A substance for which ? is a constant undergoes an isothermal, mechanically reversible process from initial state (P1, V1) to final state (P2, V2), where V is molar volume.
(a) Starting with the definition of ?, show that the path of the process is described by: V = A (T) exp (??P )
(b) Determine an exact expression which gives the isothermal work done on 1 mol of this constant-? substance.
(a) If ? is constant, then we can simply rearrange its definition, 
(at constant T). This can then be integrated directly to get 
or solving for V, we get 
. This is of the form 
which is a function only of T, since it is for a particular value of V and P.
(b) As in the previous problem, we have dW = -PdV = ?PVdP. Substituting into this the expression for V obtained above, we have
We could simplify this by recognizing that 
so this is
for the work in going from (P1, V1) to (P, V).
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