Question

The specific heat capacity of a monatomic gas at constant volume is \( x % \) of its specific heat capacity at constant pressure. Then \( x \) =

• A. 40
• B. 50
• C. 60
• D. 75

Hint:

For monatomic gases, the ratio of specific heat capacities at constant pressure and constant volume is always \( \frac{5}{3} \), and the relation between \( C_p \) and \( C_v \) can help solve such problems.

Answer 1

The Correct Option is C

For a monatomic ideal gas, the ratio of specific heat capacities at constant pressure \( C_p \) and constant volume \( C_v \) is given by: \[ \gamma = \frac{C_p}{C_v} = \frac{5}{3} \] Now, we are told that the specific heat capacity at constant volume is \( x % \) of the specific heat capacity at constant pressure. Therefore, we can express the relationship as: \[ C_v = \frac{x}{100} \times C_p \] Substitute \( C_p = \frac{5}{3} C_v \): \[ C_v = \frac{x}{100} \times \frac{5}{3} C_v \] Solving for \( x \): \[ 1 = \frac{x}{100} \times \frac{5}{3} \] \[ x = 60 \] Thus, the correct answer is option (3), 60.

Answer 2

Step 1: Recall the relation between specific heats
For an ideal gas,
\[ C_p - C_v = R \]
where \( C_p \) is specific heat at constant pressure, \( C_v \) is specific heat at constant volume, and \( R \) is the gas constant.

Step 2: Specific heats for a monatomic gas
For a monatomic ideal gas:
\[ C_v = \frac{3}{2} R, \quad C_p = C_v + R = \frac{3}{2} R + R = \frac{5}{2} R \]

Step 3: Calculate \( x \) %
Given that \( C_v \) is \( x\% \) of \( C_p \):
\[ x = \frac{C_v}{C_p} \times 100 = \frac{\frac{3}{2} R}{\frac{5}{2} R} \times 100 = \frac{3}{5} \times 100 = 60 \]

Step 4: Conclusion
Therefore, \( x = 60 \% \). The specific heat at constant volume is 60% of the specific heat at constant pressure for a monatomic gas.