Zumdahl’s Chapter 16
Spontaneity, Entropy, Free Energy, and
Why All Things Happen …
“The Universe Becomes Less Predictable”
Chapter Contents
 Spontaneous Process
and Entropy, S
 “Free Energy”, G, &
Chemical Reactions
 2nd Law of Thermo-
 G’s Dependence on
dynamics, Suniv0
 Entropy’s Change
with Temperature
 Change in S During
Chemical Reactions
Pressure
 Pointing the Way to
Equilibrium
 G’s Relation to K
 Non-PV Work & G
Spontaneity
• “Sponte” is Latin for “voluntarily.”
We’re willing to concede that highly
exothermic reactions are spontaneous.
– While the First Law assures that the enthalpy
released could be used to resurrect reactants,
we know from experience that hot things cool
off, and disperse q to the environment, so that
it is unavailable to reverse the reaction.
– But why do some endothermic reactions go?
Punctuality
For that matter, why do some highly
exothermic reactions hesitate, requiring a
kick start, to do their spontaneous thing?
– Or proceed lethargically once started?
While last slide’s question is one Thermo
can address, the questions above lie in the
realm of later chemical topics, viz.,
Kinetics and Dynamics.
Norse Mythology
Valhalla is the abode of the Norse gods.
– But, contrary to many other mythologies,
Norse gods are not immortal.
– Valhalla is held up by a giant tree, the roots of
which are being gnawed by a serpent.
– The serpent will succeed, and when it does,
Valhalla and the Universe will fall.
The serpent’s name is
haos
Universal Chaos, Suniv
The Norsemen were right!
– There is Chaos growing in the Universe all the
time at the expense of Order. It is now a
fundamental principle of Science.
– It’s called “entropy,” S, and is a state function
that must always increase for the Universe as a
whole, but some System’s S may decrease.
– It is a (logarithmic) measure of the combinations
of wave functions available to the Universe!
S = k logeW (Boltzmann’s Headstone!)
S = k ln W in modern symbolism.
– W is an actual count of how many different
ways the Universe could be arranged without
being detectably different macroscopically.
• And it is usually enormous!
• For example, how many different poker hands
might be in some player’s possession?
• W  (52)(51)(50)(49)(48) / 5! or 2,598,960.
• For 4 players, that’s ~1.481024 different games.
• Over twice Avogadro’s Number!
Poker Microstates
One microstate in poker might be a flush;
all cards of the same suit.
– Wflush = 4(13)(12)(11)(10)(9) / 5! = 5148 as the
number of ways to get a flush on the deal.
– But Wflush / Wtotal gives ~505:1 odds against.
– So flushes-on-the-deal are fairly ignorable.
In k ln W, the most likely microstate is
used to calculate W*. It overwhelms others.
Chemical Microstates
Positional
– In a solid, molecules are frozen in position.
– But a liquid can swap molecular positions
without macroscopic consequence: Sliq > Ssolid
– A gas is far more chaotic: Sgas >> Sliquid!
– Therefore, it’s a safe bet that if ngas > 0 for a
reaction, so is S.
– And, of course, ngas < 0 makes S negative.
Structure and Microstates
Since the more modes of motion in a
molecule, the more places it can hide
energy (higher heat capacity), larger
molecules have higher S than smaller ones.
Still, decomposition reactions have S > 0!
– Although the products have to be smaller
molecules, there are more of them, so Nature
can fool you as to where the atoms are!
nd
2
Law of Thermodynamics
“In any spontaneous process, the
entropy of the Universe increases.”
– We must include consideration of a system’s
environment to apply this law.
• For example, condensing a gas implies a large
decrease in the system’s entropy! Ssys << 0
• Fortunately, the (latent) heat of vaporization gets
released to force the surroundings to occupy higher
energy levels, so Ssurr >> 0 and Suniv > 0!
Entropy Rules Everywhere
Photosynthesis makes few large molecules
(CH2O)n from smaller ones (CO2 & H2O).
– So definitely Ssys < 0
– But the absorption of light releases heat into
the environment. More importantly …
– It then casts many long IR photons into the
universe having absorbed fewer short VIS.
– So even growth of Life makes Suniv > 0
Perhaps even where it shouldn’t
Over a century ago, Darwin published The
Origin of Species and coined “the survival
of the fittest.” (…condemning us to Reality TV)
– Social Darwinism used that to excuse all the
excesses of predatory Capitalism.
Economists are turning to Ilya Prigogine.
– His notion that processes win that make S
grow most quickly is ripe for similar abuse.
Entropy and Temperature
– Increased heat, q, should correlate with S since
it makes available high energy states.
– But the chaos of q makes S more impressive
if initial states are more ordered ( lower T ).
And S = q / T codifies both notions. (units?)
At constant P, S = H / T
if only q happens.
– So Ssurr = – Hsys / T since exothermicity
flows into the surroundings.
th
0
Law of Thermodynamics
“If two system are in equilibrium with a
third, they are in equilibrium with one
another.”
– Take T as a measure; we presume 2 or more
systems in contact come to the same Tequil.
•
•
•
•
If T2 > T1 , then q = q1 = – q2 > 0
S1 = q / T1 > 0 by more than S2 = – q / T2 < 0
And Suniv = S1 + S2 > 0 until T2 = T1.
Whereupon Suniv = 0 and q stops flowing.
Le Châtlier Confirmed!
Suppose a reaction has an exothermicity of
H . Then a qsurr = – H > 0
And Ssurr = qsurr / T > 0 aids spontaneity.
Le Châtlier claims that higher T makes
such a reaction less spontaneous!
 Assuming q varies insignificantly with T (true), then
higher T makes Ssurr a smaller value!
Le Châtlier Confirmed!
