What is the ratio of Cₚ/Cᵥ for gas if the pressure of the gas is proportional to the cube of its temperature and the process is an adiabatic process

The correct option is 3/2.

Step 1: Understanding the adiabatic relation between Pressure and Temperature

For an ideal gas undergoing an adiabatic process, the relation between temperature ($T$) and pressure ($P$) is given by the formula:

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

where $\gamma =\frac{C_p}{C_v}$ is the ratio of specific heats.

Step 2: Expressing Pressure in terms of Temperature

We can rewrite the adiabatic equation to express pressure ($P$) in terms of temperature ($T$):

$$P^{1-\gamma }\propto T^{-\gamma }$$

Raising both sides to the power of $\frac{1}{1-\gamma }$ gives:

$$P\propto T^{\frac{-\gamma }{1-\gamma }}$$

Simplifying the exponent by multiplying the numerator and denominator by -1:

$$P\propto T^{\frac{\gamma }{\gamma -1}}$$

Step 3: Comparing with the given condition

According to the question, the pressure of the gas is proportional to the cube of its temperature during this process:

$$P\propto T^3$$

By comparing the exponents of temperature in both proportionality relations, we set them equal to each other:

$$\frac{\gamma }{\gamma -1}=3$$

Step 4: Solving for the ratio of specific heats ($\gamma$)

Now, we solve the algebraic equation for $\gamma$:

$$\gamma =3(\gamma -1)$$

$$\gamma =3\gamma -3$$

Rearranging the terms to group the $\gamma$ terms on one side:

$$3\gamma -\gamma =3$$

$$2\gamma =3$$

$$\gamma =\frac{3}{2}$$

Therefore, the ratio of specific heats $\frac{C_p}{C_v}$ for the gas is indeed 3/2.