Question

Three moles of an ideal gas being initially at a temperature $T_i=273$ K were isothermally expanded 5 times its initial volume and then isochorically heated so that the pressure in the final state become equal to that in the initial state. The total heat supplied in the process is 80 kJ. If $\gamma =\frac{C_p}{C_v}$ of the gas is $\frac{x}{10}$, find $x$.

Answer 1

For the process BC, $\Delta V=0$ thus $W_{BC}=0$

Process AB is isothermal, thus $\Delta U_{AB}=0$

$W_{AB}=nR{T}_{o}ln\frac{V_B}{V_A}=3\left(8.314\right)\left(273\right)ln5=10958.93J$

$\Delta U_{BC}=n{C}_v\Delta T=3C_v(5T_o-T_o)=3276C_v$

$Q=\Delta U+W$

$80000=3276C_v+10958.93$

$⟹C_v=21.07$

Also, $C_p-C_v=R=8.314⟹C_p=29.39$

Now $\frac{C_p}{C_v}=1.4$

Hence, $x=14$