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CHAPTER 14 Entropy and Free Energy

CHAPTER 14 Entropy and Free Energy. Review of Thermodynamics (Chapter 10) The following are general concepts from thermodynamics as applied to chemical systems. 1) First law of thermodynamics: U = q + w w = work ( = - pV for mechanical work for a system at constant pressure)

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CHAPTER 14 Entropy and Free Energy

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  1. CHAPTER 14 Entropy and Free Energy

  2. Review of Thermodynamics (Chapter 10) The following are general concepts from thermodynamics as applied to chemical systems. 1) First law of thermodynamics: U = q + w w = work ( = - pV for mechanical work for a system at constant pressure) q = heat U = Uf - Ui = change in internal energy Note that U is a state function, while q and w are not.

  3. Sign convention w > 0 work done on system w < 0 work done by system (volume decreases) (volume increases) q > 0 heat flows into system q < 0 heat flows out of system (endothermic) (exothermic) 2) Enthalpy and heat H (enthalpy) is defined as H = U + pV (and so a state function) At constant volume U = qV At constant pressure H = qp

  4. 3) Calculating the enthalpy change for a chemical reaction Hrxn = [  Hf(products) ] - [  Hf(reactants) ] The above is in terms of the enthalpy of formation. Recall that by definition the formation reaction is the reaction that produces one mole of a single product out of elements in their standard (most stable) state. For example, the formation reaction for C2H5Cl(g) is 2 C(s) + 5/2 H2(g) + 1/2 Cl2(g)  C2H5Cl(g) Hf(C2H5Cl(g)), the enthalpy of formation for C2H5Cl(g), is the value for H when one mole of C2H5Cl(g) is formed by the above reaction at standard conditions.

  5. 4) The standard state conditions for a substance are defined as follows: gas 1 atm pressure + ideal behavior liquid pure liquid (so solvents are approximately in their standard state) solid pure solid solute 1 mol/L + ideal behavior Standard states can also be defined for other cases, but the above is sufficient for our purposes. We will use the symbol “  ” to indicate the standard state. For example: Hf(C2H5Cl(g)) enthalpy of formation for chloroethane for standard conditions 2 C(s) + 5/2 H2(g) + 1/2 Cl2(g)  C2H5Cl(g)

  6. Spontaneous Processes Consider the following two processes. Why does only one occur?

  7. Entropy To determine which processes consistent with the first law of thermodynamics actually occur we must first define a new thermo-dynamic function, called entropy (S). Entropy measures the amount of “disorder” in a system (in a precise mathematical sense)*. Entropy is a state function. We may talk about the entropy change for the system (Ssyst) and the entropy change for the surroundings (Ssurr). The entropy change for the universe (Suniv) is then given by the expression Suniv = Ssyst + Ssurr We will discuss ways to find values for S for specific types of processes. Note that at constant temperature S = (q)rev/T * * In fact, S = if (đq)rev/T

  8. The Second Law of Thermodynamics The importance of entropy lies in the fact that it can be used to decide which processes that are consistent with the first law of thermodynamics will occur, and which will not. To decide, we need a new thermodynamic law, called the second law of thermodynamics Suniv  0 for any allowed process In words, what the above means is the amount of randomness or disorder in the universe (as measured by entropy) must always increase or remain constant.

  9. Comments Concerning the Second Law 1) The units for entropy are J/K (or J/mol.K) 2) There are three types of processes according to the second law Suniv > 0 , spontaneous, will eventually occur Suniv = 0 , process where equilibrium has been established Suniv < 0 , process will not occur 3) The second law of thermodynamics is based on observation. 4) Since Suniv = Ssyst + Ssurr, it is possible for Ssyst or Ssurr to be negative and still have a spontaneous process (though both cannot be negative for the same process). 5) There is a connection between entropy and the way in which states can be arranged on a molecular level, though we will not explore this (this is studied in statistical thermodynamics).

