Zumdahls Chapter 16 Spontaneity Entropy Free Energy and

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 ¬ 2 nd Law of Thermodynamics, 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

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 loge. W (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 0 Law of Thermodynamics r r u ¬“In any spontaneous process, the s S + entropy of the Universe increases. ” – We must include consideration of a system’s environment to apply ys this law. s S condensing a gas implies a large • For example, = 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! v i n Su

Entropy Rules Everywhere ¬Photosynthesis makes few large molecules (CH 2 O)n from smaller ones (CO 2 & H 2 O). – 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 T 2 > T 1 , then q = q 1 = – q 2 > 0 S 1 = q / T 1 > 0 by more than S 2 = – q / T 2 < 0 And Suniv = S 1 + S 2 > 0 until T 2 = T 1. 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. ¬ 3 rd 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 d. S = q / T = (CP / T ) d. T • Even elements have non-zero S . • Enthalpy may be relative, but Entropy is Absolute.

Imperfect Crystals ¬Imagine the molecule NH 2 D where an H has been replaced by deuterium, i. e. , 2 H. ¬The deuteroammonia has the same crystal structure as regular NH 3, 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(3 Nav) = 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. , NH 2 D at 0 K has Smix = – R ln(1/3) – Since Xi = 1/3 for all 3 “kinds” of NH 2 D

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 2 nd 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 k. J / mol – Sfusion ~ 9. 3 J / mol K – Gfusion = Hfusion – T Sfusion = – 6. 11 k. J – 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 ¬C 2 H 4(g) + H 2(g) C 2 H 6(g) – We’re not sanguine about this since ngas < 0. – Indeed S = S (ethane) – S (ethene) – S (H 2) • S = (270) – (219) – (131) = – 120 J/mol K but… – H = Hf (ethane) – Hf (ethene) – Hf (H 2) • H = (– 84. 7) – (52) – (0) = – 137 k. J/mol and • G = (– 32. 9) – (68) – (0) = – 101 k. J/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 d. G = n. P R T l ¬d. E = q + w = Td. S – Pd. V • But H = E + PV so d. H = d. E + Pd. V + Vd. P ¬d. H = Td. S + Vd. P (used before with fixed P, so d. P=0) • But G = H – TS so d. G = d. H – Td. S – Sd. T ¬d. G = Vd. P – Sd. T or, at fixed T, d. G = Vd. P ¬G – G = d. G = Videald. P = RT P– 1 d. P ¬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 k. J/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(ln. K) = + ( H°/R)T – 2 d. T (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 = Vd. P – Sd. T + 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!”