 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/17/a/solution

Treat

R

and

a

as constants and use the total differential in the independent variables

n, V, T

d p = (\frac{RT}{V} - \frac{2 na}{V ^{2}}) d n + (- \frac{n RT}{V ^{2}} + \frac{2 n ^{2} a}{V ^{3}}) d V + \frac{n R}{V} d T .

Each coefficient differentiates with respect to its displayed variable while holding the other two fixed; the formula applies where

V \neq = 0

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