  10. Spontaneous Processes A process will occur spontaneously if Suniv > 0. If Suniv < 0, then a process will not occur. Suniv = 0 indicates we are at equilibrium. T = 20. C Suniv,melt > 0 Suniv,freeze < 0 T = 0. C Suniv,melt = 0 Suniv,freeze = 0 T = - 20. C Suniv,melt < 0 Suniv,freeze > 0

  11. Entropy Change For a Chemical Reaction (System) When a chemical reaction takes place, Ssyst = Srxn. The expression for the change in entropy, for standard conditions, is Srxn = [  S(products) ] - [  S(reactants) ] Note that elements in their standard state must be included in the above calculation. The above method works because of the fact that entropy, like enthalpy, is a state function.

  12. Molar Entropy The standard molar entropy of a substance is the value for the entropy of 1 mole of the substance for standard pressure (1 atm) and some tempera-ture T. We usually choose T = 25. C.

  13. Entropy Change For a Chemical Reaction (Surroundings) For the entropy change for the surroundings, we may show that (for constant pressure and standard conditions) Ssurr = qsurr/T = - qsyst/T = - Hrxn/T where Hrxn is the enthalpy change for the chemical reaction. The entropy change for the universe is then Suniv = Ssyst + Ssurr = Srxn - (Hrxn/T)

  14. Example Consider the following chemical reaction N2(g) + 3 H2(g)  2 NH3(g) What are Ssyst, Ssurr, and Suniv for the above reaction when carried out under standard conditions (pgas = 1.0 atm, T = 25.C)? Is the reaction spontaneous for these conditions?

  15. Consider the following chemical reaction N2(g) + 3 H2(g)  2 NH3(g) What are Ssyst, Ssurr, and Suniv for the above reaction when carried out under standard conditions (pgas = 1.0 atm, T = 25. C)? Is the reaction spontaneous for these conditions? (Appendix 2 for data) Substance H°f(kJ/mol) G°f(kJ/mol) S°(J/mol.K) N2(g) 0.0 0.0 191.6 H2(g) 0.0 0.0 130.7 NH3(g) - 45.9 - 16.4 192.8

  16. Consider the following chemical reaction N2(g) + 3 H2(g)  2 NH3(g) What are Ssyst, Ssurr, and Suniv for the above reaction when carried out under standard conditions (pgas = 1.0 atm, T = 25. C)? Is the reaction spontaneous for these conditions? (Appendix 2 for data) Substance H°f(kJ/mol) G°f(kJ/mol) S°(J/mol.K) N2(g) 0.0 0.0 191.6 H2(g) 0.0 0.0 130.7 NH3(g) - 45.9 - 16.4 192.8 Srxn = [  S(products) ] - [  S(reactants) ] = [2 (192.8)] - [(191.6) + 3(130.7)] = - 198.1 J/mol.K Hrxn = [  Hf(products) ] - [  Hf(reactants) ] = [(2 (-45.9)] - [ 0 ] = - 91.8 kJ/mol

  17. Srxn = - 198.1 J/mol.K Hrxn = - 91.8 kJ/mol So Ssyst = Srxn = - 198.1 J/mol.K Ssurr = - Hrxn/T = - (- 91800. J/mol) = 307.8 J/mol.K (298.2 K) Suniv = Ssyst + Ssurr = (- 198.1 J/mol.K) + (307.8 J/mol.K) = 109.7 J/mol.K Note that even though Ssyst < 0, the reaction is still spontaneous for standard conditions.

  18. Entropy For Pure Substances The more random a substance is, the larger the value for entropy. So Sgas >> Sliquid > Ssolid S°(Ca(s)) = 41.6 J/mol.K S°(H2O()) = 69.9 J/mol.K S°(Ca(g)) = 154.9 J/mol.K S°(H2O(g)) = 188.8 J/mol.K

  19. Predictions For Entropy Changes For Chemical Reactions Since Sgas >> Sliquid > Ssolid gases are far more disordered than solids and liquids. Based on that, we would predict the following If ngas > 0 then Srxn > 0 If ngas = 0 then Srxn ~ 0 If ngas < 0 then Srxn < 0 Example: Predict the value for Srxn for the following reactions CaCO3(s)  CaO(s) + CO2(g) C(s) + 2 H2(g)  CH4(g) 2 HCl(g) + Br2(g)  2 HBr(g) + Cl2(g)

