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Transcript
Chm 118 Fall 2015, Exercise Set 4 Quantum Levels, Populations, Entropy, Enthalpy, and Free Energy Mr. Linck Version 3.8 November 10, 2015 4.1 Probability Craps is a game in which rolling a “7” as the sum of the two dice has consequences. What is the probability that one will roll a “7” with two dice? What is the probability that one will roll a “2” with two dice? 4.2 Probability What is most probable in craps, rolling a “7” or rolling a “2” with two dice? 4.3 Definitions What is the difference between a microstate and a configuration? 4.4 Number of Microstates NOTA BENE: In this problem and those that directly follow the symbol 9/3/1/0/1 means nine particles in the lowest level, 3 in the first excited level, 1 in the second excited level, etc., as illustrated in Figure 1. Energy spacing is always one unit unless stated otherwise. Find the number of microstates in the configuration 9/0/0/1/0. What is the energy of this configuration? 4.5 Number of Microstates Find the number of microstates in the configuration 9/1/1/0/0. What’s its energy? 4.6 Number of Microstates Are the configurations of the last two problems different? Why? 4.7 Number of Configurations Is there any other configuration with the same energy as found in the last two problems? If so, what is it? What is the predominant configuration? 4.8 Number of Microstates, W Find the number of microstates in the configuration 5/1/1/1/0, a quantity we call W. 4.9 Number of Microstates, W Find the number of microstates in the configuration 6/3/1/0/0, a quantity we call W. 1.16 2 Figure 1: An Example of a Configuration 4.10 W Depends on N Consider 10 particles with a total energy of 3 units. Find all the configurations and the W of each. Identify the most probable configuration and note W for it. 4.11 W Depends on N Consider 20 particles with a total energy of 3 units. Find all the configurations and the W of each. Identify the most probable configuration and note W for it. 4.12 W Depends on N From the last two problems, what do you conclude about how W for the most probable configuration changes as N goes up? What would you do to be sure of your conclusion? 4.13 Most Probable Configuration Imagine twelve particles in the set of energy levels. Find the configuration that is most probable for a system with a total of two units of energy. 4.14 Most Probable Configuration Same number of particles as in problem 13. Find the configuration that is most probable for a system with a total of three units of energy. 4.15 Most Probable Configuration Same number of particles as in problem 13. Find the configuration that is most probable for a system with a total of four units of energy. Chm 118 Exercise Set 4 1.27 3 4.16 Changing Configurations Show that W increases in going from the most probable configuration of problem 13 to that of 14. 4.17 Changing Configurations Show that the ratio of W15 /W14 is less than W14 /W13 , where the subscripts refer to problem numbers, and the W’s are for the most probable configuration, hereafter Wmpc , and then latter still, just W for reasons that will become clear. 4.18 Changing Configurations How do you state the conclusion of the last problems in general terms? HINT: There are two issues concerning the Wmpc , how it changes with an increase in E by a constant amount, and how it changes at a low E (temperature) versus a high E (temperature). 4.19 Adding Heat Changes W In the last several problems we have been talking about energy levels that remain equally spaced. If we consider these levels for translation, then what parameters must be constant in order to keep the energy levels equally spaced? HINT: Think about the POP problem which is a reasonable model for translational motion. 4.20 Work and Heat What is work? What is heat? Be careful to distinguish between work, heat, and internal energy, the E of our problems above. HINT: Because we are looking for a precise definition, it might be advisable to look these up. 4.21 Predominant Configuration Find the predominant configuration for 6 particles with 2 units of energy. 4.22 Predominant Configuration Find the predominant configuration for 26 particles with 7 units of energy. This takes some trial and error, but is worth the effort if you examine the nature of your answer. 4.23 Changing Configurations If one of the particles of the last problem is taken from the v = 0 level and raised to the v new = 1 level, does the value of W go up or down? What is the value of W Wold ? HINT: Since we are going to use the letter “n” to represent something in the following development, I have labeled the quantum levels with a different letter, “v”. 4.24 Predominant Configuration Find the predominant configuration for 26 particles with 2 units of energy. HINT: Relatively to problem 22 this is easy. 4.25 Changing Configurations If one of the particles of the last problem is taken from the v = 0 level and raised to the v new = 1 level, does the value of W go up or down? What is the value of W Wold ? 