An important concept is the characterization of thermodynamic quantities as either intensive or extensive.
Imagine two identical containers filled with the same kind and amount of of gas at the same temperature. What happens if we bring the two containers in contact and allow the gas to fill the combined containers as if it was one. Obviously temperature and pressure do not change. The quantitiesT andP therefore are called intensive.
Volume on the other hand is doubled. Mathematically this means V ∝ n. Such quantities are said to be extensive.nitself is therefore extensive. Thus far:
13In liquid water∂lnεr/∂P|o,T ≈5ã10−5/bar.
• intensive:T,P,. . .
• extensive:V,n,. . .
The ratio of two extensive quantities, e.g. n/V, again is intensive of course.
Another intensive quantity is the chemical potential,μ. Whether one mole of material is added to a large system or to twice as large a system should not matter.14 This however has implications for the free enthalpy,G. According to Eq. (2.107) we have for a one-component system
d G
T,P,... =μdn, (2.164)
where. . .stands for other intensive variables in addition toT andP. Becauseμis intensive anddnis extensive, we conclude thatd Gis extensive also. By adding (or integrating over) sufficiently many differential amounts of material,n =
dn, we find the important relation
G(T,P,n, . . . )=μn. (2.165) For aK-component system this becomes
G(T,P,n1, . . . ,nK, . . . )= K i=1
μini. (2.166)
Equatingd Gon the left withd(K
i=1μini)on the right, i.e.
−Sd T +V d P+ ã ã ã + K i=1
μidni = K i=1
(μidni +nidμi), (2.167)
yields
−Sd T +V d P+ ã ã ã − K
i=1
nidμi =0, (2.168)
the Gibbs-Duhem equation. The significance of this equation will become clear in many examples to come.
Remark 1:Suppose we consider the potential energyU of a system consisting of N pairwise interacting molecules. Disregarding their spatial arrangement we may writeU ∼(N2/2)V−1∞
a drr2−n. The factorN2≈N(N−1)/2 is the number of distinct pairs,Vis the volume,ais a certain minimum molecular separation, andr−n
14Momentarily we talk about one-component systems and not about mixtures.
is the leading distance dependence of the molecular interaction. Ifn>3 the integral is finite and consequently U/V ∝ ρ2, whereρ is the number density molecules.
This means thatU (and also E) is extensive by our above definition. However, if n ≤3 the situation is more complex. Now it is necessary to include the spatial and orientational correlations between the molecules. In general these conspire to yield an extensiveU. An exception is gravitation, where U/V ∼ ρ2V2/3including an additional shape dependence.
Remark 2:Looking at the two Eqs. (2.164) and (2.165) we may wonder whether one can apply the same argument to
d F
T,V,...=μdn,
valid according to Eq. (2.106). This immediately leads toF =G, which clearly is incorrect! The point is that we cannot keep the volume constant and simultaneously add up increments dn to the full n. Therefore this procedure does not work for d F
T,V,....
Example: Partial Molar Volume. Consider a binary liquid mixture (AandB) at constant temperature and pressure. The volume change due to a differential change of the composition is
d V= ∂V
∂nA
T,P,nB
dnA+ ∂V
∂nB
T,P,nA
dnB=vAdnA+vBdnB.(2.169) The quantitiesvAandvBare called partial molar volumes. In this senseμiis the partial molar free enthalpy of componenti.
Now we argue as in the case of Eq. (2.164). V is extensive and so isn. Therefore we may ad upd V’s to the fullV at constantT andPand thus
V =vAnA+vBnB. (2.170)
However, the quantity of interest here is not V but the volume difference upon mixing, V. The volumes of the pure substances are v∗ini, where vi∗=∂V/∂ni|T,P,nj=0(i=j)(i,j =1,2) and thus
V =(vA−v∗A)nA+(vB−v∗B)nB. (2.171) Notice thatvAandvBare not independent. To see this we simply must realize that we can carry out the steps from Eq. (2.166) to the Gibbs-Duhem equa- tion (2.168) withGreplaced byV andμi replaced byvi. At constantT and Pthis means
dvAnA+dvBnB=0. (2.172)
In addition we notice that this may immediately be extended to more than two components. And this is not the end, because the same reasoning applies to every extensive quantity=(T,P,n1,n2, . . . ), i.e.
=
K i
(φi−φ∗i)ni, (2.173)
and
K i=1
nidφi =0 (T,P=constant), (2.174) whereφi = ∂/∂ni|T,P,nj(i=j)are the respective partial molar quantities. Here stands for extensive thermodynamic quantities likeV,H,CP,. . ..
2.4.1 Homogeneity
Before leaving this subject we look at it briefly from another angle. In mathematics a function f(x1,x2, . . . ,xn)is said to be homogeneous of ordermif the following condition is fulfilled:
f(λx1, λx2, . . . , λxn)=λmf(x1,x2, . . . ,xn). (2.175) Thus we may consider the extensive quantities free energy and free enthalpy as first-order homogeneous functions inV,niandni respectively, i.e.
F(T, λV, λn1, λn2, . . . )=λF(T,V,n1,n2, . . . ) (2.176) G(T,P, λn1, λn2, . . . )=λG(T,P,n1,n2, . . . ). (2.177) Differentiating Eq. (2.176) on both sides with respect toλyields
d F
dλ = ∂F
∂(λV)
T,n1,n2,...V + K i=1
∂F
∂(λni)
T,V,nk(=i)ni =F. (2.178) Forλ=1 this becomes
F= ∂F
∂V
T,n1,n2,...V+ K
i=1
∂F
∂ni
T,V,nk(=i)ni(2.109),(=2.166)−P V +G (2.179)
in agreement with Eq. (2.114). This also implies μi = ∂F
∂ni
T,V,nk(=i), (2.180) i.e. the generalization of Eq. (2.110) to more than one component. Differentiating Eq. (2.177) on both sides with respect toλand settingλ=1 reproduces Eq. (2.166).
Clearly, we may apply the same idea to other extensive thermodynamic functions like S(E,V,n1,n2, . . .), i.e. λS(E,V,n1,n2, . . .) = S(λE, λV, λn1, λn2, . . .), E(T,V,n1,n2, . . .), i.e.λE(T,V,n1,n2, . . .)=E(T, λV, λn1, λn2, . . .), or others.
Likewise we may consider the intensive quantities as zero-order homogeneous functions in their extensive variables, e.g.
P(T,V,n1,n2, . . .)= P(T, λV, λn1, λn2, . . .). (2.181) Differentiating with respect toλon both sides and subsequently settingλ=1 yields
0=∂P
∂V
T,n1,n2,...V + K i=1
∂P
∂ni
T,V,nk(=i)ni. (2.182) UsingP = −∂F/∂V|T,n1,n2,...and changing the order of differentiation we find
∂G
∂V
T,n1,n2,...= − 1
κT. (2.183)
This easily is verified by insertion of Eq. (2.114) and subsequent differentiation.
The concept of homogeneity does not produce otherwise unattainable relations, but it is an elegant means to compute them. We revisit homogeneity in a generalized form on p. 140 in the context of continuous phase transitions, where again it proves useful.
Chapter 3
Equilibrium and Stability