The Question is from neet-physics thermodynamics , and it was created by the author VENKATESWARARAO

Question

A diatomic ideal gas undergoes a thermodynamic process in which its molar heat capacity is $C = \frac{3R}{2}$.  Which of the following processes could this represent?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [C]
## CONCEPT FRAMEWORK
	**Primary Concept**: Molar heat capacity of gases
	**Related Topics**: Thermodynamics, Ideal gas behavior, Heat transfer
	**Prerequisites**: Understanding of heat capacities for different thermodynamic processes
## SOLUTION PATHWAY
**Given Information**:
		- Molar heat capacity $C = \frac{3R}{2}$
		- Ideal gas is diatomic
**Method & Steps**:
	1. Determine the nature of the process based on the given heat capacity.
		Reasoning: The molar heat capacity for processes can indicate whether heat is being added under constant pressure, volume, or some polytropic relation.
		Formula/Rule: For a diatomic ideal gas, the heat capacities are as follows:
		- Isobaric: $C_p = \frac{7R}{2}$
		- Isochoric: $C_v = \frac{5R}{2}$
		- Adiabatic: $C_{ad} = C_v$ (no heat transfer)
		- Polytropic: $C = \frac{nR}{n-1}$ where $n$ is the polytropic index.
	2. Identify the polytropic process that corresponds to the given heat capacity.
		Working: Setting $C = \frac{nR}{n-1}$ to $\frac{3R}{2}$:
		- $\frac{3R}{2} = \frac{nR}{n-1}$
		- Simplifying gives $3(n-1) = 2n$ leading to $3n - 3 = 2n$ resulting in $n = 3$.
		Key Point: This indicates the process is a polytropic process with $n = 3$.
	3. Confirm that this aligns with the characteristics of known processes.
		Result: The derived polytropic index $n = 3$ is characteristic of a specific type of heat transfer process.
		Verification: Other options do not yield the same molar heat capacity for a diatomic gas.
## OPTION ANALYSIS
[A] :x: Isobaric process has $C_p = \frac{7R}{2}$, which does not match.
[B] :x: Adiabatic process has $C = C_v = \frac{5R}{2}$, which does not match.
[C] ✓: Correct, corresponds to a polytropic process with $n=3$.
[D] :x: Isothermal process has $C = \infty$, which does not match.
## LEARNING AIDS
	**Common Mistakes**:
		1. Mixing up heat capacities for different processes: Always check the definitions.
		2. Misapplying formulas: Ensure you correctly understand the relationships between $C$, $n$, and the type of thermodynamic process.
	**Quick Tips**:
		1. Remember the specific heat capacities for diatomic gases: $C_v$ and $C_p$.
		2. Use the polytropic process formula to identify the type of process based on the value of $n$.

VENKATESWARARAO · Answer

Isobaric process

VENKATESWARARAO · Answer

Adiabatic process

VENKATESWARARAO · Answer

The correct Option is "Polytropic process with $n=3$"

VENKATESWARARAO · Answer

Isothermal process

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