An ideal gas has specific heat capacity at constant pressure $\frac{11}{10} R$. If one mole of this ideal…
An ideal gas has specific heat capacity at constant pressure $\frac{11}{10} R$. If one mole of this ideal gas at $125^{\circ} \mathrm{C}$ does $83 \mathrm{~J}$ of work adiabatically, then the final temperature of the gas would be (Universal gas constant, $R=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ )
$25^{\circ} \mathrm{C}$
$50^{\circ} \mathrm{C}$
$75^{\circ} \mathrm{C}$
$100^{\circ} \mathrm{C}$
Solution
Work done by gas in adiabatic process,
$\Delta W=+83 \mathrm{~J}$
As in adiabatic process, $\Delta Q=0$,
By first law of thermodynamics,
$\Delta Q=\Delta U+\Delta W$
so change in internal energy of gas,
$\Delta U=-83 \mathrm{~J}$
Also, $\Delta U=n C_V \Delta T=83$
Here, $n=1$ mole
$\Rightarrow$ specific heat at constant volume
$C_V=C_p-R$
$\Rightarrow \quad C_V=\frac{11}{10} R-R=\frac{1}{10} R$
$\begin{aligned} & \text { so, } \quad-83=1 \times \frac{1}{10} R \times\left(T_f-125\right) \\ & \Rightarrow T_f-125=\frac{-830}{8.3}=-100\end{aligned}$
or $\quad T_f=125-100=25^{\circ} \mathrm{C}$