# Kinetic Theory of an Ideal Gas and Degree of freedom

• Kinetic theory of gas equation \tt PV=\frac{1}{3}mN\ V^2_{RMS} where P is the pressure exerted by the gas and V is the volume “m” is the mass of molecule. N is the total number of molecules VRMS is Rms velocity.
• If volume and temperature of a gas one constant P ∝ mN if mass of the gas is increased, number of molecule and hence number of collisions per second increase and pressure will increase.
• If mass and volume of gas are constant \tt P\propto V_{RMS}^2\propto T if temperature increases, the mean square speed of gas molecules will increase and gas molecules move faster.
• Total kinetic energy of all the molecules of a gas is given by \tt ET=\frac{3}{2}KTN (N=Number of gas molecules).
• Average translational KE of a gas molecules depends only on its temperature and is independent of its nature.
• The total number of independent modes in which a particle can posses energy is called Degree of freedom.
• The number of degrees of freedom of monoatomic gas is '3', diatomic gas '5' polyatomic '6'
• The internal energy of 'n' moles of a gas in which each molecule has "f" degree of freedom will have internal energy \tt U=\frac{f}{2}\left(nRT\right)=\frac{f}{2}KT
• Only average translational kinetic energy of a gas contributes to its temperature. Two gases with the same average translational KE have the same temperature even if one has greater rotational energy and then greater internal energy.
• Variation of degree of freedom of a diatomic gas with temperature. At very low temperature only translation is possible. As the temperature increases rotational motion can begin at still higher temperatures vibratory motion can begin.

### View the Topic in this video From 1:01 To 58:24

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1. Kinetic energy of gas molecules.
Pressure exerted by a gas is \tt p = \frac{1}{3}\frac{M}{V}v^{2}_{rms} = \frac{1}{3}pv^{2}_{rms}

2. The rms velocity is related to density (ρ) or molar mass (M) by the relation
\tt v_{rms} = \sqrt{\overline{v}^{2}} = \sqrt{\frac{3p}{\rho}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_{B}T}{m}}

3. Kinetic energy of gas per unit volume \tt E = \frac{1}{2}\frac{M}{V}v^{2}_{rms} \Rightarrow E \propto v^{2}_{rms} \ or \ E \propto T
Also, \tt pV = \frac{2}{3}E

4. The average kinetic energy of one mole of an ideal gas is \tt E = \frac{3}{2} RT = \frac{3}{2}k_{B}N_{A}T

5. A monoatomic gas molecules has only translational kinetic energy
\tt E_{t} = \frac{1}{2} mv^{2}_{x} + \frac{1}{2}mv^{2}_{y} + \frac{1}{2} mv^{2}_{z}

6. A part from translations kinetic energy, a diatomic molecule has two rotational kinetic energies
\tt E_{t} + E_{r} = \frac{1}{2} mv^{2}_{x} + \frac{1}{2}mv^{2}_{y} + \frac{1}{2} mv^{2}_{z} + \frac{1}{2}l_{y}\omega_{y}^{2} + \frac{1}{2}l_{x}\omega_{x}^{2}

7. Diatomic molecule like CO even at moderate temperature have a mode of vibration. Its atom vibrate along the interatomic axis and contribute vibrational energy terms Ev to the total energy
\tt E = E_{t} + E_{r} + E_{v} = \frac{1}{2} mv^{2}_{x} + \frac{1}{2}mv^{2}_{y} + \frac{1}{2} mv^{2}_{z} + \frac{1}{2}l_{y}\omega_{y}^{2} + \frac{1}{2}l_{z}\omega_{x}^{2} + \frac{1}{2} m \left[\frac{dy}{dt}\right]^{2} + \frac{1}{2}Ky^{2}

8. The energy per molecule per degree of freedom = \tt \frac{1}{2}k_{B}T. Where, kB = Boltzmann's constant