Question 256: Thermodynamics & KTG

Question

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of $$\frac{\mathrm{Cp}}{\mathrm{Cv}}$$ for the gas is :

Accepted Answer

For an adiabatic process, the following relation holds:
$$P V^{\gamma} = 	ext{constant}$$
where P is the pressure, V is the volume, and $$\gamma = \frac{C_p}{C_v}$$.
We are given that the pressure is proportional to the cube of the absolute temperature:
$$P \propto T^3$$.
Using the ideal gas law, $$PV = nRT$$, we can rewrite this as:
$$V \propto \frac{T}{P} \propto \frac{T}{T^3} \propto \frac{1}{T^2}$$.
Substituting this into the adiabatic relation, we get:
$$P \left( \frac{1}{T^2} ight)^{\gamma} = 	ext{constant}$$.
Simplifying, we have:
$$P^{1-\gamma} T^{2\gamma} = 	ext{constant}$$.
Since P is proportional to $$T^3$$, we can write:
$$(T^3)^{1-\gamma} T^{2\gamma} = 	ext{constant}$$.
This simplifies to:
$$T^{3-3\gamma + 2\gamma} = 	ext{constant}$$.
For this equation to hold, the exponent of T must be zero. Therefore:
$$3 - 3\gamma + 2\gamma = 0$$.
Solving for $$\gamma$$, we get:
$$\gamma = \frac{C_p}{C_v} = \boxed{\frac{3}{2}}$$.
Therefore, the correct answer is
Option B
.

Answer

$$\frac{7}{5}$$

Answer

$$\frac{3}{2}$$

Answer

$$\frac{9}{7}$$

Answer

$$\frac{5}{3}$$

Thermodynamics & KTG

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of $\frac{\mathrm{Cp}}{\mathrm{Cv}}$ for the gas is :

Choose the correct answer: