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

Step 1: Understanding the Concept:
Specific heat $C$ is zero when there is no heat exchanged with the surroundings during a temperature change. This is the definition of an adiabatic process.
Step 2: Key Formula or Approach:
Polytropic Process: $PV^n = K$.
Specific heat: $C = C_v + \frac{R}{1-n}$.
Adiabatic process: $dQ = 0 \implies C = 0$ and $n = \gamma$.
Step 3: Detailed Explanation:
The process described is $PV^n = K$.
Heat exchanged in such a process is $dQ = nCdT$.
If specific heat $C = 0$, then $dQ = 0$ regardless of temperature change.
A thermodynamic process where no heat is exchanged ($dQ = 0$) is an adiabatic process.
For an ideal gas, the equation of an adiabatic process is $PV^\gamma = \text{constant}$.
Comparing $PV^n = K$ with $PV^\gamma = K$, we get $n = \gamma$.
Step 4: Final Answer:
The value of $n$ is $\gamma$.