4.26 Changing Configurations new How do your results for the ratio of W Wold from problems 23 and 25? What do you conclude? HINT: I am trying to get you to restate something you should have already stated because it is important. Chm 118 Exercise Set 4 1.32 4 4.27 Ratio of Particles in the MPC Does the ratio of particles in the most predominant configuration (MPC) between two v values depend on the energy difference between those v values? You might look at problem 22 for instance and compare the ratio nv=1 /nv=0 with nv=2 /nv=0 , where ni is the number of particles in the ith level. 4.28 Finding the Energy Dependence of nv=1 /nv=0 In the last problem, you determined (presumably) that the ratio nv=1 /nv=0 depended in some way on the energy. Energy is a quantity for which we can arbitrarily set a zero. Try the following: If nv=1 /nv=0 depends on the energy of the lower level, and we change the zero of energy, then the ratio would change. That situation is impossible! So let’s see what happens if the ratio is dependent on the energy difference: nv=1 = f (e1 − e0 ) nv=0 Now change the zero of energy. What happens to the ratio? Is that better? 4.29 The Ratio Must Depend on the Exponential of the Energy Difference Given the result of the last problem and that nnv=2 = f (e2 − e1 ), show that since v=1 nv=2 nv=2 nv=1 = nv=0 nv=1 nv=0 if follows that f (e2 − e0 ) = f (e2 − e1 )f (e1 − e0 ) which in turm requires that f (ei − ej) = eβ(ei −ej) where β is some constant (at least at a given temperature); please NOTE there are two uses of “e” in this equation, distinguished by the presence or lack of subscripts. This leads us to the Boltzmann equation, ni = n0 e−(ei −e0 )/(kb T ) (1) where β has been evaluated as 1/(kb T). 4.30 Boltzmann Distribution Show that you get the same answer for nv=2 /nv=1 from direct use of equation 1 as you get if you calculate the ratio in level 2 relative to that in level 0 and the ratio in level 1 relative to level 0 and use those two numbers to compute nv=2 /nv=1 . HINT: I am simply asking you to verify that this form for our function in equation 1 is consistent with the equation: f (e2 − e0 ) = f (e2 − e1 )f (e1 − e0 ) 4.31 Boltzmann Distribution You have a set of equally spaced energy levels, the lowest of which has zero units of energy and each energy level gap is one unit. If this system is in the predominant configuration and there are 1920 particles in the e=0 level and 480 in the e=1 level, how many are in the e=5 level? HINT: Don’t try to calculate this by evaluating factorials! Chm 118 Exercise Set 4 1.41 5 4.32 Boltzmann’s Law If you have 1000 particles in the n = 0 level, the spacing between the levels is 4.14×10−21 J/K, and the temperature is 300 K, what is the number of particles in n = 1? in n = 2? 4.33 Changing Energy Levels and the Predominant Configurations Imagine a system with energy spacing of two units, comprised of 20/2/0 particles for a total energy of 4 units. Hold the energy constant, but increase the size of the container such that the spacing between the levels becomes one unit. What configuration is roughly consistent with 4 units of energy now? 4.34 Changing Energy Levels and the Predominant Configurations What is W for each of the configurations of problem 33? 4.35 Changing Energy Levels and the Predominant Configurations Does W increase in going from the small container of problem 33 to the large one? 4.36 Changing Energy Levels and the Predominant Configurations Will a gas expand to occupy a larger volume if given the opportunity? In view of the last problem, do you see why? 4.37 Energy Levels and Entropy The last few problems illustrate how important energy level spacing is in determining entropy. There are generally four kinds of energy level spacings that we divide molecular energy into. What are they? 4.38 Energy Levels and Entropy Which of the four kinds of energy level spacings (see last problem) has the most closely spaced energy levels? the most widely spaced? 4.39 Macroscopic versus Microscopic Views Describe a gas in terms of the behavior of individual particles, their motions and their relative motions. Temperature is roughly a measure of the energy of the particles, but given populations of various levels yielding an average energy, what does the temperature of a gas represent in terms of individual particles? 4.40 Macroscopic versus Microscopic Views Imagine a set of billiard balls rolling around on a table with random speeds distributed among them. Upon the collision of two balls the speed of each could be changed. However, these are magic billiard balls, which can take up energy from a collision and store it internally. But they can do this only if the energy of the collision is sufficiently high. When the billiard balls are moving gently on the table, is it likely that any given collision will be able to excite the internal energy of the ball? What happens if the average velocity on the table increases? Chm 118 Exercise Set 4 1.48 6 Figure 2: The Folly of Using “Disorder” after Lambert, F. L.; J. Chem. Ed, 2002, 79, 187-192. 