Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 36E / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii
- iii b
  - Solution iii

i

Solution

The ideal gas free energy contributes

- N k_{B} T lo g V

. With

V = 4 π r^{3} /3

and surface energy

4 π t r^{2}

, the radius-dependent free energy is therefore

F (r) = - N k_{B} T lo g V + 4 π t r^{2} + constant .

For equilibrium against external pressure

P_{0}

, minimize

F + P_{0} V

.

ii

Solution

Differentiation gives

- \frac{3 N k _{B} T}{r} + 8 π t r + 4 π P_{0} r^{2} = 0,

equivalently the Laplace-pressure condition

P_{in} = \frac{N k _{B} T}{V} = P_{0} + \frac{2 t}{r} .

iii

Solution

Let

r_{0} = (3 N k_{B} T / (4 π P_{0}))^{1/3}

be the zero-tension radius. From

P_{0} r^{3} + 2 t r^{2} = P_{0} r_{0}^{3}

, put

r = r_{0} + δ

and retain first order:

3 P_{0} r_{0}^{2} δ + 2 t r_{0}^{2} = 0, r = r_{0} - \frac{2 t}{3 P _{0}} + O (t^{2}) .

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