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

Table of contents
- Solution 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 .

Path (a) first changes temperature at pressure

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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