Question

For a gas in a thermodynamic process, the relation between internal energy (U), the pressure (P), and the volume (V) is given by: \[ U = 3 + 1.5PV \] The ratio of the specific heat capacities of the gas at constant volume and constant pressure is: 

• A. \( \frac{5}{3} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{4}{3} \)
• D. \( \frac{3}{4} \)

Hint:

For thermodynamic problems, always use the relation: \[ \gamma = \frac{C_p}{C_v} \] and apply the first law of thermodynamics to derive the expressions for specific heat capacities.

Answer 1

The Correct Option is A

Step 1: Expression for \( C_v \) and \( C_p \) From the first law of thermodynamics, the change in internal energy \( dU \) is related to heat added at constant volume and work done: \[ dU = C_v dT \] The relation between internal energy and pressure-volume work is given by: \[ dU = 1.5 P dV \] Now, considering the general thermodynamic relation: \[ C_p = C_v + R \] where \( R \) is the universal gas constant. 

Step 2: Finding the ratio \( \frac{C_p}{C_v} \) We can use the thermodynamic identity for the ratio of specific heat capacities: \[ \gamma = \frac{C_p}{C_v} \] From the given equation \( U = 3 + 1.5 PV \), and the fact that \( \gamma \) is related to the equation \( \gamma = \frac{C_p}{C_v} \), we substitute the values and derive the relation: \[ \gamma = \frac{C_p}{C_v} = \frac{5}{3} \] Thus, the correct answer is: \[ \mathbf{\frac{5}{3}} \]