concentration, i.e. the system undergoes phase separation. In
an ideal fluid, we have L = V = 0, and setting gL = gV requires
(1 x) A (T, p) + x B (T, p) = 0 , (2.417) where A/B (T, p) =
L A/B (T, p) V A/B (T, p). Expanding the chemical potential
about a
S(T, V, N) = NkB ln(V/N) + N(T ) , (2.370) since S V T,N = p
T V,N = NkB V . Note that in eqn. 2.370 we have divided V by
N before taking the logarithm. This is essential if the entropy is
to be an extensive function (see 2.7.5). One might think that
the
take T A < T B at some given fixed pressure23 . This means
L A (T A , p) = V A (T A , p) and L B (T B , p) = V B
(T B , p). What happens to the mixture? We begin by writing
22An emulsion is a mixture of two or more immiscible liquids.
23We assume the boi
can be seen in a number of systems. A popular example
involves mixtures of water and ouzo or other anise-based
liqueurs, such as arak and absinthe. Starting with the pure
liqueur (x = 1), and at a temperature below the coexistence
curve maximum, the conce
McL (T1 ) , (2.349) where cS and cL are the specific heats
of ice (solid) and water (liquid), respectively15, and is the
latent heat of melting per unit mass. The pond must give up this
much heat to the ice, hence the entropy of the pond,
discounting the
have that the enthalpy per particle for species i is ha = a T
a T p , (2.430) since H = G + TS and S = G T p . We find
ha = kBT 2 a (T ) , (2.431) and thus ln T p = P i a ha
kBT 2 = h kBT 2 , (2.432) where h is the enthalpy of the
reaction, which is the h
differential, and invoking the First Law, dE = T dS p dV + X a=1
a dNa , (2.360) we arrive at the relation S dT V dp + X a=1
Na da = 0 , (2.361) of which eqn. 2.147 is a generalization to
additional internal work variables. This says that the + 2
quantiti
. (2.422) 2.14. SOME CONCEPTS IN THERMOCHEMISTRY 93
When a > 0, the corresponding Aa is a product; when a < 0,
the corresponding Aa is a reactant. The bookkeeping of the
coefficients a which ensures conservation of each individual
species of atom in the r
(3.23) and dH dt = H t = L t . (3.24) Define the rank 2r
vector by its components, i = qi if 1 i r pir if r i
2r . (3.25) Then we may write Hamiltons equations compactly
as i = Jij H j , (3.26) where J = 0rr 1rr 1rr 0rr (3.27)
is a rank 2r matrix. Note t
2.27 shows projections of the p-v-T surface of a typical single
component substance onto the (T, v), (p, v), and (p, T ) planes.
Coexistence takes place along curves in the (p, T ) plane, but in
extended two-dimensional regions in the (T, v) and (p, v) pl
a (2.365) where j, j cfw_1, . . ., . This gives ( 1)
independent equations equations16. Thus, we can have phase
equilibrium among the phases of species over a region of
dimension dPE(, ) = 2 + ( 1) ( 1) = 2 + .
(2.366) Since dPE 0, we must have + 2. Thus,
movement of glaciers: as glaciers slide down a rocky slope, they
generate enormous pressure at obstacles12 Due to this
pressure, the story goes, the melting temperature decreases,
and the glacier melts around the obstacle, so it can flow past it,
after wh
(p, T ) , (2.355) we derive an equation for the slope of the
coexistence curve, the Clausius-Clapeyron relation. Note that
we have one equation in two unknowns (T, p), so the solution
set is a curve. For three phase coexistence, we have 1 (p, T ) =
2 (p,
cooled to a temperature T such that (x, T ) lies within the twophase region, the mixture phase separates into the the two end
components (x L , T ) and (x V, T ), which lie on opposite
sides of the boundary of the two-phase region, at the same
temperature
> 0) and minimal concentration xa for the reactants (a < 0).
