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(A) The internal energy $U$ of an ideal gas is given by the formula $U = \mu C_v T$,where $\mu$ is the number of moles,$C_v$ is the molar heat capacit

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$U = \frac{PV}{\gamma - 1}$

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(A) The internal energy $U$ of an ideal gas is given by the formula $U = \mu C_v T$,where $\mu$ is the number of moles,$C_v$ is the molar heat capacity at constant volume,and $T$ is the absolute temperature.For an ideal gas,the equation of state is $PV = \mu RT$,which implies $\mu T = \frac{PV}{R}$.The molar heat capacity at constant volume is given by $C_v = \frac{R}{\gamma - 1}$,where $\gamma$ is the adiabatic index.Substituting these expressions into the internal energy formula:$U = \mu \left( \frac{R}{\gamma - 1} \right) T = \left( \frac{\mu RT}{\gamma - 1} \right)$.Since $\mu RT = PV$,we get $U = \frac{PV}{\gamma - 1}$.

What is the relationship between the internal energy $U$ of an ideal gas and its pressure $P$ and volume $V$?

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