Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Statement I: Change in internal energy of a system containing \(n\) mole of ideal gas can be written as \(\Delta U = n C_v (T_f - T_i) = \dfrac{nR}{\gamma - 1}(T_f - T_i)\), where \(\gamma = \dfrac{C_p}{C_v}\), \(T_i =\) initial temperature, \(T_f =\) final temperature.
Statement II: Relation between degree of freedom \(f\) and \(\gamma (= C_p/C_v)\) is \(\left(\gamma = 1 + \dfrac{2}{f}\right)\)
Choose the correct answer from the options given below

Statement I is true because for an ideal gas, the change in internal energy is given by \(\Delta U = n C_v \Delta T\). Using Mayer's relation \(C_p - C_v = R\) and the ratio of specific heats \(\gamma = \dfrac{C_p}{C_v}\), we can write \(C_v = \dfrac{R}{\gamma - 1}\). Substituting this, we get \(\Delta U = \dfrac{nR}{\gamma - 1}(T_f - T_i)\).

Statement II is true because according to the law of equipartition of energy, the molar heat capacity at constant volume is \(C_v = \dfrac{f}{2}R\) and at constant pressure is \(C_p = \left(\dfrac{f}{2} + 1\right)R\). Therefore, \(\gamma = \dfrac{C_p}{C_v} = 1 + \dfrac{2}{f}\).

Statement II does not explain Statement I, as Statement I is derived purely from macroscopic thermodynamic relations (Mayer's relation) and is independent of the microscopic concept of degrees of freedom.

Hence, both statements are true but Statement II is NOT the correct explanation of Statement I.

Answer: Both A and R are true but R is NOT the correct explanation of A