Question

A gas undergoes a change in which its pressure ' $P$ ' and volume ' $V$ ' are related as $PV^n = \text{constant}$, where $n$ is a constant. If the specific heat of the gas in this change is zero, then the value of $n$ is ($\gamma = \text{adiabatic ratio}$)}

• A. $1 - \gamma$
• B. $\gamma + 1$
• C. $\gamma - 1$
• D. $\gamma$

Hint:

Zero specific heat means no heat exchange for an infinitesimal change, which corresponds to an adiabatic process.

Answer 1

The Correct Option is D

Concept:
Specific heat of a process is zero when any small heat supplied changes only internal energy and work in such a way that: \[ \delta Q=0 \] That means the process is adiabatic. For an adiabatic process: \[ PV^\gamma=\text{constant} \] ip

Step 1: Identify the process.
Specific heat of the process is zero. So the process is adiabatic. ip

Step 2: Compare with given polytropic form.
Given: \[ PV^n=\text{constant} \] For adiabatic process: \[ PV^\gamma=\text{constant} \] Therefore, \[ n=\gamma \] ip Hence, the correct answer is:
\[ \boxed{(D)\ \gamma} \]