Question 685: Thermodynamics & KTG

Question

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

Accepted Answer

We are asked:
A gas undergoes a polytropic process given by
$ P V^n = 	ext{constant}. $
The specific heat in this process is zero. We want the value of $n$ in terms of the adiabatic index $\gamma = \frac{C_p}{C_v}$.
Step 1: Recall general formula for specific heat in polytropic process
For any polytropic process $P V^n = 	ext{constant}$, the
effective molar specific heat
is:
$ C = \frac{C_p - n C_v}{1 - n} $
Step 2: Condition for zero specific heat
We want $C = 0$. That means:
$ \frac{C_p - n C_v}{1 - n} = 0 $
$ C_p - n C_v = 0 \quad \Rightarrow \quad n = \frac{C_p}{C_v} = \gamma. $
Step 3: Verify
Thus, when $n = \gamma$, the specific heat is zero. This has physical meaning: in such a process all heat supplied goes into expansion work, with no temperature change in terms of heat capacity.
Final Answer:
$ \boxed{\gamma} $
Correct option: D.

Answer

$1-\gamma$

Answer

$\gamma+1$

Answer

$\quad \gamma-1$

Answer

$\gamma$

Thermodynamics & KTG

A gas undergoes a change in which its pressure ' P ' and volume ' V ' are related as \mathrm{PV}^{\mathrm{n}}= 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= adiabatic ratio)

Choose the correct answer: