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

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

At equal pressure and temperature, ideal gases have equal molecular number density. The mean air molecular mass is

.8 (28) + .2 (32) = 28.8

, while helium has mass

4

, so

ρ_{He} / ρ_{air} = 4/28.8 = 5/36.

ii

Solution

Monatomic helium has molar

C_{p} = 5 R /2

and diatomic air has

7 R /2

. Dividing by molar masses,

\frac{c _{p}^{He}}{c _{p}^{air}} = \frac{( 5/2 ) R /4}{( 7/2 ) R /28.8} = \frac{36}{7} .

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