During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of Cp/Cv for the gas is

Question

The quantity which changes with temperature:

Answer 1

To solve this problem, we need to understand the relationship between the pressure and temperature of a gas during an adiabatic process and its connection to the specific heat capacities C_p and C_v.

In an adiabatic process, the relation between pressure P and temperature T for an ideal gas can be expressed using the adiabatic equation:

P \cdot V^\gamma = \text{constant}

where \gamma (gamma) is the adiabatic index or the ratio of specific heats C_p / C_v.

The given condition states that pressure P is proportional to the cube of temperature T:

P \propto T^3

From the ideal gas law, we know:

PV = nRT

Thus, for an adiabatic process, combining both equations, we have:

PV^\gamma = \text{constant} \quad \Rightarrow \quad P \cdot \left(\frac{nRT}{P}\right)^\gamma = \text{constant}

Simplifying gives:

P^{1-\gamma} \cdot T^\gamma = \text{constant}

With P \propto T^3, we equate the exponents:

1-\gamma + 3\gamma = 0

1+2\gamma = 0 \quad \Rightarrow \quad \gamma = \frac{3}{2}

Therefore, the ratio C_p/C_v = \gamma = \frac{3}{2}.

Thus, the correct answer is 3/2.

Answer 2