Prove that the average kinetic energy of a molecule of an ideal gas in directly proportional to the absolute temperature of the gas.

Following is the deduction of kinetic theory in terms of pressure:

\(p=frac{mnv^{2}}{3}\)

Where,

m is the mass of the gas molecule

n is the number of molecules per unit volume

v is the rms speed

So, n = \(frac{N}{V}\)

Where N is the number of molecules

Substituting for n we get,

\(pV=frac{mNv^{2}}{3}\) \(frac{mv^{2}}{2}=E\) is the kinetic energy of the molecule

Therefore, \(pV=frac{2NE}{3}\)

From ideal gas equation,

\(pV=mu RT\)

Where \(mu =frac{N}{N_{A}}\) \(mu RT=frac{2NE}{3}\) \(frac{NRT}{N_{A}}=frac{2NE}{3}\) \(E=frac{3kT}{2}\)

Where \(k=frac{R}{N_{A}\)

Therefore, it can be said that kinetic energy is proportional to temperature

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