Question

The pressure of an ideal gas is proportional to the cube of its temperature (on absolute scale) in an adiabatic process. Then the value of the ratio $C_p/C_v$ is

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

Hint:

In adiabatic processes, the exponent in $P \propto T^n$ helps determine the value of $\gamma$.

Answer 1

The Correct Option is D

Step 1: In an adiabatic process, pressure ($P$) and temperature ($T$) follow the relation: \[ P \propto T^n \] where $n$ is a constant. Given that pressure is proportional to the cube of the temperature, \[ P \propto T^3 \] thus, $n = 3$. 
Step 2: The adiabatic relation between pressure and temperature for an ideal gas is given by: \[ P T^{-\frac{\gamma}{\gamma - 1}} = {constant} \] where $\gamma = \frac{C_p}{C_v}$ is the heat capacity ratio. 
Step 3: Comparing with the given relation $P \propto T^3$, we equate: \[ -\frac{\gamma}{\gamma - 1} = 3 \] 
Step 4: Solving for $\gamma$: \[ \gamma = \frac{3}{2} \] 
Step 5: Therefore, the correct answer is (D). \bigskip