4.41 Macroscopic versus Microscopic Views We have the same situation as in the last problem, but the billiard balls now have two different (quantum) methods of storing internal energy. One method, call it A, requires a minimum energy of x; the second, call it B, requires 10x. As temperature is increased (what does he mean by that?) which internal system, A or B, becomes activated first? If a given amount of heat is added to the table (what does he mean by that?) does it go entirely to translational energy? If not, why not? What can you say about the change in translational energy as heat is added at low temperatures? At high temperatures? 4.42 Entropy Let’s define entropy, S, as kb ln W, where kb is Boltzmann’s constant, 1.38 x 10−23 J /K, and W is the number of microstates in the predominant configuration. For real systems, W is a very large number but S is considerably smaller. Why? 4.43 Energy Levels and Entropy At room temperature, are vibration energy level spacings (of about 2×10−22 J) important in determining entropy? Why or why not? 4.44 Don’t Use “Disorder” in Your Arguments Look at Figure 2 and determine which side is most “disordered”. If you say that the right hand picture has roughly even spacing of the particles, then it is “ordered” and the left hand picture is “disordered”. Now consider the real situation where there is a barrier at the midpoint of the container to keep all the particles on the left hand side in the initial state. Use the population of energy levels to assess which picture with the highest entropy. 4.45 The Change in Entropy of the System Give a complete and logical statement about how you would assess the entropy change of some system. 4.46 The Change in Entropy of the Surroundings Give a complete and logical statement about how you would assess the entropy change of the surroundings of some system. HINT: The surroundings are large. To visualize them on a molecular level is difficult. Also, since the surroundings are large, the temperature of the same changes little when some heat is taken into it. 4.47 Review What the relative spacing of energy levels for electronic motion, vibration, rotation, and translation? Chm 118 Exercise Set 4 1.59 7 4.48 Entropy Determine which system has the highest entropy, Ag(s) at 298K or Ag(s) at 340K? How did you reach your conclusion? 4.49 Atomic Motion in the Gas Phase If you have a bottle of Li(g) and a bottle of the same size of Cs(g), both at the same temperature, determine which is moving more rapidly. HINT: All the energy is kinetic and total energy is proportional to temperature. 4.50 Another Example Taken from Lambert–see Figure 2 Given your answer to the last problem, which would you imagine is more chaotic, less ordered, Li(g) or Cs(g). What does “disorder” predict for the highest entropy? 4.51 Another Example Taken from Lambert–see Figure 2 Now consider Li(g) and Cs(g) from the point of view of microstates, not disorder. If you use a POP model, which has energy levels more closely spaced (in the same size container), Li(g) or Cs(g)? Which do you predcit has the highest W and the highest entropy? 4.52 Entropy Look up some data to determine which system has the highest entropy Cs(g) or Li(g) at 298K, both at the same pressure? 4.53 Entropy Determine which system has the highest entropy Na(s) at 371K or Na(`) at 371K? How did you reach your conclusion? 4.54 Entropy Determine which system has the highest entropy NO(g) or NO2 (g) at 298K and the same pressure? How did you reach your conclusion? HINT: There are a couple of things going on here. 4.55 Entropy Determine which system has the highest entropy KClO3 (s) and a liter of water or KClO3 (aq) (in that liter of water)? HINTS: Ionic solution. Also, maybe ambiguous? 4.56 Extensive and Intensive Properties An extensive property is one that depends on the amount of material. Give some examples. 4.57 Extensive and Intensive Properties An intensive property is one that does not depend on the amount of material. Give some examples. 4.58 Extensive and Intensive Properties Is energy extensive or intensive? How about volume? Temperature? Entropy? Chm 118 Exercise Set 4 1.66 8 4.59 Thermodynamic Properties of Reactions As chemists we most often want to express the value of a thermodynamic quantity for a reaction of some sort. Since the quantity of material that we deal with is variable, the most useful kind of quantity is an intensive one. What kind of property is entropy? What would happen if we expressed entropy per mole of material transferred from the left of our equation to the right? What kind of property—intensive or extensive—would that quantity be? 4.60 Thermodynamic Properties of Reactions We have a symbol for the change in entropy of a substance, dS or ∆S. Now we need one for the quantity we just invented in the last problem, one to be used with reactions. There are several suggestions for such symbols in use, including ∆r S, which is used in advanced g which was suggested but never adopted, books, but is often regarded as rather clumsy, ∆S, and even ∆S, which is often used in introductory texts (and courses) and which leaves it up to the user to understand the context–probably seldom the case. What would I mean if I told you the hunk of gold in going from state a to state b has ∆S of -20 J/K? 4.61 Thermodynamic Parameters of Reactions Referring to the last problem, what would I mean if I told you that ∆r S for H2 O(s, T=273K) = H2 O(`, T=273K) was 22 J/(mole K)? 