Question 32

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

Let’s check both statements clearly.

Statement I

$$ΔU=nCᵥ(Tf−Ti)$$ is correct.

Also, using relation:

$$C_v=\frac{R}{\gamma-1}$$

so,

$$\Delta U=n\frac{R}{\gamma-1}(T_f-T_i)$$

So Statement I is correct.

Statement II

Relation:

$$\gamma=1+\frac{2}{f}$$

This is correct for an ideal gas (from kinetic theory).

Now relation between them

Statement II helps derive relations between Cₚ, Cᵥ, and γ, which is used in Statement I.

So II correctly explains I.

final answer:
Both statements are correct and II is the correct explanation of I

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