Question

Sol. [Key Concept: A diatomic gas with rotational modes has $C_{v} = \frac{5}{2}\ R$; at constant pressure $Q = {nC}_{p}\Delta\ T$ with $C_{p} = \frac{7}{2}\ R$, giving about $25$ J.]

I $\ ^{\text{st}\text{~}}$ Method

$${Q = \Delta U + W }{Q = \frac{f}{2}nR\Delta T + nR\Delta T = \left( \frac{f}{2} + 1 \right)nR\Delta T }$$if $f = 2 \Rightarrow Q = 2nR\Delta T \Rightarrow 2 \times 1 \times 8.3 \times 1.2 = 19.92$

(No option is matching)

if $f = 5 \Rightarrow Q = 3.5nR\Delta T$

$$\Rightarrow 3.5 \times 1 \times 8.3 \times 1.2 = 34.86 $$(No option is matching)

Method-2

$${Q = \Delta U + W }{Q = \frac{f}{2}nR\Delta T + P\Delta V }$$if $f = 2$

$${\Rightarrow Q = nR\Delta T + P\Delta V }{= 1 \times 8.3 \times 1.2 + 10^{5} \times 25 \times 10^{- 3} \times 4 \times 10^{- 4} }{= 10.96 }$$(No option is matching)

if $f = 5$

$${\Rightarrow Q = 2.5nR\Delta T + P\Delta V }{2.5 \times 1 \times 8.3 \times 1.2 + 10^{5} \times 25 \times 10^{- 3} \times 4 \times 10^{- 4} }{= 25.9 }$$(closest is 25 )

Note: As $P\Delta V \neq nR\Delta T$, given state is inconsistent

Hence we can opt the most suitable option as (2).