Starting at temperature \(300\; \mathrm{K},\) one mole of an ideal diatomic gas \((\gamma=1.4)\) is first compressed adiabatically from volume \(\mathrm{V}_{1}\) to \(\mathrm{V}_{2}=\frac{\mathrm{V}_{1}}{16} .\) It is then allowed to expand isobarically to volume \(2 \mathrm{V}_{2} \cdot\) If all the processes are the quasi-static then the final temperature of the gas (in \(\left. \mathrm{K}\right)\) is (to the nearest integer)

\(\mathrm{TV} ^{\gamma-1}=\mathrm{C}\)

\(300 \times \mathrm{V}^{\frac{7}{5}-1}=\mathrm{T}_{2}\left(\frac{\mathrm{V}}{16}\right)^{\frac{7}{5}-1}\)

\(300 \times 2^{4 \times \frac{2}{5}}=\mathrm{T}_{2}\)

Isobaric process

\(\mathrm{V}=\frac{\mathrm{nRT}}{\mathrm{P}}\)

\(\mathrm{V}_{2}=\mathrm{kT}_{2}\)

\(2 \mathrm{V}_{2}=\mathrm{KT}_{\mathrm{f}}\)

\(\frac{1}{2}=\frac{\mathrm{T}_{2}}{\mathrm{T}_{\mathrm{f}}} \Rightarrow \mathrm{T}_{\mathrm{f}}=2 \mathrm{T}_{2}\)

\(\mathrm{T}_{\mathrm{f}}=2 \times 300 \times 2^{\frac{8}{5}}=1818.85\)