**Part D**

In a thermodynamic process, the work is done is:

$\overline{){\mathbf{W}}{\mathbf{=}}{{\mathbf{\int}}}_{{\mathbf{V}}_{\mathbf{i}}}^{{\mathbf{V}}_{\mathbf{f}}}{\mathbf{p}}{\mathbf{d}}{\mathbf{V}}}$

An ideal monatomic gas is contained in a cylinder with a movable piston so that the gas can do work on the outside world, and heat can be added or removed as necessary. The figure shows various paths that the gas might take in expanding from an initial state whose pressure, volume, and temperature are p_{0}, V_{0}, and T_{0} respectively. The gas expands to a state with final volume 4V_{0}. For some answers it will be convenient to generalize your results by using the variable R_{v} = V_{final}/V_{initial}, which is the ratio of final to initial volumes (equal to 4 for the expansions shown in the figure.)

The figure shows several possible paths of the system in the *pV* plane. Although there are an infinite number of paths possible, several of those shown are special because one of their state variables remains constant during the expansion. These have the following names:

*Adiabiatic*: No heat is added or removed during the expansion.*Isobaric*: The pressure remains constant during the expansion.*Isothermal*: The temperature remains constant during the expansion.

Part D

Graphically, the work along any path in the *pV* plot ____________.

a) is the area to the left of the curve from P_{0} to P_{final.}

b) is the area under the curve from V_{0} to V_{final}

c) requires knowledge of the temperature T(V)