4.62 Thermodynamic Parameters of Reactions Given the equation in the last problem, how would you interpret the following sentence: “The entropy change, ∆S, for melting water at 273K is 11 J/K.” 4.63 Thermodynamic Parameters of Reactions What would you conclude with the information that “∆S for the melting of water at 273K is 22 J/(mole K)”? 4.64 Reference or Standard State A second issue we need to address is that changes in chemical reactions depend upon the conditions of the reagents and products, their pressures, temperatures, etc. Generally, it has been agreed that in the standard state all materials are pure, gases are at one atmosphere pressure and all concentration are equal to one molar, and, usually, the temperature is 298.15K If this is so, the thermodynamic parameter is given a “super o”. Would the entropy change of the following reaction at 298.15K be characterized by a ∆S o ? C(s) + 2Cl2 (g, P = 0.1 atm) = CCl4 (`) 4.65 Entropy Change Use a table of absolute entropies to predict the value of ∆S o for 2K(s) + F2 (g) = 2KF(s) Give a rationalization for your answer. HINT: If the text you are using does not have a good table of thermodynamic values, try the web page at http://chemistrytable.webs.com/enthalpyentropyandgibbs.htm or other sites found by searching for “enthalpy of formation” or “absolute entropy”. Chm 118 Exercise Set 4 1.76 9 4.66 Entropy Change Use a table of absolute entropies to predict the value of ∆S o for NH3 (g) + HBr(g) = NH4 Br(s) Give a rationalization for your answer. 4.67 Entropy Change Use a table of absolute entropies to predict the value of ∆S o for CH3 CHCH2 (g) = cyclic-C3 H6 (g) Give a rationalization for your answer. 4.68 Entropy Change Use a table of absolute entropies to predict the value of ∆S o for 3 H2 (g) + Fe2 O3 (s) = 2Fe(s) + 3H2 O(g) Give a rationalization for your answer. 4.69 Entropy Change and Temperature Comment on which direction the reaction 2NO(g) + Cl2 (g) = 2NOCl(g) will proceed if all reagents are under standard conditions and you are at some “high enough” temperature. HINT: You need to apply your thoughts about ∆Suniv since you are being asked about spontaneity. 4.70 Entropy Change and Temperature Comment on which direction the reaction SrSO4 (s) = SrO(s) + SO3 (g) will proceed at some “high enough” temperature. 4.71 Entropy Change and Temperature Comment on which direction the reaction SO3 (g) = SO2 (g) + 21 O2 (g) will proceed at some “high enough” temperature. 4.72 Entropy The entropy of CCl4 is about 210 J/(mole K) at 65o C, 214 J/(mole K) at 75o C, but is 309 J/(mole K) at 80o C. Explain these data. 4.73 Entropy Which would have the higher entropy at 298o K, NH3 (g) or Ne(g)? Why? 4.74 Entropy Differences The entropy of CO2 is 213.7 J/(mole K) and that of SO2 is 248.2 J/(mole K), both at 298o K. Explain the difference. Chm 118 Exercise Set 4 1.84 10 4.75 Entropies The entropies of NaF(s), MgO(s), and AlN(s) are, respectively, 51.5, 26.8, and 20.2 J/(mole K) at 298o K. Give a rationalization. 4.76 Entropy with Pressure Change Justify why one mole of a gas has a higher entropy if it is changed from a pressure of 1.0 atm to a pressure of 0.1 atm at constant temperature. 4.77 Work on a Gas Calculate the work done on an ideal gas (in a piston and cylinder) when 1.0 moles at a pressure of 20.0 atm. and T = 200K is expanded against an external pressure of 1.0 atm isothermally. Find the heat. 4.78 Work Depends on Path Do the expansion in the last problem, but in this case let the external pressure be 10.0 atm until the piston stops moving, then reduce the external pressure to 1.0 atm. Find the work and heat for the overall process. 4.79 Path Dependent Processes Are work and heat dependent upon the path? 4.80 Reversible Work Expand the gas in problem 77 from the same initial state to the same final state, but do it reversibly. What is the work? What is the reversible heat? What can you say about the reversible work compared to that found in problem 77 and problem 78? 4.81 Total Energy of a Collection of Particles Write the total energy, E, in terms of the energy of the various quantum levels and the number of particles in each level. HINT: This is a generic problem; just give the various levels some label and sum over those that are occupied times the number of particles in each level. 4.82 Total Energy for a Collection of Particles Use the Boltzmann equation, equation 1, to revise your equation of the last problem so that the number of particles in the various levels are removed. 4.83 Total Energy for a Collection of Particles The sum in the equation you got for the last problem becomes an integral if the levels are close enough together (as they are for translation, for instance). If we evaluate that integral the result is the equation 3 3 E = N kb T = RT 2 2 (2) for the translational energy of an ideal gas. In general terms, where does this temperature dependence originate? Chm 118 Exercise Set 4 1.95 11 4.84 First Law at Constant Pressure Use the first law of thermodynamics, dE = δw + δq = −P dV + δq (3) set the pressure to a constant, and solve for the heat, called here dqP (in a clever attempt to indicate constant pressure). Is this heat a “state” function (and hence should have a “d” instead of a “δ”? HINT: It is if it equals a collection of state functions. 