 Solution

ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2017/paper-2/section-c/c10/state-temperatures/solution

For one mole,

p V = RT

, and constant heat capacities satisfy

C_{p} - C_{V} = R

. The specified diatomic-gas model has

γ = 1.4 = \frac{7}{5}, C_{V} = \frac{R}{γ - 1} = \frac{5}{2} R, C_{p} = \frac{7}{2} R .

A reversible adiabatic process satisfies

T V^{γ - 1} = constant

. Constant-pressure expansion satisfies

T \propto V

. Applying these in order gives

T_{2} T_{3} T_{4} T_{final} = T_{1} (\frac{V _{1}}{V _{2}})^{γ - 1}, = T_{2} \frac{V _{3}}{V _{2}} = T_{1} (\frac{V _{1}}{V _{2}})^{γ - 1} \frac{V _{3}}{V _{2}}, = T_{3} (\frac{V _{3}}{V _{1}})^{γ - 1} = T_{1} (\frac{V _{3}}{V _{2}})^{γ}, = T_{1} .

Every volume ratio is dimensionless, so the expressions have units of temperature. For the intended volume ordering, the minimum temperature is

T_{1}

and the maximum is

T_{3}

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