The Question is from neet-chemistry , and it was created by the author Mr. S M HUSSAIN

Question

A gas undergoes a process where its pressure $P$ and volume $V$ are related by $PV^n = constant$, where $n$ is a constant.  The molar heat capacity $C$ for this process is given by which expression?

Mr. S M HUSSAIN · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Thermodynamics - First Law and Specific Heats
	**Related Topics**: Polytropic Processes, Adiabatic Processes, Isothermal Processes, Isobaric Processes, Isochoric Processes
	**Prerequisites**: Understanding of heat capacity at constant volume ($C_v$), heat capacity at constant pressure ($C_p$), and the adiabatic index ($γ = C_p/C_v$).
## SOLUTION PATHWAY
**Given Information**:
	$PV^n = constant$
	$n$ is a constant.
**Method & Steps**:
	1. **Relationship between $C$, $C_v$, $C_p$, and $n$**: 
		Reasoning: For a polytropic process, the molar heat capacity $C$ is given by a specific formula relating it to $C_v$, $C_p$, and the polytropic index $n$.
		Formula/Rule: $C = C_v + R/(1-n)$
	2. **Expressing $R$ in terms of $C_p$ and $C_v$**:
		Working: We know that $C_p - C_v = R$.
		Key Point: This relationship is crucial for simplifying the expression.
	3. **Substituting and Simplifying**:
		Result: Substitute $R = C_p - C_v$ into the formula for $C$: $C = C_v + (C_p - C_v)/(1-n) = C_v[1 + (γ - 1)/(1 - n)] = C_v[(1 - n + γ - 1)/(1 - n)] = C_v[(γ - n)/(1 - n)]$
		Verification: The final expression matches option A.
## OPTION ANALYSIS
[A] ✓: $C = C_v [(γ - n) / (1 - n)]$. This is the correct expression derived above.
[B] :x: $C = C_p [(γ - n) / (1 - n)]$. This incorrectly uses $C_p$ instead of $C_v$ in the final expression.
[C] :x: $C = C_v [(n - γ) / (1 - n)]$. This has the wrong sign in the numerator.
[D] :x: $C = R [(γ - n) / (n - 1)]$. This uses $R$ and has the denominator inverted and sign changed.  While it's possible to manipulate the correct answer into a form involving only $R$, $γ$ and $n$, this option doesn't represent the standard form for $C$ in terms of $C_v$, and is thus incorrect.
## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrectly using $C_p$ instead of $C_v$: Carefully remember the derivation starts with the relationship between $C$ and $C_v$.
		2. Sign errors: Pay close attention to the signs during algebraic manipulation.
	**Quick Tips**:
		1. Remember $C_p - C_v = R$ and $γ = C_p/C_v$.
		2. For an adiabatic process ($n = γ$), $C = 0$, which is consistent with the derived formula.  This can be a useful check.

Mr. S M HUSSAIN · Answer

The correct Option is "C = Cv [(γ - n) / (1 - n)]"

Mr. S M HUSSAIN · Answer

C = Cp [(γ - n) / (1 - n)]

Mr. S M HUSSAIN · Answer

C = Cv [(n - γ) / (1 - n)]

Mr. S M HUSSAIN · Answer

C = R [(γ - n) / (n - 1)]

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