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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

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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, 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!”


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