Question

During an adiabatic process, the pressure of a gas found to be proportional to the cube of its temperature. The ratio of $\frac{C_p}{C_v}$ for the gas is -

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

Answer 1

The correct option is A $\frac{3}{2}$

For an adiabatic process,

$T^{\gamma }{P}^{(1-\gamma )}=\text{constant}$

$\Rightarrow P^{\left(\frac{1-\gamma }{\gamma }\right)}\propto \frac{1}{T}$

$\Rightarrow P\propto T^{\left(\frac{\gamma }{\gamma -1}\right)}\dots \left(1\right)$

Given, $P\propto T^3\dots \left(2\right)$

Comparing (1) and (2) we get,

$3=\left(\frac{\gamma }{\gamma -1}\right)$

$\Rightarrow \gamma =\frac{C_p}{C_v}=\frac{3}{2}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(A\right)$ is the correct answer.