4.85 Defining Enthalpy We define a function which we call the enthlapy, H, as H = E + PV Write the differential of this expression and then apply a constant pressure restriction. Does qP from problem 84 equal dH (or ∆H for a finite change)? 4.86 Enthalpy What is the change in enthalpy? 4.87 Enthalpy, Exothermicity, and Spontaneity The enthalpy change for a reaction is -5.4 kJ/mole. Is the process exothermic? Is the process spontaneous? 4.88 Enthalpy Change Use tables of enthalpies of formation to determine the enthalpy change for the process 2SO2 (g) + O2 (g) = 2SO3 (g) under standard conditions. What do those numbers tell you about the spontaneity of this reaction? 4.89 Rationalizing an Enthalpy Change Give a reason why the enthalpy change is what it is in the reaction in the last problem. HINT: Think about what is happening. 4.90 Enthalpy of Vaporization What is the sign of the enthalpy of vaporization of a liquid, any liquid? 4.91 Enthalpy of Vaporization Give a molecular explanation for your answer to the last problem. 4.92 Enthalpy of Vaporization The enthalpy of vaporization of water is 40.7 kJ/mole and that of fusion of water is 6.01 kJ/mole. Assuming these values are independent of temperature, what is the enthalpy of sublimation of ice to water vapor at -10o C? 4.93 Enthalpy Change and Hess’s Law State Hess’s Law. Be articulate. 4.94 Enthalpy of Combustion Describe the process of combustion. What is the enthalpy of combustion? Chm 118 Exercise Set 4 1.104 12 4.95 Enthalpy of Combustion The standard enthalpy of combustion of propane is -2220; of C(gr), -394; and of dihydrogen, -286; all in kJ/mole of the substance named. Find the enthalpy change for the conversion of graphite into propane by reaction with dihydrogen. 4.96 Enthalpy Change Find the enthalpy change for the reaction (not balanced) B2 O3 (s) + CaF2 (s) = BF3 (g) + CaO(s) HINT: You will need to look up some numbers. 4.97 Enthalpy Change and Calorimetry Describe how a calorimeter works. 4.98 Enthalpy Change and Calorimetry How would you calibrate a calorimeter? 4.99 Quantitative Calculation of Entropy Change; Review of Concepts What is our equation relating the change in energy to the change in the number of microstates in the predominant configuration? There were two assumptions made during the derivation of this equation: what are they? 4.100 Quantitative Calculation of Entropy Change We have dE dlnW = kb T How is entropy related to W? How is entropy related to dE and T? 4.101 Changes During a Rapid Expansion To calculate the entropy change we need to evaluate dE = q when the temperature is constant. Imagine one mole of an ideal gas in a cyclinder with a piston at a pressure of 10 atm and a temperature of 298K held in place with a pin. We remove the pin and let it expand against an external pressure of one atm in a large temperature bath at 298K. Does this occur quickly? How much work is done? When (rapidly or slowly) is it done? Does the energy of the gas decrease because of the work done? Does heat flow from the temperature bath to the system? When (rapidly or slowly)? Is the temperature of the gas constant as this heat flows? Make an approximate plot of Tgas versus t. Can you use the heat that the gas absorbs to calculate the entropy? Why or why not? 4.102 Entropy is a State Function “Entropy is a state function.” Use our statistical definition in terms of the number of microstates in the predominant configuration to justify this. HINT: A state function is independent of path. 4.103 Finding the Correct Path to Calculate Entropy Since entropy is a state function, if we can find a path in which we can calculate the entropy change, and that path connects the same initial and final states, then we have the entropy change. Our task is then to find the appropriate path. What must be true for this path, given the issues raised in problem 101. What would that path be called. Chm 118 Exercise Set 4 1.113 13 4.104 Quantitative Calculation of Entropy Change Do a reversible expansion of the gas in problem 101 to the same final state and determine the heat. Use that to find the entropy change of the system. 4.105 Analysis of the Large Surroundings Our surroundings in the expansion in problem 101 are large. If they start at T = 298K and they are really large, say roughly the size of the Pacific Ocean, what will the final temperature be? Do the surroundings absorb heat at constant temperature? 4.106 Quantitative Calculation of Entropy Change Calculate the entropy change of the surroundings in the expansion in problem 101. Find the entropy change of the universe. Is the process spontaneous? 4.107 Calculation of Entropy Change for Vaporization of Water Calculate the entropy change of the universe when a mole of liquid water is converted to a mole of gaseous water at a pressure of one atm against an external pressure of 1 atm and a temperature of 298K. HINTS: Remember that the temperature does not change during such a conversion. You will have to look up a number. 