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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 317

Assume the material gas is monatomic. Its specific enthalpy together with the radiation enthalpy is
$$
h=\frac52\mathcal RT+\frac{4a_{\rm rad}T^4}{3\rho}
=\mathcal RT\left(\frac4\beta-\frac32\right).
$$
At constant total pressure, logarithmic differentiation of
$P=\mathcal R\rho T+a_{\rm rad}T^4/3$ gives
$$
\frac{d\beta}{dT}\bigg|_P=-\frac{4(1-\beta)}T.
$$
Consequently the <specific heat capacity at constant pressure> is
$$
\boxed{
c_P=\left(\frac{\partial h}{\partial T}\right)_P
=\frac{\mathcal R}{2\beta^2}
(32-24\beta-3\beta^2)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 3 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 3 / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 3 / i / Solution

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