Zumdahl

1
Zumdahls Chapter 16

• Spontaneity, Entropy, Free Energy, and
• Why All Things Happen
• The Universe Becomes Less Predictable

2
Chapter Contents

• Spontaneous Process and Entropy, S
• 2nd Law of Thermo-dynamics, ?Suniv?0
• Entropys Change with Temperature
• Change in S During Chemical Reactions

• Free Energy, G, Chemical Reactions
• Gs Dependence on Pressure
• Pointing the Way to Equilibrium
• Gs Relation to K
• Non-PV Work G

3
Spontaneity

• Sponte is Latin for voluntarily.
• Were 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?

4
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 slides question is one Thermo can
  address, the questions above lie in the realm of
  later chemical topics, viz., Kinetics and
  Dynamics.

5
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 serpents name is

6
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.
• Its called entropy, S, and is a state function
  that must always increase for the Universe as a
  whole, but some Systems S may decrease.
• It is a (logarithmic) measure of the combinations
  of wave functions available to the Universe!

7
S k logeW (Boltzmanns 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 players possession?
• W ? (52)(51)(50)(49)(48) / 5! or 2,598,960.
• For 4 players, thats 1.48?1024 different games.
• Over twice Avogadros Number!

8
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 5051 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.

9
Chemical Microstates

• Positional
• In a solid, molecules are frozen in position.
• But a liquid can swap molecular positions without
  macroscopic consequence Sliq gt Ssolid
• A gas is far more chaotic Sgas gtgt Sliquid!
• Therefore, its a safe bet that if ?ngas gt 0 for
  a reaction, so is ?S.
• And, of course, ?ngas lt 0 makes ?S negative.

10
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 gt 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!

11
2nd Law of Thermodynamics

• In any spontaneous process, the entropy of the
  Universe increases.
• We must include consideration of a systems
  environment to apply this law.
• For example, condensing a gas implies a large
  decrease in the systems entropy! ?Ssys ltlt 0
• Fortunately, the (latent) heat of vaporization
  gets released to force the surroundings to occupy
  higher energy levels, so ?Ssurr gtgt 0 and ?Suniv gt
  0!

?Suniv ?Ssys ?Ssurr ? 0
12
Entropy Rules Everywhere

• Photosynthesis makes few large molecules (CH2O)n
  from smaller ones (CO2 H2O).
• So definitely ?Ssys lt 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 gt 0

13
Perhaps even where it shouldnt

• 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.

14
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.

15
0th 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 gt T1 , then q q1 q2 gt 0
• ?S1 q / T1 gt 0 by more than ?S2 q / T2 lt 0
• And ?Suniv ?S1 ?S2 gt 0 until T2 T1.
• Whereupon ?Suniv 0 and q stops flowing.

16
Le Châtlier Confirmed!

• Suppose a reaction has an exothermicity of ?H .
  Then a qsurr ?H gt 0
• And ?Ssurr qsurr / T gt 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!
17
S, an Extensive State Function

• ?Srxn? ? np Sproducts? ? nr Sreactants?
• where ?s seem to be missing on the right side!
• This version of Hesss 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.

18
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
• Thats per molecule. Per mole WNav instead.
• ln(3Nav) NAv ln 3, so S(0 K) 1.099 R

19
Perfect Solutions

• Assuming no molecular interactions differ between
  pure solutions, they mix perfectly.
• The Entropy of Mixing quantifies Natures 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

20
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!
• Gibbs Free Energy, G ? H TS

21
Spontaneity and Equilibrium

• ?G lt 0 betokens a spontaneous process since it
  means that T ?Suniv gt 0.
• ?G gt 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.

22
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, thats 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.

23
Hydrogenation of Ethene

• C2H4(g) H2(g) ? C2H6(g)
• Were not sanguine about this since ?ngas lt 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 lt 0
• So reaction is spontaneous at std. conditions.

24
Improving Le Châtliers Odds

• Since ?H? lt 0, we dont want to heat the
  reaction, or wed reduce spontaneity.
• We would expect ?G? to be increased.
• But since ?ngas lt 0, we do want to apply
  additional pressure to drive it to products.
• Wed expect ?G? to become more negative.
• So what was that again about Gs pressure
  dependence?

25
Gs 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
  dP0)
• But G H TS so dG dH TdS SdT
• dG VdP SdT or, at fixed T, dG VdP
• G G? ? dG ? VidealdP RT ? P1dP
• G G? RT ln(P / P?) RT ln P

26
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

27
G and Reaction Progress, ?
G
G minimizes at equilibrium. ?G0 for any small
variation there.
??
0 (pure reactants)
1 (pure products)
28
Equilibrium Constant

• K e ?G / RT is that relations inverse.
• For the hydrogenation, ?G 101 kJ/mol
• K e101,000 J / 8.314 J/K (298 K) 5.1?1017
• well and truly spontaneous!
• Remember, while K is clearly dependent upon T, it
  is independent of Ptotal. Its the partial Ps
  that adjust to render ?G 0.

29
Ks 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.
• Thats if H and S are weak functions of T
  themselves. True if we dont change T much.
• d(lnK) (?H/R)T 2 dT (vant 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.

30
Maximizing Work

• ?G ?VdP ?SdT wnon-PV
• Weve been ignoring the non-PV work all this
  time, but its 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!