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(A) For an ideal gas,the molar heat capacity at constant volume is given by $C_{V} = \frac{dU}{dT}$.For $1$ mole of an ideal gas,$U = \frac{f}{2}RT +

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$p^{5} V^{7} = 	ext{constant}$

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(A) For an ideal gas,the molar heat capacity at constant volume is given by $C_{V} = \frac{dU}{dT}$.For $1$ mole of an ideal gas,$U = \frac{f}{2}RT + C$,where $f$ is the degree of freedom.Given $U = \frac{5}{2}pV + C$. Since $pV = RT$ for $1$ mole,we have $U = \frac{5}{2}RT + C$.Comparing this with $U = \frac{f}{2}RT + C$,we get $\frac{f}{2} = \frac{5}{2}$,which implies $f = 5$.The adiabatic index $\gamma$ is given by $\gamma = 1 + \frac{2}{f} = 1 + \frac{2}{5} = \frac{7}{5}$.The equation for an adiabatic process is $pV^{\gamma} = 	ext{constant}$.Substituting $\gamma = \frac{7}{5}$,we get $pV^{7/5} = 	ext{constant}$.Raising both sides to the power of $5$,we obtain $p^{5}V^{7} = 	ext{constant}$.

For an ideal gas,the internal energy is given by $U = 5pV/2 + C$,where $C$ is a constant. The equation of the adiabats in the $pV$-plane will be

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