Question

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

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

Answer 1

Answer: (A)

In an adiabatic process, $\left(\textit{T}\right)^{\gamma } = \left(\text{constant}\right) \left(\textit{P}\right)^{\gamma - 1}$
or $\left(\textit{T}\right)^{\gamma / \gamma - 1} = \left(\text{constant}\right) \textit{ P}$
Given $\left(\textit{T}\right)^{3} = \left(\text{constant}\right) \textit{ P}$
$\therefore $ $\frac{\gamma }{\gamma - 1} = 3 \Rightarrow 3 \gamma - 3 = \gamma $
or $2 \gamma = 3 \Rightarrow \gamma = 3 / 2$
For monoatomic gas, $\gamma = \frac{5}{3}$ = 1.67
For diatomic gas, $\gamma = \frac{7}{5}$ = 1.4
when $\gamma $ = 1.5, the gas must be a suitable mixture of monoatomic and diatomic gases
$\therefore $ $\gamma $ = 3/2.