4.108 Calculation of Entropy Change for Vaporization of Water Repeat the last problem—find the entropy change of the unverse–at a temperature of 373K assuming that the enthalpy and entropy of the conversion are not a function of temperature. 4.109 Entropy and Pressure, a Review Does the entropy of a mole of gas depend on the pressure? State why. 4.110 Analysis of Spontaneity in a Real System with All Species in Their Standard State Seldom are we interested in gases expanding. In most instances of interest to a chemist, we are focussed on a reaction. The concept of spontaneity in a reaction depends, as we shall see, on the states of the various components. For a while we will deal with the question of spontaneity when all chemicals are in their standard states, one atm pressure and pure. To do this, we can proceed in one of two equivalent thinking processes: (1) We can imagine a very large vessel of reactants and products, with all species at one atm pressure and then allow a mole of reactants to move to products without that conversion causing any significant change in pressure of any reagent or product; or (2) we can move a small amount of reactants to products so that pressures do not change, compute the entropy change for that small amount, and convert it to a “per mole” number. The description of this system is “all species in their standard state”. Compute the enthalpy change when CaCO3 decomposes into CaO and CO2 (g) with all species in their standard state. If this occurs at 25o C, what is the change in the entropy of the surroundings? 4.111 Analysis of Spontaneity under Standard Conditions Find the entropy change of the system for the reaction in the last problem. 4.112 Analysis of Spontaneity and Temperature with All Species in Their Standard State Is the reaction of the last two problems spontaneous at room temperature? What would happen at a higher temperature? Show your answer to this second question mathematically. Chm 118 Exercise Set 4 1.124 14 4.113 Entropy of Surroundings’ Change with All Species in Their Standard State The reaction to produce formaldehyde is H2 (g) + CO(g) = H2 CO(g) At T = 25o C, ∆H o = 1.96 kJ/mole and ∆S o = -109.6 J/mole. Find the change in entropy of the surroundings when one mole of formaldehyde is produced at 25o C under standard conditions. 4.114 Entropy of Surroundings’ Change with All Species in Their Standard State If the temperature for the formation of formaldehyde (see last problem) is raised to 50o C, what will happen to the value of the change in entropy of the surroundings provided ∆H o and ∆S o are approximately independent of temperature over this range. 4.115 Spontaneity with All Species in Their Standard State Is the production of formaldehyde (see problem 113) spontaneous at 25o C? 4.116 Spontaneity with All Species in Their Standard State Will KClO3 (s) decompose into KClO4 (s) and KCl(s) at 25o C? Consider the change in entropy of the universe and be quantitative. 4.117 Spontaneity and Temperature with All Species in Their Standard State What, if any, will be the effect of a higher temperature upon the decomposition of KClO3 (see last problem)? 4.118 Enthalpy Change Find ∆H o for the reaction 2Mg(s) + O2 (g) = 2MgO(s). 4.119 Sign of Entropy Change Estimate the sign of ∆S o for the reaction in problem 118. 4.120 Entropy Change Find the exact magnitude of ∆S o for the reaction in problem 118. 4.121 Thermodynamic Cycle There are a number of thermodynamic functions that are “state” functions. What does this mean? Give some examples. 4.122 Thermodynamic Cycle What is sublimation? Write a thermodynamic cycle that relates sublimation to two other well-known processes. HINT: The final state of a sublimation is a gas; the initial is a solid. Think about an indirect way to get from solid to gas. 4.123 Entropy Change of Sublimation Predict the sign and magnitude of the entropy of sublimation relative to the entropy of fusion (melting) of the same substance. Chm 118 Exercise Set 4 1.131 15 4.124 Definition of a New Function of Great Use Let’s invent a new function called G that is composed of previously defined functions: G = E + PV − TS (4) Find the total differential of this equation. Then assert the conditions that T and P are constant to get an expression for dG. Finally insert the definition of dH from problem 85 and show that dGP,T = dHP − T dST or, for a finite change in the standard state ∆Go = ∆H o − T ∆S o (5) 4.125 The Equivalence of dG the Entropy of the Universe We can take equation 5 and change the enthalpy change from a system value to one for the surroundings, then divide all terms by T. Show that this equates -∆Go /T with ∆S o univ . 4.126 Spontaneity What is the criterion for spontaneity in terms of WP C ? 4.127 Spontaneity What is the criterion for spontaneity in terms of ∆S univ ? 