Question

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

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

Answer 1

Answer: (C)

$\frac{3}{2}$

Given :

Pressure, P$\alpha T^3$

Solution:

In an adiabatic process,

$T^{\gamma }{P}^{1-\gamma }=constant$ ($\gamma =\frac{C_p}{C_v})$

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

$\Rightarrow T\alpha P^{\frac{(\gamma -1)}{\gamma }}$

$\Rightarrow P\alpha T^{\frac{\gamma }{(\gamma -1)}}$

Comparing the two relations, we get

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

$\Rightarrow 3(\gamma -1)=\gamma$

$\Rightarrow 3\gamma -3=\gamma$

$\Rightarrow 2\gamma =3$

Therefore, $\gamma =\frac{3}{2}$

Hence, correct option is C.