Question

The specific heat capacity at constant volume \(C_v\) of a mole of an ideal gas is related to the gas constant \(R\) and the ratio of the specific heats \(\gamma\) as

• A. \(\frac{R}{\gamma-1}\)
• B. \(\frac{R\gamma}{1-\gamma}\)
• C. \(\frac{R(1-\gamma)}{\gamma}\)
• D. \(\frac{R}{1-\gamma}\)
• E. \(\frac{R}{1+\gamma}\)

Hint:

Always remember the two key relations: \(C_p-C_v=R\) and \(\gamma=\frac{C_p}{C_v}\). These are enough to derive any required formula.

Answer 1

The Correct Option is A

Step 1: Recall the definition of \(\gamma\).
The ratio of specific heats is defined as: \[ \gamma=\frac{C_p}{C_v} \]

Step 2: Recall the relation between \(C_p\), \(C_v\), and \(R\).
For an ideal gas: \[ C_p-C_v=R \]

Step 3: Express \(C_p\) in terms of \(C_v\).
From \[ \gamma=\frac{C_p}{C_v}, \] we get: \[ C_p=\gamma C_v \]

Step 4: Substitute into the relation \(C_p-C_v=R\).
\[ \gamma C_v - C_v = R \] \[ C_v(\gamma-1)=R \]

Step 5: Solve for \(C_v\).
\[ C_v=\frac{R}{\gamma-1} \]

Step 6: Verify the expression.
This is the standard relation for molar specific heat at constant volume for an ideal gas.

Step 7: State the final answer.
\[ \boxed{\frac{R}{\gamma-1}} \] which matches option \((1)\).