + \frac 75 = 1.40 \end{align*}
   The value of r is found to be true experimentally for diatomic gas like hydrogen, oxygen, nitrogen etc.
3. Triatomic Gas
   Each molecule of a triatomic gas possesses 6 degrees of freedom i.e. f = 6
   \begin{align*} C_v &= \frac f2R = \frac 62R = 3R \\ C_p &= \left (\frac f2 + 1\right ) R = 4R \\ \therefore r &= 1 + \frac 2f = \frac 86=\frac 43 = 1.33 \end{align*}
   It may be noted that more complex the molecule, greater will be the degree of freedom and value of two specific heats, but the ratio of the two specific heats goes on decreasing, approaching unity.

   Bibliography

   S.S. Singhal, J.P. Agarwal, Satya Prakash. heat and thermodynamics and statistical physics. pragati prakashan, 2010.

   —. Heat and Thermodynamics and Statistical Physics. Pragati Prakashan, 2010.

   Vatsyayan, Dr. Rakesh Ranjan. Refresher Course in Physics. kathmandu: Surya Book Traders, 2015.