Question

Starting with the virial theorem for an equilibrium configuration show that: (a) the total kinetic energy of a finite gaseous configuration is equal to the total internal energy if $\gamma=C_p / C_v=5 / 3$, where $C_p$ and $C_v$ are the molar specific heats of the gas at constant pressure and at constant volume, respectively, (b) the finite gaseous configuration can be in Newtonian gravitational equilibrium only if $C_p / C_v>4 / 3$.

   Starting with the virial theorem for an equilibrium configuration show that:
(a) the total kinetic energy of a finite gaseous configuration is equal to the total internal energy if $\gamma=C_p / C_v=5 / 3$, where $C_p$ and $C_v$ are the molar specific heats of the gas at constant pressure and at constant volume, respectively,
(b) the finite gaseous configuration can be in Newtonian gravitational equilibrium only if $C_p / C_v>4 / 3$.

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 2, Problem 162

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Step 1

The virial theorem states that for a system in equilibrium, the time average of the virial $ \langle G \rangle $ is related to the total kinetic energy $ T $ and the potential energy $ U $ of the system. In a gaseous configuration, this can be expressed Show more…

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Starting with the virial theorem for an equilibrium configuration show that: (a) the total kinetic energy of a finite gaseous configuration is equal to the total internal energy if $\gamma=C_p / C_v=5 / 3$, where $C_p$ and $C_v$ are the molar specific heats of the gas at constant pressure and at constant volume, respectively, (b) the finite gaseous configuration can be in Newtonian gravitational equilibrium only if $C_p / C_v>4 / 3$.

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