Question

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

• A. $p^{5}V^{7}$ = constant
• B. $p^{7}V^{5}$ = constant
• C. $p^{3}V^{5}$ = constant
• D. $p^{5}V^ {2}$ = constant

Answer 1

Answer: (A)

For an ideal gas
$C_{V}=\frac{\partial U}{\partial T}|_{v = \text{constant}}$
or $C_{V}=\frac{dU}{dT}$
Also, for $1$ mole of gas,
$C_{V}=\frac{f}{2}.R$
where, $f =$ degree of freedom.
Hence, we have
$\frac{f}{2}R=\frac{dU}{dT}$
Here, $U=\frac{5}{2}pV + C=\frac{5}{2}RT+C$
[$\therefore $ one mole of gas is considered]
So, $\frac{f}{2}R= \frac{d}{DT} \left( \frac{5}{2} RT + C\right)$
$\Rightarrow \frac{f}{2} R = \frac{5}{2}R\Rightarrow f=5$
Now, using $\gamma=1+\frac{2}{\gamma}$
We have, $\gamma =1+\frac{2}{5}$
$\Rightarrow \gamma=\frac{7}{5}$
So, equation of adiabats can be written $pV^{\gamma}=$constant
$\Rightarrow pV^{1 5}=$ constant
$\Rightarrow p^{5}V^{7}=$ constant