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### Zumdahl’s Chapter 16

Spontaneity, Entropy, Free Energy, and

Why All Things Happen …

“The Universe Becomes Less Predictable”

Spontaneous Process and Entropy, S

2nd Law of Thermo-dynamics, Suniv0

Entropy’s Change with Temperature

Change in S During Chemical Reactions

“Free Energy”, G, & Chemical Reactions

G’s Dependence on Pressure

Pointing the Way to Equilibrium

G’s Relation to K

Non-PV Work & G

Chapter ContentsSpontaneity

- “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.

haosNorse 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, Valhallaand the Universe will fall.
- The serpent’s name is

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 Smay 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 ~1.481024 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!

2nd 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!

Suniv = Ssys + Ssurr 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 Smore impressiveif initial states are more ordered ( lower T ).
- And S = q/T codifies both notions. (units?)
- At constant P, S = H/Tif only q happens.
- So Ssurr = –Hsys/T since exothermicity flows into the surroundings.

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>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 higherT makes Ssurr a smaller value!

Le Châtlier Confirmed!

S, an Extensive State Function

- Srxn= npSproducts– nrSreactants
- 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 = – RXi ln Xi(mole fractions)
- which isentirely 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–TSsys 0 is too.
- Gsys Hsys–TSsys 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 – TSfusion= – 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?

dG = RT lnPG’s Pressure Dependence

- dE=q + w=TdS–PdV
- But H=E + PV so dH= dE + PdV + VdP
- dH=TdS + VdP (used before with fixedP, so dP=0)
- But G=H–TS so dG= dH–TdS–SdT
- dG=VdP–SdTor, at fixed T, dG=VdP
- G–G= dG= VidealdP=RT P–1dP
- G–G=RT ln(P/P) =RT ln P

Mass Action

Quotient

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

equilibrium

equilibrium

G°

G and Reaction Progress, G

G minimizes at equilibrium.

G=0 for any small variation there.

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) = 5.110+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 fixedP & 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!”

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