Question

If during an adiabatic process, the pressure of a gas is found to be proportional to the square root of the $5^{\text{th}}$ power of its absolute temperature, the ratio $C_p/C_v$ for the gas is

• A. $\frac{5}{3}$
• B. $\frac{3}{2}$
• C. $\frac{4}{3}$
• D. $\frac{7}{5}$

Answer 1

The correct option is A $\frac{5}{3}$

From the data given in the question,
$P\propto T^{5/2}$
$\Rightarrow P{T}^{-5/2}=k......\left(1\right)$
Equation of state of a gas during an adiabatic process can be given by
$P^{\gamma -1}{T}^{-\gamma }=k$
$\Rightarrow P{T}^{\frac{-\gamma }{\gamma -1}}=k.....\left(2\right)$
Comparing $\left(1\right)$ and $\left(2\right)$, we get
$\frac{-\gamma }{\gamma -1}=-\frac{5}{2}\Rightarrow \gamma =\frac{5}{3}$
Thus, $\gamma =\frac{C_p}{C_v}=\frac{5}{3}$
Hence, option (a) is the correct answer.