Question

For a gas, \( C_p - C_v = R \). If \( C_v = 2R \), then \( C_p \) is:

• A. \( R \)
• B. \( 2R \)
• C. \( 3R \)
• D. \( 4R \)

Hint:

Specific heat at constant pressure (\( C_p \)) is always greater than specific heat at constant volume (\( C_v \)) because at constant pressure, heat is used not only to increase internal energy but also to do work during expansion.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The relationship between the specific heat capacities of a gas is known as Mayer’s Relation.
\(C_p\) is the molar specific heat at constant pressure, and \(C_v\) is the molar specific heat at constant volume.
For an ideal gas, \(C_p\) is always greater than \(C_v\) because when heat is added at constant pressure, the gas expands and does work, requiring more heat to raise the temperature compared to the constant volume case.
Key Formula or Approach:
The core relationship is:
\[ C_p - C_v = R \]
Where \(R\) is the universal gas constant.
Step 2: Detailed Explanation:
We are given two pieces of information:
1. The standard relation: \(C_p - C_v = R\).
2. The specific value for the gas: \(C_v = 2R\).
Our goal is to find the value of \(C_p\).
Substitute the given value of \(C_v\) into Mayer’s Relation:
\[ C_p - (2R) = R \]
To isolate \(C_p\), add \(2R\) to both sides of the equation:
\[ C_p = R + 2R \]
\[ C_p = 3R \]
Step 3: Final Answer:
The molar specific heat capacity at constant pressure for this gas is \(3R\).
This matches option (C).