The Direction of Spontaneous Change: Entropy and Free Energy

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1 The Direction of Spontaneous Change: Entropy and Free Energy Reading: from Petrucci, Harwood and Herring (8th edition): Required for Part 1: Sections 20-1 through Recommended for Part 1: Sections 7-1 and 7-5. Examples for Part 1: 20-1, 20-2, 20-4 Problem Set for Part 1: Chapter 20 questions: 29, 33, 39, 41 Additional problems from Chapter 20: 13, 37 York University CHEM Spontaneous Change - 1

2 Examples of Spontaneous Processes When a hot object is placed in contact with a cold object, the hot object gets colder and the cold one gets hotter. Water flows downhill. Ice melts in summer. Water freezes in winter. Water evaporates from lakes and oceans. Water condenses to form clouds and rain. Gases expand to fill the available volume. Many liquids mix to form solutions. If Q C < K C, a reaction will proceed to the right. If Q C > K C, a reaction will proceed to the left. York University CHEM Spontaneous Change - 2

3 Internal Energy Energy is contained within materials in a variety of forms. Nuclear energy - associated with interactions among protons and neutrons in atomic nuclei. Chemical energy - associated with interactions among electrons and nuclei in forming chemical bonds. Molecular kinetic energy - associated with the motions of atoms and molecules. Directly related to temperature. Molecular potential energy - associated with the interactions between molecules. Potential energy associated with external fields: electrostatic, magnetic, and gravitational. York University CHEM Spontaneous Change - 3

4 Measuring Internal Energy Measuring internal energy is impossible. We can only measure changes in internal energy. Two basic ways to measure changes in internal energy: Heat is the transfer of internal energy due to a difference in temperature. Work is any transfer of internal energy that could have the sole effect of doing mechanical work. Heat and work are processes. A hot object contains more molecular kinetic energy than a cold one. It does not contain "more heat". York University CHEM Spontaneous Change - 4

5 Heat Amount of energy transferred as heat can be determined from temperature changes: York University CHEM Spontaneous Change - 5

6 Work Work is any transfer of internal energy that could have the sole effect of doing mechanical work. Mechanical work = force distance = F x Pressure-Volume work = F x = PA x= P V Electrochemical work: Use a battery to run an electric motor and lift a weight. York University CHEM Spontaneous Change - 6

7 The First Law of Thermodynamics Energy is conserved. It is neither created or destroyed. Some definitions: The system is that part of the universe in which we are interested. U sys = internal energy of the system The surroundings are everything else. U sur = internal energy of the surroundings According to the First Law: U sys + U sur = 0 York University CHEM Spontaneous Change - 7

8 Heat, Work, and the First Law Heat and work are the only known ways to transfer internal energy. q = internal energy transferred as heat w = internal energy transferred as work (all forms) Then U sys = q sys + w sys and U sur = q sur + w sur From the First Law: U sys + U sur = 0 (q sys + w sys ) = -(q sur + w sur ) This does not require q sys = -q sur or w sys = -w sur. York University CHEM Spontaneous Change - 8

9 Turning "Work" into "Heat" Water flows over a waterfall. As it falls, its gravitational potential energy decreases. This could be used to do mechanical work. If no work is done, the potential energy turns into molecular kinetic energy. The water gets warmer. The energy is transferred to the surroundings as heat. If it is possible to transfer energy as work, it is possible to transfer the same amount of energy as heat. York University CHEM Spontaneous Change - 9

10 Heat is not Work Heat can be used to do work: (1) Place a hot object in contact with a cold gas. (2) The temperature of the gas rises, the gas expands. (3) Use the expanding gas to do mechanical work. Does this mean that heat is a form of work? No! The work done will not be the sole effect of the transfer of internal energy. Some of the energy removed from the hot object as heat will be transferred to the surroundings as heat. York University CHEM Spontaneous Change - 10

11 Why Heat is not Work Work is any transfer of internal energy that could have the sole effect of doing mechanical work. What if we could turn "heat" entirely into "work"? (1) Remove energy from a cold object as heat. (2) Turn the "heat" entirely into "work". (3) Turn the "work" entirely into "heat" within a hot object. Result: The cold object gets colder and the hot one gets hotter with no other change. Impossible! Conclusion: Heat is fundamentally different from work. York University CHEM Spontaneous Change - 11

12 Heat, Work, and Internal Energy - Summary Energy is conserved (First Law of Thermodynamics). Internal energy is contained within materials. Internal energy may be transferred as heat or as work. All forms of work are fundamentally equivalent. "Work" can be converted entirely to "heat". "Heat" can not be converted entirely to "work". The last two statements imply something about the direction of spontaneous change. York University CHEM Spontaneous Change - 12

