Question

# In a cylinder, 1.20 mol of an ideal monatomic gas, initially at $3.60 \times 10^{5} \mathrm{Pa}$ and 300 K, expands until its volume triples. Compute the work done by the gas if the expansion is (a) isothermal; (b) adiabatic; (c) isobaric. (d) Show each process in a pV-diagram. In which case is the absolute value of the work done by the gas greatest? Least? (e) In which case is the absolute value of the heat transfer greatest? Least? (f) In which case is the absolute value of the change in internal energy of the gas greatest? Least?

Solution

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## Given

We are given an ideal monatomic gas with a number of moles $n$ = 1.20 mol at initial pressure $p_{1} = 2.60 \times 10^{5} \,\text{Pa}$ and initial temperature $T_{1}$ = 300 K. The gas expands until its volume triples which means $V_{2} = 3 V_{1}$. We want to calculate the work done of the process if the expansion is (a) isothermal, (b) adiabatic, and (c) isobaric.

## Solution

(a) Isothermal process means the temperature is constant during the process where $\Delta T$ = 0 while the pressure and the volume are changed. So the work done for the isothermal process is given by

$W = \int_{V_{1}}^{V_{2}} p dV$

Where $p$ could be extracted from ideal gas law by $\left(p=\dfrac{nRT}{V}\right)$. Now plug this value of $p$ into equation (1) and take the constant term $nRT$ outside the integration and complete the integration where equation (1) will be

$\begin{gathered} W = nRT \int_{V_{1}}^{V_{2}} \dfrac{1}{V} dV \text{\int \left(\dfrac{1}{V}\right) = \ln V}\\ W = nRT \ln \left(\dfrac{V_{2}}{V_{1}} \right) \tag{1*} \end{gathered}$

Now we can plug our values for $n, R, T$ and $V_{2}$ into equation (1*) to get the work done during the isothermal process where $R$ is the gas constant and equals $8.314 \mathrm{~J/mol\cdot K}$ (See Appendix F) and $V_{2} = 3V_{1}$

\begin{aligned} W &= nRT \ln \left(\dfrac{V_{2}}{V_{1}} \right) \\ &= 1.20 \,\text{mol} \times 8.314 \mathrm{~J/mol\cdot K} \times 300 \,\text{K} \ln \left(\dfrac{3 \cancel V_{1}}{\cancel V_{1}} \right)\\ & = \boxed{3288 \,\text{J}} \end{aligned}

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