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Question
Explain in detail the kinetic interpretation of temperature.
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Solution
To understand the microscopic origin of temperature in the same way, from pressure exerted by a gas,
P = `1/3 "N"/"V""m"bar("v"^2)`
PV = `1/3"Nm"bar("v"^2)` ............(1)
Comparing the equation (1) with the ideal gas equation PV = NkT,
NkT = `1/3"Nm"bar("v"^2)`
kT = `1/3"m"bar("v"^2)` .........(2)
Multiply the above equation by `3/2` on both sides,
`3/2"kT" = 1/2"m"bar("v"^2)` ...........(3)
R.H.S. of the equation (3) is called average kinetic energy of a single molecule `(bar"KE")`.
The average kinetic energy per molecule
`bar"KE"` = ∈ = `3/2"kT"` ...........(4)
Equation (3) implies that the temperature, of a gas, is a measure of the average translational kinetic energy per molecule of the gas.
Equation (4) is a very important result of the kinetic theory of gas. We can infer the following from this equation.
(i) The average kinetic energy of the molecule is directly proportional to the absolute temperature of the gas. Equation (3) gives the connection between the macroscopic world (temperature) to the microscopic world (motion of molecules).
(ii) The average kinetic energy of each molecule depends only on the temperature of the gas hot on the mass of the molecule. In other words, if the temperature of an ideal gas is measured using the thermometer, the average kinetic energy of each molecule can be calculated without seeing the molecule through the naked eye.
By multiplying the total number of gas molecules with the average kinetic energy of each molecule, the internal energy of the gas is obtained.
Internal energy of ideal gas U = `"N"(1/2"m"bar("v"^2))"`
By using equation (3), U = `3/2"NkT"` ..........(5)
Here, we understand that the internal energy of an ideal gas depends only on absolute temperature and is independent of pressure and volume.
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