In thermodynamics adiabatic process, pressure is directly proportional to cube of absolute temperature. Find $\frac{C_P}{C_V} $for the gas.

Question

Given $C_p = a + bT$
$a = 19.5$, $b = 0.042$, $n = 1$ mole
Temperature changes from $27^\circ C$ to $327^\circ C$.
Find $\Delta H$.

Answer 1

To solve this problem, we need to determine the ratio of specific heats $\frac{C_P}{C_V}$ for a gas undergoing an adiabatic process.

For an adiabatic process in an ideal gas, the relation between pressure and volume is:

$P V^{\gamma} = \text{constant}$, where $\gamma = \frac{C_P}{C_V}$.

Using the ideal gas equation:

$P V = n R T$

We can eliminate $V$ by writing $V = \frac{nRT}{P}$ and substituting into the adiabatic equation.

This gives the temperature–pressure relation for an adiabatic process:

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

Rearranging, we get:

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

According to the question, pressure is directly proportional to the cube of absolute temperature:

$P \propto T^3$

Comparing the powers of $T$:

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

Solving:

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

Therefore, the ratio of specific heats is:

Correct Answer: $\frac{C_P}{C_V} = \frac{3}{2}$

The Correct Option is C