 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/section-b/b3/b/solution

Within the stated small-volume-change approximation, treat

V, κ_{T}, α

in the differential constitutive law as constants. Then

δ W = V_{1} κ_{T} P d P - V_{1} α P d T .

P_{1}

, then changes pressure at fixed temperature

T_{2}

. The two contributions are

- V_{1} α P_{1} (T_{2} - T_{1})

and

V_{1} κ_{T} (P_{2}^{2} - P_{1}^{2}) /2

. Therefore

W_{a} = \frac{1}{2} V_{1} κ_{T} (P_{2}^{2} - P_{1}^{2}) - V_{1} α P_{1} (T_{2} - T_{1}) .

Work is path dependent even though the endpoints and approximate total volume change are the same.

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