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Question
Show that the average energy per molecule is directly proportional to the absolute temperature ‘T’ of the gas.
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Solution
Consider n moles of an ideal gas in a container of volume V. If m is the mass of a gas molecule and vrms is the root-mean-square speed of the gas molecules, then, by the kinetic theory, the pressure exerted by the gas is
P = `1/3 (Nm)/V v_(rms)^2` ....(1)
where N is the number of molecules of the gas; N = nNA, where NA is the Avogadro number.
∴ PV = `1/3 Nmv_(rms)^2 = 2/3 N (1/2 mv_(rms)^2)` ....(2)
The equation of state of an ideal gas is
PV = nRT .....(3)
∴ `2/3 N (1/2 mv_(rms)^2) = nRT`
∴ `1/2 mv_(rms)^2 = 3/2 n/N RT = 3/2 ((N//N_A)/N) RT = 3/2 R/N_A T` ....(4)
The left-hand side is the average kinetic energy per molecule and `R/N_A = k_B`, the Boltzmann constant.
∴ Average KE per molecule = `3/2 k_B T` .....(5)
Thus, the average kinetic energy per molecule of an ideal gas is proportional to its absolute temperature.
This equation describes the relationship between a gas’s average kinetic energy per molecule and its absolute temperature, which is a macroscopic characteristic. The absolute temperature of a gas is defined as its average kinetic energy per molecule. This finding is known as the kinetic interpretation of temperature, or temperature as interpreted by the kinetic theory of gases.
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