  20. Example: Predict the value for Srxn for the following reactions CaCO3(s)  CaO(s) + CO2(g) ng = +1, so S > 0 (S = + 160.2 J/mol.K) C(s) + 2 H2(g)  CH4(g) ng = -1, so S < 0 (S = - 80.8 J/mol.K) 2 HCl(g) + Br2(g)  2 HBr(g) + Cl2(g) ng = 0, so S ~ 0 (S = + 1.2 J/mol.K)

  21. Other Trends in Entropy 1) Entropy generally increases for similar molecules as the number of atoms per molecule increases. S°(CH4(g)) = 186.3 J/mol.K S°(C2H6(g)) = 229.2 J/mol.K S°(C3H8(g)) = 270.3 J/mol.K 2) Entropy generally increases when molecular solids or liquids are dissolved in solvents. For ionic compounds, entropy usually (but does not always) also increases. S°(NaCl(s)) = 72.1 J/mol.K S°(NaCl(aq)) = 115.0 J/mol.K

  22. Third Law of Thermodynamics The entropy of a pure substance increases as temperature increases. This is because the degree of randomness in the substance increases. This is particularly true if a phase transition occurs.

  23. Based on low temperature experiments, it appears that the entropy of every pure substance approaches the same value as T  0. K. Third law of thermodynamics: The absolute entropy (S) of a perfect crystal of any pure substance at absolute zero is 0.0 J/mol.K. Because there are standard ways of find the change in entropy for a pure substance as we change the temperature of the substance at constant pressure, the third law of thermodynamics allows us to assign values for entropy for pure substances at any temperature. Note that because for a pure chemical substance the entropy always increases when temperature increases, the third law implies that the absolute entropy of a pure chemical substance at any temperature above absolute zero is a positive number.

  24. Standard molar entropy (S) – The value for the entropy of one mole of a pure substance at p = 1.00 atm and a particular temperature. We usually choose T = 25. C = 298. K. Note - The above does not apply to ions in aqueous solution. In that case, the entropy of H+ ion at T = 25. C is defined as 0.0 J/mol.K. Substance S (J/molK) Ca2+(aq) - 55.2 Cs+(aq) 133.05 H+(aq) 0.00 Mg2+(aq) - 117.99 NO3-(aq) 146.4 Rb+(aq) 124.27 SO42-(aq) 17.15

  25. Free Energy - Definition While the second law of thermodynamics is true for all processes, it is difficult to use because it requires combining the results of several calculations. To help us around this problem we define a new thermodynamic function, called the free energy (G). G = H - TS (a state function) Note that for a process taking place at a particular fixed temperature G = H - (TS) = H - TS

  26. Free Energy and the Second Law For a process taking place at a particular fixed temperature G = H - (TS) = H - TS If we divide both sides of the above equation by –T, we get - (G)/T = - (H)/T + S = Ssurr + Ssyst = Suniv Since the second law requires that Suniv 0, and since we know that T > 0, it follows that the sign for G for a process where pressure and temperature do not change tells us whether or not a process is consistent with the second law of thermodynamics. Since we require Suniv  0, it follows that for processes at a fixed temperature and pressure, G  0.

  27. Free Energy and the Second Law For a process where temperature and pressure are constant, we may say the following: G < 0 process is spontaneous G = 0 system is at equilibrium G > 0 process is not spontaneous (will not occur) Notice this means we need only do one calculation to find out if a process is spontaneous for conditions of constant temperature and pressure. In fact, since free energy is a state function, we only require that the initial and final temperature and pressure are the same (so they can change during the process, as long as they return to their initial values at the end).

  28. Finding Gsyst For a Chemical Reaction For a chemical reaction Gsyst = Grxn is given by the expression Grxn = [  Gf(products) ] - [  Gf(reactants) ] Note Gf = 0 for an element in its standard state. Example: What are Hrxn, Srxn, and Grxn for the following process 4 Fe3O4(s) + O2(g)  6 Fe2O3(s) Is the reaction spontaneous for standard conditions?