4.128 Spontaneity What is the criterion for spontaneity in terms of ∆G? 4.129 The Pressure Dependence of the Free Energy Change of a Substance Go back to equation 4 and take the total differential. Now substitute into this expression the value for dE from the first law where we specify a reversible process so that TdS = δq/T. Show that you get dG = V dP − SdT (6) 4.130 Pressure Dependence of the Free Energy Change of a Substance Take equation 6 and integrate it between the limits of pressure = 1 atm and some arbitrary pressure, P at constant T, for an ideal gas. Note that you must substitute V for the equivalent perfect gas expression nRT/P. Show that your answer is P o (7) G = G + RT ln Po where Go is the G when P = Po , the pressure in some “standard” state. Chm 118 Exercise Set 4 1.138 16 4.131 The Change in Free Energy of a Mixture The function G is a state property and hence must be defined by the conditions of the state. If we consider a mixture of chemicals, say A, B, and C, then G must be a function of T and P, as always, but also the number of moles of each of the various components. We could write this as G = G(T, P, nA , nB , nC ). Show that at constant T and P that the change in G would be given by X ∂G X dG = dni = µi dni (8) ∂n i nj ,T,P i i where i runs over, in our specific case, the three components A, B, and C, and nj is all of those n except ni . In the second equality I have inserted, as is conventionally done, the ∂G chemical potential, µi , which is simply the name for the value of ( ∂n )nj ,T,P , which is also i called a partial molar free energy. 4.132 Partial Molar Free Energy Look at the definition of µi in the last problem and see if you can justify the following verbal description of a partial molar quantity, stated for reasons of concrete appreciation in terms of a partial molar volume: “The partial molar volume of substance i is the change in volume (dV) of the mixture (of which it is a component) when a small number of moles (dni ) of substance i is added.” 4.133 Extent of Reaction For clarity, let’s consider a specific reaction: A + B = C. and let’s define a parameter ξ that we define as the extent of reaction dξ = 1 dni νi where νi is the stoichiometric coefficient of reagent i and is a negative quantity for reactants. Show that for the reaction above that a value of ξ = 0.5 corresponds to the correct number of moles for the point where the reaction is half over if the initial number of moles of A and B are both 1. 4.134 Extent of Reaction For the reaction A + 3B = 2C, show that the extent of reaction varies from zero at the beginning of the reaction to 1 at the end (no limiting reagent left) if the initial number of moles of A, B, and C are respectively 1, 4, and 1. 4.135 Extent of Reaction For the reaction A + 3B = 2C, show that the extent of reaction varies from zero at the beginning of the reaction to 13 at the end (no limiting reagent left) if the initial number of moles of A, B, and C are respectively 3, 1, and 2. 4.136 Change in Moles Expressed with dξ Whatever the number of moles, ξ is a single parameter that tells us how far we have gone from reactants toward products. We can then express the change in any component’s number of moles with dξ. Write such an expression for dni . HINT: An exceptionally easy problem given the definition of ξ in problem 133. Chm 118 Exercise Set 4 1.147 17 4.137 dG for a Specific Problem Write the second of the two equations labeled 8 for the specific reaction A + B = C. 4.138 Finding the Change in G Use your result from the last problem for dG and your result for dni from problem 136 and evaluate the second of the two equations labeled 8 for dG in terms of µA , etc., and dξ. 4.139 An Expression for the dG/dξ Rearrange your result from the last problem to obtain dG/dξ, whose sign gives us the spontaneity of a reaction at the conditions specified by the various µi . 4.140 The Pressure Dependence of the Chemical Potential The last problem suggests our job now is to find how the µi vary with conditions. Since µi is a partial molar quantity, the variation with conditions is the same as that for G, which we found in problem 130 except that the variables need to refer to the individual species i. For a gaseous example, we get Pi o µi = µi + RT ln (9) Po where Pi is the pressure of substance i and Po is the standard state, 1 atm. When is µi equal to µoi ? 4.141 A Final Version of the Free Energy Change as ξ changes Insert the value of µi from equation 9 for i = A, B, and C, respectively, into your answer in problem 139 and rearrange. Show the result is PC /PCo dG o (10) = ∆G = ∆G + RT ln dξ PA PB /(PAo PBo ) where ∆Go = µoC - µoA - µoB and where all the Po could be set to unity. 4.142 Free Energy Change Find ∆Go for the reaction 2Mg(s) + O2 (g) = 2MgO(s). 4.143 Entropy Change in the Universe Give an explanation of the spontaneity of the reaction in problem 142 in terms of the entropy of the universe and its components. Be complete. 