5 The isobaric ensembles
5.1 Why constant pressure?
Standard handbooks of thermodynamic data report numerical values of physical prop-
erties, including standard enthalpies, entropies and free energiesof formation, redox
potentials, equilibrium constants (such as acid ionization constants, solubility prod-
ucts, inhibition constants) and other such data, under conditionsof constant tem-
perature and pressure. This makes the isothermal-isobaric ensemble one of the most
important ensembles since it most closely reflects the conditions under which many
condensed-phase experiments are performed.
In order to maintain a fixed internal pressure, the volume of a system must be
allowed to fluctuate. We may therefore view an isobaric system as coupled to an
isotropic “piston” that compresses or expands the system uniformly in response to
instantaneous internal pressure fluctuations such that the average internal pressure
is equal to an external applied pressure. Remember that an instantaneous pressure
estimator is the total force exerted by the particles on the walls oftheir container, and
the average of this quantity gives the observable internal pressure. Coupling a system to
the piston leads to an ensemble known as theisoenthalpic-isobaricensemble, since the
enthalpy remains fixed as well as the pressure. Recall that the enthalpy isH=E+PV.
If the system also exchanges heat with a thermal reservoir, whichmaintains a fixed
temperatureT, then the system is described by theisothermal-isobaricensemble.
In this chapter, the basic thermodynamics of isobaric ensembles willbe derived by
performing a Legendre transformation on the volume starting withthe microcanonical
and canonical ensembles, respectively. The condition of a fluctuating volume will be
seen to affect the ensemble distribution function, which must be viewed as a function
of both the phase space vector x and the volumeV. Indeed, when considering how the
volume fluctuates in an isobaric ensemble, it is important to note thatboth isotropic
and anisotropic fluctuations are possible. Bulk liquids and gases in equilibrium only
support isotropic fluctuations. However, in any system that is notisotropic by nature,
anisotropic volume fluctuations are possible even if the applied external pressure is
isotropic. For solids, if one is interested in structural phase transitions under an ex-
ternal applied pressure or in mapping out the space of crystal structures of complex
molecular systems that exhibit polymorphism, it is often critical to include anisotropic
shape changes of the containing volume or supercell. Other examples that support
anisotropic volume changes include biological membranes, amorphous materials, and
interfaces, to name a few.
After developing the basic statistical mechanics of the isobaric ensembles, we will
see how the extended phase space techniques of the previous chapter can be adapted