Question

One mole of ${{O}_{2}}$ gas having a volume equal to 22.4 litres at ${{0}^{\circ }}C$ and 1 atmospheric pressure is compressed isothermally so that its volume reduces to 11.2 litres. The work done in this process is then
A) $\text{1672}\text{.5 J}$
B) $\text{1728 J}$
C) $-1728\text{ J}$
D) $-1572.5\text{ J}$

Answer 1

Since it is mentioned that the process is isothermal, therefore temperature remains constant, i.e. ${{0}^{\circ }}C$. So, work done for an isothermal process is given as: $W=nRT\ln \left( \dfrac{{{V}_{2}}}{{{V}_{1}}} \right)$ where W is work done, n is number of moles, R is universal gas constant, T is the constant temperature, ${{V}_{1}}$ and ${{V}_{2}}$ are initial and final volume.

Complete step by step answer:
Since we have:
$\begin{aligned} & T={{0}^{\circ }}C \\\ & n=1 \\\ & P=1atm \\\ & {{V}_{1}}=22.4l \\\ & {{V}_{2}}=11.2l \\\ \end{aligned}$
So, for isothermal process, by using the formula: $W=nRT\ln \left( \dfrac{{{V}_{2}}}{{{V}_{1}}} \right)$
We get:

& W=1\times 8.314\times 273\ln \left( \dfrac{11.2}{22.4} \right) \\\ & =8.314\times 273\ln \left( \dfrac{1}{2} \right) \\\ & =-1572.5J \end{aligned}$$ **So, the correct answer is “Option D”.** **Additional Information:** An isothermal process is a thermodynamic process in which the temperature of a system remains constant. The transfer of heat into or out of the system happens so slowly that thermal equilibrium is maintained. The PV graph of an isothermal process is a rectangular hyperbola. Different examples of isothermal process are, 1\. The Carnot engine is an industrial example of isothermal process. 2\. Refrigerator also works isothermally as the temperature inside the refrigerator remains constant through a set of changes taking place in the mechanism. 3\. Heat pump is also an example of isothermal process as the heat is removed from the house and dumped outside or vice versa. **Note:** Remember that we have temperature in terms of $^\circ {C}$. The PV graph of an isothermal process is a rectangular hyperbola. So, firstly convert the temperature in terms of degree Kelvin and then solve the question.