S, an Extensive State Function
Srxn =  np Sproducts –  nr Sreactants
• where ’s seem to be missing on the right side!
– This version of Hess’s Law is correct for S.
3rd Law: S for perfect crystal at 0 K is 0.
– W = 1 since all atoms frozen in fixed places!
–  S   0 since we can warm solids up from 0
to 298 K via dS = q / T = (CP / T ) dT
• Even elements have non-zero S .
• Enthalpy may be relative, but Entropy is Absolute.
Imperfect Crystals
Imagine the molecule NH2D where an H
has been replaced by deuterium, i.e., 2H.
The deuteroammonia has the same crystal
structure as regular NH3, but each D can be
in one of three possible places at random.
S(0 K) = k ln W = k ln(3) = 1.099 k
– That’s per molecule. Per mole: WNav instead.
– ln(3Nav) = NAv ln 3, so S(0 K) = 1.099 R
Perfect Solutions
– Assuming no molecular interactions differ
between pure solutions, they mix perfectly.
The Entropy of Mixing quantifies Nature’s
need to scramble stuff to confuse you:
Smix = – R  Xi ln Xi (mole fractions)
– which is entirely consistent with R ln W
– E.g., NH2D at 0 K has Smix = – R ln(1/3)
– Since Xi = 1/3 for all 3 “kinds” of NH2D
Hiding the Surroundings
Since Ssurr = – Hsys / T, and
Suniv = Ssys + Ssurr  0, and therefore
T Suniv = T Ssys + T Ssurr  0, then
T Ssys – Hsys  0 is also the 2nd Law.
Hsys – T Ssys  0 is too.
Gsys  Hsys – T Ssys  0 is our choice!
Gibb’s Free Energy, G  H – TS
Spontaneity and Equilibrium
G < 0 betokens a spontaneous process
since it means that T Suniv > 0.
G > 0 means that the reverse process is
the spontaneous one!
But G = 0 means neither the process nor
its reverse is spontaneous. So
G = 0 means EQUILIBRIUM.
Freezing Point of Mercury
Hg(solid)  Hg(liquid)
– Hfusion ~ 2.16 kJ / mol
– Sfusion ~ 9.3 J / mol K
– Gfusion = Hfusion – TSfusion = – 6.11 kJ
– OK, that’s spontaneous; Hg should be liquid at 298 K.
– Tfusion  Hfusion / Sfusion since Gfusion = 0
– Tfusion ~ Hfusion / Sfusion = 232 K = – 41ºC
– The actual Tfusion = – 39ºC so H and S are T-dependent.
Hydrogenation of Ethene
C2H4(g) + H2(g)  C2H6(g)
– We’re not sanguine about this since ngas < 0.
– Indeed S = S(ethane) – S(ethene) – S(H2)
• S = (270) – (219) – (131) = – 120 J/mol K but…
– H = Hf(ethane) – Hf(ethene) – Hf(H2)
• H = (– 84.7) – (52) – (0) = – 137 kJ/mol and
• G = (– 32.9) – (68) – (0) = – 101 kJ/mol < 0
– So reaction is spontaneous at std. conditions.
Improving Le Châtlier’s Odds
Since H < 0, we don’t want to heat the
reaction, or we’d reduce spontaneity.
– We would expect G to be increased.
But since ngas < 0, we do want to apply
additional pressure to drive it to products.
– We’d expect G to become more negative.
So what was that again about G’s pressure
dependence?
G’s Pressure Dependence
dE = q + w = TdS – PdV
• But H = E + PV so dH = dE + PdV + VdP
dH = TdS + VdP
(used before with fixed P, so dP=0)
• But G = H – TS so dG = dH – TdS – SdT
dG = VdP – SdT or, at fixed T, dG = VdP
G – G =  dG =  VidealdP = RT  P–1dP
G – G = RT ln(P / P) = RT
ln P
G and K (equilibrium constant)
G – G° =  n Gproducts –  m Greactants
G – G° = RT [  n ln Pp –  m ln Pr ]
(G – G°) / RT =  ln Ppn –  ln Prm ]
(G – G°) / RT = ln Ppn – ln Prm
(G – G°) / RT = ln (Ppn / Prm) = ln Q
– But Q  K when G  0 so
+ G° = – RT ln K
Mass Action
Quotient
G and Reaction Progress, 
G
G minimizes at equilibrium.
G=0 for any small variation there.
G°
equilibrium
0
(pure reactants)

1
(pure products)
Equilibrium Constant
K = e –G° / RT is that relation’s inverse.
For the hydrogenation, G° = – 101 kJ/mol
K = e+101,000 J / 8.314 J/K (298 K) = 5.110+17
– well and truly spontaneous!
Remember, while K is clearly dependent
upon T, it is independent of Ptotal. It’s the
partial Ps that adjust to render G = 0.
K’s Temperature Dependence
ln K = – G°/RT = – H°/RT + S°/R
ln K = – (H°/R)T –1 + (S°/R)
– We expect a plot of ln K vs. 1/T to be ~ linear.
• That’s if H and S are weak functions of T
themselves. True if we don’t change T much.
d(lnK) = + (H°/R)T –2 dT
(van’t Hoff)
• It says that ln K increases with T when the reaction
is endothermic; decreases otherwise. – Le Châtlier!
• But the increase becomes less impressive at high T.
Maximizing Work
G = VdP – SdT + wnon-PV
• We’ve been ignoring the non-PV work all this
time, but it’s really been there in E, H, and G.
– Here it means that at fixed P & T, the first two
terms vanish, and G = wnon-PV, the maximum
(non-PV) work of which the system is capable.
• If you want maximum total w, the physicists need
to tell you about A. (A = E – TS, the “work
function.”) In either case, we must be so gentle as
to be at equilibrium all the time; “reversible work!”