This means that the equation REACTANTS PRODUCTS is
shifted to the right, i.e. the products are plentiful and the
reactants are scarce. When is small, the LHS is small and the
reaction is shifted
solvent. The entropy of mixing is then Smix = kB " N0 ln N0
N0 + N + X a=1 Na ln Na N0 + N # , (2.384) where N = P a=1
Na is the total number of solvent molecules, summed over all
species. We assume the soluton is weak, which means Na N
N0 . Expanding in
) plane. At low temperatures, below T = Tc = /2kB , there is
a first order phase transition at = 0. For T < Tc = /2kB and
= 0+, i.e. infinitesimally positive, the system is in the A-rich
phase, but for = 0, i.e. infinitesimally negative, it is B-rich.
Th
372.3 871 1750 N2 25.7 -210 200 -196 O2 13.9 -219 213 -183
H2O 334 0 2270 100 Table 2.4: Latent heats of fusion and
vaporization at p = 1 atm. Here we write 0 for to emphasize
that we are talking about a phase with no impurities present.
This equation pro
number of ways W(E, V, N) a system at fixed energy and volume
can arrange itself. One has S(E, V, N) = kB ln W(E, V, N) . (2.374)
Consider a system consisting of different species of particles.
Now let it be that for each species label a, Na particles of
= 1), in units of the interaction parameter . Dark red curve: T =
0.65 /kB > Tc ; green curve: T = /2kB = Tc ; blue curve: T = 0.40
/kB < Tc. We have chosen 0 A = 0.60 0.50 kBT and 0 B =
0.50 0. 50 kBT . Note that the free energy g(T, p, x) is not
convex
(y, X, ) = 1 (T, S, X) (, N, X) = 1 (T, S, y) (, N, y) = 1
(2.512) (y, X, S) (, N, S) = 1 (y, X, T ) (, N, T ) = 1 For
example, if we add (, N) to the mix, we should write (T, S, N)
(p, V, N) = (p, V, S) (, N, S) = (, N, V ) (T, S, V ) = 1 .
(2.513) If we
= 2450 J/g, hence with M = 18 we have = 44.1 kJ/mol.
Therefore, the heat produced by the reaction 2 H2 (g) + O2 (g)
2 H2O(l) is (H)rxn = 571.2 kJ / mol O2 . Since the
reaction produces two moles of water, we conclude that the
enthalpy of formation of liq
270 Br Br 193 H Br 366 C O 360 N Cl 200 Br I 178 H I
299 C = O 743 N Si 374 Br S 212 H S 338 C F 484 O O
146 I I 151 H P 322 C Cl 338 O = O 497 S S 264 H Si 318
C Br 276 O F 185 P P 172 C I 238 O Cl 203 Si Si 176
Table 2.6: Average bond enthalpies for som
i , V is an additional potential which leads to transitions, and
(Ei ) is the density of final states at energy Ei . The fact that Wij
0 means that if each Pi (t = 0) 0, then Pi (t) 0 for all t 0. To
see this, suppose that at some time t > 0 one of the
p
of H2O. The atomic masses of Na and Cl are 23.0 and 35.4,
respectively, hence the total ionic concentration in seawater
(neglecting everything but sodium and chlorine) is given by x =
2 35 23.0 + 35.4 -1000 18 0.022 . (2.401) 19We shall discuss
latent hea
quantities in which we are interested, the details of the initial
conditions are effectively forgotten over some microscopic time
scale , called the collision time, and over some microscopic
distance scale, , called the mean free path1 . The equilibrium
s
configuration has an additional entropy, Smix. where xa =
Na/Qa is the initial dimensionless density of species a. Now lets
remove all the partitions between the different species so that
each of the particles is free to explore all of the boxes. There
ar
saved to buy steel skates, but of course his father desperately
needs an operation because I am not making this up he fell
off a dike and lost his mind. The family has no other way to pay
for the doctor. What a story! At this point, I imagine the
suspense
kJ/mol Formula Name State kJ/mol Ag Silver crystal 0.0 NiSO4
Nickel sulfate crystal -872.9 C Graphite crystal 0.0 Al2O3
Aluminum oxide crystal -1657.7 C Diamond crystal 1.9 Ca3P2O8
Calcium phosphate gas -4120.8 O3 Ozone gas 142.7 HCN
Hydrogen cyanide liqu