$\overline{){\mathbf{W}}{\mathbf{=}}{\mathbf{p}}{\mathbf{(}}{{\mathbf{V}}}_{{\mathbf{2}}}{\mathbf{-}}{{\mathbf{V}}}_{{\mathbf{1}}}{\mathbf{)}}}$

An ideal monoatomic 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.)

Calculate W_{A}, the work done by the gas as it expands along path A from V_{0} to V_{A} = R_{v}V_{0}.

Express W_{A} in terms of p_{0}, V_{0}, and R_{v}.

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