The Question is from neet-physics thermodynamics cp-and-cv-their-relations, and it was created by the author Mr. ANIL KUMAR.G

Question

For an ideal gas undergoing an isochoric process, the change in internal energy ($\Delta U$) is related to the heat supplied (Q) and the specific heat at constant volume ($C_v$) by which equation?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: First law of thermodynamics and isochoric processes
 **Related Topics**: Internal energy, heat, work, specific heat capacities
 **Prerequisites**: Understanding of thermodynamic processes, ideal gas laws
## SOLUTION PATHWAY
**Given Information**:
  * Isochoric process (constant volume, $\Delta V = 0$)
  * Ideal gas
  * Change in internal energy ($\Delta U$)
  * Heat supplied (Q)
  * Specific heat at constant volume ($C_v$)
**Method & Steps**:
 1. Recall the first law of thermodynamics: $\Delta U = Q - W$, where W is the work done by the system.
  Reasoning: The first law states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
  Formula/Rule: $\Delta U = Q - W$
 2. For an isochoric process, the volume remains constant.  Therefore, no work is done by the gas ($W = 0$).
  Working: Since $W = P\Delta V$, and $\Delta V = 0$, then $W = 0$.
  Key Point:  Work done is zero in an isochoric process.
 3. Substitute $W = 0$ into the first law equation:
  Result: $\Delta U = Q$
  Verification: This shows that all the heat supplied goes into increasing the internal energy of the gas.  For an ideal gas, the change in internal energy is directly related to the change in temperature: $\Delta U = nC_v\Delta T$.  Since $\Delta U = Q$ for an isochoric process, we can say $Q = nC_v\Delta T$.
 ## OPTION ANALYSIS
A ✓: $\Delta U = nC_v\Delta T$ is the correct equation relating the change in internal energy to the specific heat at constant volume and the change in temperature for an ideal gas. Since $Q = \Delta U$ in an isochoric process, this is consistent with the first law.
B :x: $\Delta U = nC_p\Delta T$ uses the specific heat at constant pressure ($C_p$), which is incorrect for a constant volume process. 
C :x: $\Delta U = Q - W$ is the general form of the first law, but it's not the simplified form specific to an isochoric process.
D :x: $\Delta U = Q + W$ is incorrect. The work done by the system is subtracted from the heat added, not added to it.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing $C_v$ and $C_p$: Remember that $C_v$ applies to constant volume processes, and $C_p$ applies to constant pressure processes.
  2.Ignoring the work done: In an isochoric process, the work done is zero, but this must be explicitly considered when applying the first law.
 **Quick Tips**:
  1.For isochoric processes, remember that $\Delta U = Q$ and $W = 0$.
  2.Visualize the process: Imagine a sealed container; no volume change means no work is done.

Mr. ANIL KUMAR.G · Answer

The correct Option is "$\Delta U = nC_v\Delta T$"

Mr. ANIL KUMAR.G · Answer

$\Delta U = nC_p\Delta T$

Mr. ANIL KUMAR.G · Answer

$\Delta U = Q - W$

Mr. ANIL KUMAR.G · Answer

$\Delta U = Q + W$

NEET Physics Thermodynamics Cp And Cv Their Relations MCQs

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