Question

A hypothetical gas expands adiabatically such that its volume changes from 08 litres to 27 litres. If the ratio of final pressure of the gas to initial pressure of the gas is $\frac{16}{B 1}$. Then the ratio of $\frac{C p}{C v}$ will be

• A. $\frac{3}{1}$
• B. $\frac{3}{2}$
• C. $\frac{1}{2}$
• D. $\frac{4}{3}$

Hint:

The ratio \( \gamma = \frac{C_P}{C_V} \) is constant for an ideal gas during adiabatic processes.

Answer 1

The Correct Option is D

The correct answer is (D) : $\frac{4}{3}$

Answer 2

For an adiabatic process, the relation between pressure and volume is given by: \[ P_1 V_1^\gamma = P_2 V_2^\gamma \] where \( \gamma = \frac{C_P}{C_V} \) is the adiabatic index. Taking the ratio of the final and initial pressures: \[ \frac{P_2}{P_1} = \left( \frac{V_1}{V_2} \right)^\gamma \] Substitute the values: \[ \frac{16}{81} = \left( \frac{8}{27} \right)^\gamma \] \[ \left( \frac{8}{27} \right)^\gamma = \frac{2}{9} \] Solving for \( \gamma \), we get \( \gamma = \frac{4}{3} \). Thus, the ratio of \( C_P \) to \( C_V \) is \( \frac{4}{3} \).