13 State Functions A finite number of quantities are needed to fully specify the state of a system. For a system at equilibrium, the most convenient set is: temperature, pressure, and moles of each component in each phase. A state function is any quantity that depends only on the state of the system. Examples: temperature, pressure, volume, internal energy, moles of a component in a particular phase Specifying the state fixes the values of all state functions. York University CHEM Spontaneous Change - 13

14 Enthalpy The enthalpy, H, is a state function with the property that H = q P + w E where q P and w E are energy transferred to the system as heat and non-pv work during a process at constant pressure. q and w are not state functions. They depend on the path. H depends only on the initial and final states. Chemical reactions are usually at constant pressure. If w E =0, H = q P. Enthalpy is sometimes called "heat content". York University CHEM Spontaneous Change - 14

15 Reversible Processes A reversible process can be made to occur in exactly the opposite direction, so that both the system and the surroundings can be returned to their original states. During a reversible process, the system and surroundings are always in equilibrium. A reversible process is an idealization; it can not occur in the real world. Reversible processes define the limits of what can occur in the real world. York University CHEM Spontaneous Change - 15

16 Spontaneous Processes A spontaneous process is one that may occur in a system left to itself. A non-spontaneous process requires some external action. The reverse of a spontaneous process is always a non-spontaneous process. A spontaneous process is not reversible. Returning the system to its original state leaves the surroundings in a different state. A reversible process is neither spontaneous or nonspontaneous. It defines the boundary between them. York University CHEM Spontaneous Change - 16

17 Heat and the Direction of Change "Work" can be converted entirely to "heat". "Heat" can not be converted entirely to "work". Implication: Some processes can only occur in one direction. They are irreversible. Transferring energy as heat is closely related to irreversibility. By definition, heat is the transfer of internal energy due to a difference in temperature. This is an irreversible process. York University CHEM Spontaneous Change - 17

18 The Second Law of Thermodynamics When a hot object is placed in contact with a cold object, the hot object gets colder and the cold one gets hotter. It is impossible to remove energy as heat from a system in such a way that the sole effect is to do mechanical work. The entropy of the universe increases during any spontaneous process. During a reversible process, S univ = 0. These are all statements of the Second Law. They are exactly equivalent. York University CHEM Spontaneous Change - 18

19 Entropy The entropy, S, is a state function with the property that during any reversible, isothermal process. q is not a state function. It depends on the path taken. S depends only on the initial and final states. A reversible path is used to calculate the entropy change. Note: The formula ds = dq rev /T may be used for non-isothermal processes. York University CHEM Spontaneous Change - 19

20 Calculating an Entropy Change Phase transitions take place at constant temperature and pressure with the transfer of energy as heat. H 2 O(s) H 2 O(l) H tr = 6.01 kj mol -1 H 2 O(l) H 2 O(g) H tr = 40.7 kj mol -1 We can imagine carrying out a phase transition reversibly at constant temperature and pressure. Then: For melting ice: S = (6010 J mol -1 ) / ( K) S = 22.0 J mol -1 K -1 York University CHEM Spontaneous Change - 20

21 Irreversible Processes To get S for an irreversible process, use q for a reversible process that produces the same change in state. Consider a process that turns "work" into "heat": Examples: friction, water flowing down hill. Energy removed from a system as work, S sys = 0. Same amount of energy delivered to surroundings as heat, q sur > 0 and S sur > 0. The entropy of the universe increases: S univ = S sys + S sur > 0 Common observation - things proceed to lower energy. York University CHEM Spontaneous Change - 21

22 The Direction of Spontaneous Change Ice is in equilibrium with water at T = T fus = K. Ice melts spontaneously if T > T fus. q sys = H fus > 0 S sys = q sys /T = H fus /T fus q sur = - H fus < 0 S sur = q sur /T fus = - H fus /T S univ = S sys + S sur = H fus ( 1/T fus - 1/T ) > 0 if T > T fus Water freezes spontaneously if T < T fus. q sys = - H fus < 0 S sys = q sys /T = - H fus /T fus q sur = H fus < 0 S sur = q sur /T fus = H fus /T S univ = S sys + S sur = - H fus ( 1/T fus - 1/T ) > 0 if T < T fus York University CHEM Spontaneous Change - 22

23 A More Subtle Example A gas will spontaneously expand into a vacuum. If the gas is ideal, q sys = q sur = 0. So S sur = 0. Is S sys = 0? No! The process is not reversible. If the same change in state were done reversibly, q sys > 0. So S sys > 0. S univ = S sys + S sur > 0 York University CHEM Spontaneous Change - 23

24 Microscopic View of Entropy Entropy is related to the number of equivalent microstates (configurations) that a system can occupy. S = k lnw k = R / N A = Boltzmann's constant W = number of available microstates W increases when: the number of molecules in the system increases the quantum energy levels become closer the temperature increases York University CHEM Spontaneous Change - 24

