Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 47

At fixed particle number and pressure, the joint isothermal-isobaric ensemble weight for a microstate of energy $E$ and volume $V$ is $e^{-\beta (E+PV)}$. Write its enthalpy as $H=E+PV$. Differentiating the isothermal-isobaric partition function with a temperature-independent volume measure givesAfter division by $Q_P$ this is the normalized ensemble average, soThe differentiation assumes that the microscopic Hamiltonian and the integration measure have no additional explicit temperature dependence. An extra temperature-dependent normalization factor must be differentiated separately.

Differentiate the normalized average, including the derivative of its denominator. The enthalpy fluctuation formula follows fromSince $d\beta /dT=-1/(k_B{T}^2)$,Thus a larger equilibrium spread of enthalpy corresponds to a larger response to heating at fixed pressure. The variance is nonnegative, so this finite-system equilibrium description has $C_P\ge 0$. The relation connects a response coefficient to fluctuations of the appropriate exchanged quantity; at fixed volume the corresponding quantity is internal energy.

Away from singular regimes, assume enthalpy and heat capacity at constant pressure are extensive: $⟨H⟩=Nh$ and $C_P=N{c}_P$, with nonzero intensive $h$. The enthalpy fluctuation formula then givesAbsolute fluctuations grow as $\sqrt{N}$, while the mean grows as $N$. In the thermodynamic limit, enthalpy per particle therefore concentrates around its mean; it is relative, rather than absolute, fluctuations that vanish. This is the usual relative fluctuations of an extensive observable scaling. It assumes ordinary extensive heat capacity and a nonzero mean with a stated energy reference; a critical divergence or an artificially zero mean requires separate analysis.

The volume integral is a gamma function integral,For the prescribed bridge relation, $\beta P{Q}_P=(\beta P{\Lambda }^3{)}^{-N}$, so $G=N{k}_{B}T\ln (\beta P{\Lambda }^3)$. Since $\Lambda \propto T^{-1/2}$, this gives the macroscopic ideal-gas resultHeating at fixed pressure requires both an increase in internal energy and expansion work; at fixed volume the latter contribution is absent.

The finite-size normalization of the isothermal-isobaric ensemble matters here. Applying part (a) literally to the unprefactored $Q_P$ instead gives $⟨H{⟩}_P=(5N/2+1)k_{B}T$ and $C_P=(5N/2+1)k_B$. The two answers differ because $-{\partial }_{\beta }\ln (\beta P{Q}_P)=-{\partial }_{\beta }\ln Q_P-k_{B}T$. The normalized continuous-volume distribution has $⟨PV⟩=(N+1)k_{B}T$; its variance agrees with the unprefactored result in part (b). The bridge convention subtracts an order-one contribution and recovers the usual extensive thermodynamic answer. Both agree to leading order in the thermodynamic limit, but their exact finite-$N$ derivatives should not be mixed.