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Question
Show that the average energy of the molecules of a gas is directly proportional to the absolute temperature of the gas.
Solution
- From kinetic theory of gases pressure of the gas is given as,
P = `1/3 "N"/"V""m"overline("v"^2)`
∴ PV = `1/3"N""m"overline("v"^2)`
∴ PV = `2/3"N"(1/2"m"overline("v"^2))` ......(1) - The quantity `1/2"m" overline("v"^2)` represents the average translational kinetic energy of an ideal gas molecule.
- Therefore, the total energy of the gas is,
E = N`(1/2"m" overline("v"^2))` - Substituting in equation (1),
PV = `2/3`E - From ideal gas equation,
PV = NkBT = `2/3`E ......(2)
∴ E = `3/2` NkBT ......(3) - From equation (3), it can be concluded that the average energy of the molecules of a gas is directly proportional to the absolute temperature of the gas.
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