A three$-$step process is used to convert heat into work. Three moles of methane at $P_1=400$kPa

and $T_1=27C$ in a piston are heated to $T_2=177C$ with no change in volume. Work is then

obtained as the piston expands isothermally back to $400$kPa of pressure. In a final step, the piston

is returned to its initial state via a constant pressure process involving a combination of work and$/$or

heat being added or removed. Assume that all steps involving work being added or removed from

the gas occur by mechanically reversible processes and that methane behaves as an ideal gas

throughout the process. Heat capacity data for methane as an ideal gas may be found in Appendix

A$\mathrm{.}2\mathrm{.}1$ in your text. Note that the expression for $\frac{C_p}{R}$ is dimensionless and must be multiplied by $R$

to obtain $C_p.$ Also, a value of $"$B x ${10}^{33}"$ equaling $9\mathrm{.}081$ means that $B=9\mathrm{.}081{10}^{-3}.$

i$)$ Sketch the process on a P$-$v set of axes. Start by drawing isotherms for $27$ and $177C.$

Then, identify the starting and ending points for each step in the process and draw the paths

for the three steps. Note values of pressure on your y$-$axis for each end point in the process.

ii$)$ Calculate the heat and work flows $($in $J)$ for the first step.

iii$)$ Calculate the heat and work flows $($in J$)$ for the second step.

iv$)$ Calculate the heat and work flows $($in $J)$ for the third step.

v$)$ Determine the net work $($in $J)$ obtained from the process.