Question

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 $\Delta U = n C_v (T_f - T_i) = \frac{nR}{\gamma - 1}(T_f - T_i)$, where $\gamma = \frac{C_p}{C_v}, T_i = \text{initial temperature}, T_f = \text{final temperature}$.
Statement II: Relation between degree of freedom $f$ and $\gamma (= C_p/C_v)$ is $\gamma = 1 + \frac{2}{f}$}
Choose the correct answer from the options given below

• 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

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Thermodynamics and kinetic theory of gases.
We need to verify the validity of the internal energy formula for an ideal gas and investigate the link between the ratio of specific heats ($\gamma$) and degrees of freedom ($f$).
Step 2: Key Formula or Approach:
The change in internal energy is given by $\Delta U = n C_v \Delta T$.
Molar specific heat at constant volume is $C_v = \frac{f}{2}R$.
The ratio of specific heats is $\gamma = \frac{C_p}{C_v} = 1 + \frac{R}{C_v}$.
Step 3: Detailed Explanation:
For any ideal gas, the change in internal energy is defined as:
\[ \Delta U = n C_v (T_f - T_i) \]
From Mayer's relation $C_p - C_v = R$, we can write $\gamma = 1 + \frac{R}{C_v} \implies C_v = \frac{R}{\gamma - 1}$.
Substituting this into the internal energy equation gives:
\[ \Delta U = \frac{nR}{\gamma - 1}(T_f - T_i) \]
Thus, Statement I is true.
Now, for degrees of freedom $f$, the molar specific heat is $C_v = \frac{f}{2}R$.
Substitute this into the expression for $\gamma$:
\[ \gamma = 1 + \frac{R}{\frac{f}{2}R} = 1 + \frac{2}{f} \]
Thus, Statement II is true.
Since Statement II directly provides the relationship used to determine $C_v$ in Statement I, it acts as the correct explanation.
Step 4: Final Answer:
Both statements are true and Statement II is the correct explanation of Statement I.