Explain the kinetic interpretation of temperature. (or) Prove that the average kinetic energy of a molecule of an ideal gas is directly proportional to the absolute temperature of gas.

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Detailed Solution

$\begin{array}{l} p = \frac{1}{3} \frac{M {\bar{V}}^{2}}{V} \Rightarrow PV = \frac{1}{3} M {\bar{V}}^{2} \\ \Rightarrow PV = \frac{1}{3} mN {\bar{V}}^{2} (∵ M = mN) \\ \Rightarrow RT = \frac{1}{3} {mN}^{2} (∵ PV = RT) \\ \Rightarrow 3 \frac{RT}{N} = m {\bar{V}}^{2} \Rightarrow \frac{1}{2} m {\bar{V}}^{2} = \frac{3}{2} \frac{RT}{N} \\ (∵ K_{B} = \frac{R}{N}) \\ \frac{1}{2} m {\bar{V}}^{2} = \frac{3}{2} K_{B} T \Rightarrow KEαT \end{array}$
* K is Boltzman constant

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