  29. Example: What are Hrxn, Srxn, and Grxn for the following process 4 Fe3O4(s) + O2(g)  6 Fe2O3(s) Is the reaction spontaneous for standard conditions? As before, we need a table of thermodynamic data top do this problem. Such data are given in Appendix 2 of the textbook. Substance H°f(kJ/mol) G°f(kJ/mol) S°(J/mol.K) Fe3O4(s) - 1118.4 - 1015.4 146.4 O2(g) 0.0 0.0 205.2 Fe2O3(s) - 824.2 - 742.2 87.4

  30. Example: What are Hrxn, Srxn, and Grxn for the following process 4 Fe3O4(s) + O2(g)  6 Fe2O3(s) Hrxn = [6 Hf(Fe2O3(s))] - [4 Hf(Fe3O4(s))] = [6 (- 824.2 kJ/mol)] - [4 (- 1118.4 kJ/mol)] = - 471.6. kJ/mol Srxn = [6 S(Fe2O3(s))] - [4 S(Fe3O4(s)) + S(O2(g))] = [6 (87.4 J/mol.K)] - [4 (146.4 J/mol.K) + (205.2 J/mol.K)] = - 266.4 J/mol.K Grxn = [6 Gf(Fe2O3(s))] - [4 Gf(Fe3O4(s))] = [6 (- 742.2 kJ/mol)] - [4 (- 1015.4 kJ/mol)] = - 391.6 kJ/mol As a check, Grxn = Hrxn - TSrxn = - 471.6 kJ/mol - (298.2 K) (- 0.2664 kJ/mol.K) = - 392.2 kJ/mol (process is spontaneous)

  31. Finding When Reactions Are Spontaneous Since G = H - TS, it follows that at a particular temperature G =  H - T S To a first approximation the values for Hrxn and Srxn (but not the value for Grxn) are independent of temperature. We can use this to find the approximate value for temperature at which equilibrium occurs for standard conditions of concentration. Grxn = 0 = Hrxn - Teq Srxn Teq  Hrxn /Srxn Example: At what temperature will H2O() and H2O(g) exist at equilibrium? (Note this will correspond to the normal boiling point of water).

  32. Example: At what temperature will H2O() and H2O(g) exist at equilibrium? (Note this will correspond to the normal boiling point of water). H2O()  H2O(g) Substance H°f(kJ/mol) G°f(kJ/mol) S°(J/mol.K) H2O() - 285.8 - 237.2 69.9 H2O(g) - 241.8 - 228.6 188.7 Hrxn = - 241.8 kJ/mol – (- 285.8 kJ/mol) = + 44.0 kJ/mol Srxn = 188.7 J/mol.K – 69.6 J/mol.K = + 118.8 J/mol.K

  33. Example: At what temperature will H2O() and H2O(g) exist at equilibrium? (Note this will correspond to the normal boiling point of water). H2O()  H2O(g) Hrxn = + 44.0 kJ/mol Srxn = + 118.8 J/mol.K Teq  Hrxn /Srxn = (44000. J/mol)/(118.8 J/mol.K) = 370.4 K Since Grxn = Hrxn - T Srxn it follows that if T > 370.4 K, Grxn < 0, and boiling is spontaneous. Of course, the true value for the normal boiling point for water is 373.2 K. The small difference between this and the value calculated above is due to the assumption that Hrxn and Srxn are independent of temperature, which is only approximately correct.

  34. Deciding When Reactions Will Be Spontaneous Recall that Grxn = Hrxn - T Srxn Grxn < 0 for a spontaneous process If we assume that both Hrxn and Srxn are approximately independent of temperature, then we have the following possibilities: Hrxn Srxn behavior positive positive spontaneous at high temperatures (spontaneous if T > Teq) positive negative reaction is never spontaneous negative positive reaction is always spontaneous negative negative spontaneous at low temperatures (spontaneous if T < Teq)

  35. End of Chapter 14 “…even though the average disorder in the universe has been increasing ever since the beginning, this increasing disorder is not occurring at every particular place in the universe. You can have local order created in certain locations at the expense of greater disorder elsewhere. Take your refrigerator as an example. You have ice cubes in the freezer that are certainly a manifestation of a great deal of order, but if you go around to the back of the refrigerator there’s a lot of hot air coming out indicating that the ‘cost’ of the order of the ice cubes is paid by the greater amount of disorder somewhere else.” Murray Gell-Mann “[Thermodynamics] is the only physical theory of universal content which, within the framework of the applicability of its basic concepts, I am convinced will never be overthrown.” Albert Einstein

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