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5. Consider the following calculation of the entropy of an ideal gas in the microcanonical ensemble. An ideal gas is placed in a container of volume $V$. The total energy of the gas is between $E$ and $E + Delta$ where $Delta ll E$. First calculate the volume of phase space occupied by an ideal gas with energy less than $E$. This is $Phi(E, V) = int dq_i int dp_i$ where the integral over the coordinates $q_i$ is $V^N$ and the integral over the momenta $p_i$ is the volume of a hypersphere satisfying $sum p_i^2 < 2mE$ (you will need to look this up). Now calculate: (a) The phase volume for the ideal gas such that the energy is between $E$ and $E + Delta$, and $Delta ll E$. This will be $Gamma(E, V) = frac{partial Phi(E, V)}{partial E} Delta$ (b) The entropy of the gas using $Gamma$.

          5. Consider the following calculation of the entropy of an ideal gas in the microcanonical ensemble.
An ideal gas is placed in a container of volume $V$. The total energy of the gas is between $E$ and
$E + Delta$ where $Delta ll E$.
First calculate the volume of phase space occupied by an ideal gas with energy less than $E$.
This is
$Phi(E, V) = int dq_i int dp_i$
where the integral over the coordinates $q_i$ is $V^N$ and the integral over the momenta $p_i$ is the
volume of a hypersphere satisfying $sum p_i^2 < 2mE$ (you will need to look this up). Now calculate:
(a) The phase volume for the ideal gas such that the energy is between $E$ and $E + Delta$, and
$Delta ll E$. This will be
$Gamma(E, V) = frac{partial Phi(E, V)}{partial E} Delta$
(b) The entropy of the gas using $Gamma$.

$5. Consider the following calculation of the entropy of an ideal gas in the microcanonical ensemble. An ideal gas is placed in a container of volume V. The total energy of the gas is between E and E + Delta where Delta ll E. First calculate the volume of phase space occupied by an ideal gas with energy less than E. This is Phi(E, V) = int dqi int dpi where the integral over the coordinates qi is V^N and the integral over the momenta pi is the volume of a hypersphere satisfying sum pi^2 < 2mE (you will need to look this up). Now calculate: (a) The phase volume for the ideal gas such that the energy is between E and E + Delta, and Delta ll E. This will be Gamma(E, V) = fracpartial Phi(E, V)partial E Delta (b) The entropy of the gas using Gamma.$

Added by F-Tima S.

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