Question

Six grams of hydrogen gas at a temperature of $273K$ was isothermally expanded to five times its initial volume and then isochorically heated so that the pressure in the final state becomes equal to that in the initial state. The total amount of heat absorbed by the gas during the entire process is:

• A. $79050J$
• B. $80000J$
• C. $7905J$
• D. $8000J$

Answer 1

The correct option is A $79050J$

The first process is an isothermal process, thus its internal energy is zero and heat absorbed is equal to the work done. Thus we get
$\Delta Q=nRT\ln \frac{V_2}{V_1}$
or
$\Delta Q=\frac{6}{2}\times 8.314\times 273\times \ln \left(\frac{5V_1}{V_1}\right)=10958.9J$.

Now
For the isothermal process we have $P_1{V}_1=P_2{V}_2$
Thus we get $P_1=5P_2$
also as $P_1=P_3$, thus $P_3=5P_2$
And for the isochoric process we have
$\frac{P_2}{T_2}=\frac{P_3}{T_3}$
or
$T_3=5T_2$
We know that $T_1=T_2=273K$
Thus we get $T_3=1365K$
Thus for isochoric $\Delta T=T_3-T_2=1092K$
Also hydrogen is a diatomic gas, its degree of freedom is 5. Thus its $C_v$ is $\frac{5}{2}R$
Thus we calculate the heat absorbed in the isochoric process as $n{C}_v\Delta T=3\times \frac{5}{2}R\times 1092=68091.66J$
Thus we get the total heat absorbed as $10958.9+68091.66\approx 79050J$
Thus the answer is A.