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 $p{V}^{\gamma }=$constant
$\Rightarrow p{V}^{15}=$ constant
$\Rightarrow p^5{V}^7=$ constant