25 Positional Microstates All four particles on one side - there are two possible arrangements. Two particles on each side - there are 12 possible arrangements. For a gas, entropy increases when volume increases. York University CHEM Spontaneous Change - 25

26 Energy Microstates At absolute zero, energy is a minimum. Only one microstate is possible. At T > 0, total energy is higher. There are more accessible microstates. W is larger and S is larger. Entropy increases with higher temperature or more closely spaced energy levels. York University CHEM Spontaneous Change - 26

27 Phase Transitions Melting: Entropy increases. Vaporization: Entropy increases. Dissolving solid: Entropy usually increases. York University CHEM Spontaneous Change - 27

28 Summary of Entropy Changes The entropy per mole increases when: the temperature increases (always) the volume of a gas increases (always) the volume of a liquid or solid increases (usually) a solid melts to form a liquid (always) a solid or liquid vaporizes to form a gas (always) gases mix (always) liquids mix (usually) solids dissolve in a liquid (usually) a reaction increases the number of moles of gas (always) York University CHEM Spontaneous Change - 28

29 Absolute Entropies The absolute value of the entropy may be determined if all the quantum states of a system are known. S = k lnw For a pure, perfect crystal at 0 K, W = 1. So S = 0. For a solid near absolute zero, S = BT 2. The constant, B, may be determined from heat capacity measurements. For gases, all the quantum states can be determined from spectroscopic data. With these as starting points, other absolute entropies can be determined. York University CHEM Spontaneous Change - 29

30 Applying the Second Law Chemical reactions commonly take place at constant temperature and pressure with no non-pv work. Then q sur = -q sys = - H sys and S sur = q sur /T, so T S sur = - H sys Therefore: T S univ = T S sur + T S sys = - H sys + T S sys Conclusion: if - H sys + T S sys > 0, then S univ > 0 spontaneous if - H sys + T S sys = 0, then S univ = 0 reversible if - H sys + T S sys < 0, then S univ < 0 non-spontaneous York University CHEM Spontaneous Change - 30

31 The Gibbs Free Energy The Gibbs free energy, G, is a state function defined such that, in a process at constant temperature G = H - T S. For a process at constant temperature and pressure with no non-pv work, T S univ = - G sys For these conditions, G can be used instead of S univ. if G < 0, then the process is spontaneous if G = 0, then the process is reversible (at equilibrium) if G > 0, then the process is non-spontaneous York University CHEM Spontaneous Change - 31

32 G - Example At K with one bar of O 2, the reaction 2Fe(s) + 3/2O 2 (g) Fe 2 O 3 (s) has H = kj mol -1 and S = J mol -1 K -1. What is the value of G? Is the reaction spontaneous under these conditions? Solution: G = H - T S = kj mol -1 The reaction is spontaneous. Note that in doing the calculation, either H must be converted to J mol -1 or S must be converted to kj mol -1 K -1. York University CHEM Spontaneous Change - 32

33 Inverse Processes Changing the direction of a process changes the signs of H, S, and G. Example: Using data on the previous slide, find H, S, and G for Fe 2 O 3 (s) 2Fe(s) + 3/2O 2 (g) at K with one bar of O 2. Is this reaction spontaneous? Solution: This reaction is the reverse of the one on the previous slide. So H = kj mol -1 S = J mol -1 K -1 G = kj mol -1 The reaction is non-spontaneous. York University CHEM Spontaneous Change - 33

34 Temperature Dependence of G G = H - T S. At low temperature, H is more important than T S. Then H < 0 implies G < 0. At high temperature, T S is more important than H. Then S > 0 implies G < 0. Conclusions: A process with H < 0 (exothermic) will be spontaneous at low temperature. A process with S > 0 will be spontaneous at high temperature. York University CHEM Spontaneous Change - 34

35 Temperature Dependence - continued If H < 0, enthalpy decreases during the process. Also, the process will be spontaneous at low temperature. Low temperature favors low enthalpy. If S > 0, entropy increases during the process. Also, the process will be spontaneous at high temperature. High temperature favors high entropy. Example: solid liquid gas low temperature high temperature low enthalpy high enthalpy low entropy high entropy York University CHEM Spontaneous Change - 35

36 Temperature Dependence of Spontaneity H < 0 spontaneous at low temperature S > 0 spontaneous at high temperature Therefore: H < 0, S > 0 spontaneous at all T H > 0, S < 0 non-spontaneous at all T H < 0, S < 0 spontaneous at low T, non-spontaneous at high T H > 0, S > 0 non-spontaneous at low T, spontaneous at high T Over-simplified since H and S depend on T and P. York University CHEM Spontaneous Change - 36

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