Question

For ideal gas: \(n\): number of moles, \(C_v\): molar specific heat at constant volume, \(\gamma\): adiabatic exponent of gas, \(T_f\): final temperature, \(T_i\): initial temperature, \(R\): gas constant. Assertion : \[ nC_v(T_f-T_i)=\frac{nR}{\gamma-1}(T_f-T_i) \] Reason : \[ \gamma = 1 + \frac{2}{f} \]

• A. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
• B. Both Assertion and Reason are true but Reason is NOT the correct explanation of Assertion.
• C. Assertion true but Reason is false.
• D. Assertion is false but Reason is true.

Answer 1

The Correct Option is B

Concept: For an ideal gas, \[ C_p - C_v = R \] and \[ \gamma = \frac{C_p}{C_v} \] From these relations, \[ C_v = \frac{R}{\gamma - 1} \] Step 1: Substitute in heat expression \[ Q = nC_v(T_f-T_i) \] \[ Q = n\left(\frac{R}{\gamma-1}\right)(T_f-T_i) \] \[ Q = \frac{nR}{\gamma-1}(T_f-T_i) \] Thus the Assertion is true. Step 2: Check the Reason \[ \gamma = 1 + \frac{2}{f} \] This relation is also correct from kinetic theory, where \(f\) is degrees of freedom. However, it does not directly explain the assertion derived above. Thus both statements are true but the reason is not the correct explanation. \[ \boxed{\text{Option (2)}} \]