Question

The difference between the two molar specific heats of a gas is 9000 J/kg K. If the ratio of the two specific heats is 1.5, calculate the two molar specific heats.

Hint:

Use \( C_p - C_v = R \) and the ratio \( \gamma = \frac{C_p}{C_v} \) to solve for molar specific heats.

Answer 1

The difference between molar specific heats at constant pressure (\( C_p \)) and constant volume (\( C_v \)) is: \[ C_p - C_v = R. \] Given: \( C_p - C_v = 9000 \, \text{J/kg K} \), and \( \frac{C_p}{C_v} = 1.5 \).
Note: The unit J/kg K suggests specific heats per unit mass, but “molar” implies per mole. Assuming a typo and interpreting as J/mol K (common in such problems).
\[ C_p - C_v = 9000 \, \text{J/mol K}, \quad \frac{C_p}{C_v} = 1.5. \] Let \( C_v = x \). Then \( C_p = 1.5x \).
\[ 1.5x - x = 9000 \quad \Rightarrow \quad 0.5x = 9000 \quad \Rightarrow \quad x = 18000. \] \[ C_v = 18000 \, \text{J/mol K}, \quad C_p = 1.5 \times 18000 = 27000 \, \text{J/mol K}. \] Answer: \( C_p = 27000 \, \text{J/mol K} \), \( C_v = 18000 \, \text{J/mol K} \).