76 Microcanonical ensemble
3.2 Basic thermodynamics, Boltzmann’s relation, and the
partition function of the microcanonical ensemble
We begin by considering a system ofNidentical particles in a container of volume
V with a fixed internal energyE. The variablesN,V, andE are all macroscopic
thermodynamic quantities referred to ascontrol variables. Control variables are simply
quantities that characterize the ensemble and that determine other thermodynamic
properties of the system. Different choices of these variables leadto different system
properties. In order to describe the thermodynamics of an ensemble of systems with
given values ofN,V, andE, we seek a unique state function of these variables.
We will now show that such a state function can be obtained from thefirst law of
thermodynamics, which relates the energyEof a system to a quantityQof heat
absorbed and an amount of workWdone on the system:
E=Q+W. (3.2.1)The derivation of the desired state function begins by examining howthe energy
changes if a small amount of heat dQis added to the system and a small amount
of work dW is done on the system. SinceEis a state function, this thermodynamic
transformation may be carried out along any path, and it is particularly useful to
consider a reversible path for which
dE= dQrev+ dWrev. (3.2.2)Note that sinceQandWare not state functions, it is necessary to characterize their
changes by the “rev” subscript. The amount of heat absorbed bythe system can be
related to the change in the entropy ∆Sof the system by
∆S=
∫
dQrev
T, dS=dQrev
T, (3.2.3)
whereTis the temperature of the system. Therefore, dQrev=TdS. Work done on the
system is measured in terms of the two control variablesVandN. LetP(V) be the
pressure of the system at the volumeV. Mechanical work can be done on the system
by compressing it from a volumeV 1 to a new volumeV 2 < V 1 :
W
(mech)
12 =−∫V 2
V 1P(V)dV, (3.2.4)where the minus sign indicates that work is positive in a compression. Asmall volume
change dV corresponds to an amount of work dWrev(mech)=−P(V)dV. Although we
will typically suppress the explicit volume dependence ofP onV and write simply,
dWrev(mech)=−PdV, it must be remembered thatPdepends not only onVbut also on
NandE. In addition to the mechanical work done by compressing a system,chemical
work can also be done on the system by increasing the number of particles. Letμ(N)
be the chemical potential of the system at particle number,N(μalso depends onV