Question

Assertion (A) In an adiabatic process, $R = \frac{5}{3} C_v$. The pressure of the gas will be proportional to $T^{7/4}$. Reason (R) In an adiabatic process system is insulated from the surrounding and heat absorbed or released is zero.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (D)

D

Answer 2

Assertion (A) Analysis:

The assertion states two conditions for an adiabatic process:

1. $R = \frac{5}{3} C_v$
2. The pressure of the gas ($P$) is proportional to $T^{7/4}$ (i.e., $P \propto T^{7/4}$)

Let's analyze each part:

Part 1: $R = \frac{5}{3} C_v$

For an ideal gas, we know the relation between the molar specific heats at constant pressure ($C_p$) and constant volume ($C_v$) and the gas constant ($R$) is given by Mayer's formula:

$C_p - C_v = R$

Also, the adiabatic index ($\gamma$) is defined as the ratio of specific heats:

$\gamma = \frac{C_p}{C_v}$

From these two equations, we can express $R$ in terms of $C_v$ and $\gamma$:

$R = C_p - C_v = \gamma C_v - C_v = (\gamma - 1) C_v$

Comparing this with the given condition $R = \frac{5}{3} C_v$:

$(\gamma - 1) C_v = \frac{5}{3} C_v$

$\gamma - 1 = \frac{5}{3}$

$\gamma = 1 + \frac{5}{3} = \frac{8}{3}$

Part 2: $P \propto T^{7/4}$ for an adiabatic process

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

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

This can be rewritten as:

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

Comparing this with the given condition $P \propto T^{7/4}$:

$\frac{\gamma}{\gamma-1} = \frac{7}{4}$

Cross-multiplying:

$4\gamma = 7(\gamma - 1)$

$4\gamma = 7\gamma - 7$

$3\gamma = 7$

$\gamma = \frac{7}{3}$

Consistency Check for Assertion (A):

From Part 1, the assertion implies $\gamma = 8/3$.

From Part 2, the assertion implies $\gamma = 7/3$.

Since $8/3 \neq 7/3$, the two conditions stated in Assertion (A) are contradictory and cannot simultaneously hold for the same gas undergoing an adiabatic process. Therefore, Assertion (A) is false.

Reason (R) Analysis:

"In an adiabatic process system is insulated from the surrounding and heat absorbed or released is zero."

This statement correctly defines an adiabatic process. By definition, an adiabatic process is one where there is no heat exchange between the system and its surroundings ($Q=0$). This is typically achieved by insulating the system. Therefore, Reason (R) is true.

Conclusion:

Assertion (A) is false. Reason (R) is true.