Kinetic Theory of Gas

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