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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 2 nd Law of Thermo-dynamics,  S univ 0 Entropy’s Change with Temperature Change in S During Chemical Reactions

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zumdahl s chapter 16

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

2nd Law of Thermo-dynamics, Suniv0

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 Contents
spontaneity
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
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
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 s univ
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 log e w boltzmann s headstone
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
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
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
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!
2 nd law of thermodynamics
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
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
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
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.
0 th law of thermodynamics
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 ch tlier confirmed
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
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
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
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 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
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
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
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
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 ch tlier s odds
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
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
g and k equilibrium constant
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
g and reaction progress
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
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
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
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!”