Zumdahl’s Chapter 16

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, 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 • “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.

haos 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, 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 ~1.48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!

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 = – 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 • 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?

dG = RT lnP G’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) = 5.1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 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!”

Zumdahl’s Chapter 16