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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 1

At constant volume, the <First law of thermodynamics> gives
$$
\delta Q=dE=C_V\,dT.
$$
At constant pressure,
$$
\delta Q=C_p\,dT=dE+p\,dV.
$$
Differentiating $pV=Nk_BT$ at fixed $p$ gives
$$
p\,dV=Nk_B\,dT.
$$
Consequently,
$$
C_p\,dT=(C_V+Nk_B)dT,
$$
and the <Mayer relation> is
$$
\boxed{C_p-C_V=Nk_B}.
$$
The <specific-heat ratio> is therefore
$$
\boxed{
\gamma=\frac{C_p}{C_V}
=1+\frac{Nk_B}{C_V}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 36C / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 36C / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 36C / b / Solution

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