4.144 Free Energy Change Find the free energy change for the reaction in problem 142 if the pressure of O2 (g) is 1.0 ×10−4 atm. Is the reaction spontaneous under these conditions? HINT: Since the two solids have rather small change in volume with a change in pressure, their Go values are nearly independent of P. 4.145 Free Energy Change Find ∆Go for the reaction CH3 OH(g) = CO(g) + 2H2 (g) at 298K. Chm 118 Exercise Set 4 1.155 18 4.146 Free Energy Change Find ∆G for the reaction in the last problem if the pressure of CH3 OH(g) is 1.0 atm and the pressures of CO(g) and H2 (g) are both (a) 1.0×10−4 atm and (b) 0.01 atm. Is the reaction spontaneous in the forward direction under any of these conditions? in the reverse? 4.147 Free Energy Change Find ∆G for the reaction in problem 145 if the pressure of CH3 OH(g) is 0.1 atm, that of CO(g) is 1.0×10−2 atm, and that of H2 is 5.0×10−3 atm. 4.148 Free Energy Change and Temperature Assuming that ∆H o and ∆S o are independent of T, find ∆Go for the reaction in problem 145 at 50o C and 150o C. 4.149 Understanding Chemistry A long tube is prepared containing SiCl4 (g); at one end ot this tube is placed some impure Si(s) and that end of the tube is heated. Over time at the cool end of the tube pure Si(s) forms. Account for this result using thermodynamic arguments. HINT: Think about the conproportionation of Si(IV) and Si(0): SiCl4 + Si = 2SiCl2 4.150 Free Energy and Pressure Write the chemical equation that represents benzene liquid vaporing to benzene gas at a pressure of one atmosphere. HINT: We can express the full conditions of a gas, say X, in a reaction via X(g, p = 0.5 atm), for instance. 4.151 Free Energy and Pressure Write the chemical equation that represents benzene liquid vaporing to benzene gas at a pressure of p atmosphere. 4.152 Free Energy and Pressure We can couple the two equations from the last two problems by letting benzene liquid at p = 1 atm. go to benzene liquid at p = p atm. What is your estimate of ∆H for this process. HINT: Think about what energy changes take place. What is your guess about ∆S for this process? HINT: Is anything happening to the benzene liquid? 4.153 Free Energy and Pressure To complete the cycle we have been working on, we could let benzene gas at p = 1 atm. go to benzene gas at p = p atm. To be concrete, if p = 0.1, and we work at constant temperature, we are changing the pressure of benzene gas at constant temperature. What does this do to the energy under the assumption of ideal gas behavior? What does it do to the entropy? 4.154 Free Energy and Pressure Write the complete thermodynamic cycle from the information in the last four problems. Let the value of ∆G at p = 1 atm. be ∆Go and the value of ∆G at p = p be ∆G. Write one in terms of the other and additional terms as appropriate. You have just produced what we derived more rigorously in problem 141. Chm 118 Exercise Set 4 1.167 19 4.155 Free Energy Change and Pressure Free energies of formation of benzene liquid and benzene gas at 298K are 124.5 and 129.7 kJ/mole, respectively. Find the value of ∆G when the pressure of benzene gas is 10 mm Hg. HINT: 760 mm Hg is one atmosphere. 4.156 Free Energy Change and Pressure Use data from the last problem to find the value of ∆G when the pressure of benzene vapor is 400 mm Hg at 25o C. 4.157 Free Energy Change and Pressure What is the value of the pressure of benzene vapor at equilibrium with benzene liquid at 25o C? 4.158 Free Energy Change and Pressure Find the value of ∆G at 298K for the reaction in problem 145 if the pressures are: PCH3 OH(g) = 0.15 atm. PCO(g) = 0.01 atm, PH2 (g) = 2.0 atm. 4.159 Free Energy Change and Pressure Find the value of ∆G at 298K for the reaction in problem 145 if the pressures are: PCH3 OH(g) = 10.0 atm. PCO(g) = 0.10 atm, PH2 (g) = 0.002 atm. 4.160 Equilibrium Constant Find the equilibrium constant at 298K for the reaction in problem 145. 4.161 Equilibrium Constant Find K at 298K for the reaction 2NO(g) + Cl2 (g) = 2NOCl(g). 4.162 Free Energy Change and Specified Pressure Find ∆G at 298K for the reaction in the last problem when all pressures are 0.001 atm. 4.163 Free Energy Change and Pressure Is the reaction I2 (g) + Cl2 (g) = 2ICl(g) spontaneous when all pressures are 1 atm? HINT: The free energy of formation of ICl(g) is -6.1 kJ/mole at 298K. 4.164 Free Energy Change and Pressure Is the reaction in problem 163 spontaneous when the pressure of ICl is 0.001 atm and the other pressures are 1.0 atm? 4.165 Free Energy Change and Pressure Is the reaction in problem 163 spontaneous when the pressure of ICl is 10.0 atm and the other pressures are 1.0 atm? 4.166 Free Energy Change and Pressure Offer a rationalization as to why the standard free energy change for the reaction in problem 163 is so small in magnitude. Chm 118 Exercise Set 4 1.176 20 4.167 Free Energy Change and Equilibrium Constant For the reaction ZnF2 (s) = Zn2+ (aq) + 2F– (aq) the value of K is 0.003. Find the value of ∆Go for this reaction. 4.168 Free Energy Change and Concentration If you make 50 mL of a 2 M solution Zn(NO3 )2 and 50 mL of a 4.0M solution of CsF, and mix them, what will happen? HINT: Ignore any reactions of nitrate ion and cesium ion, but pay attention to the last problem. 4.169 Some Thermodynamic Statements For each statement, indicate if it is true or false. If false, fix it. If the statement is just silly, say so and fix it. • All spontaneous reactions take place quickly. • If a reaction is spontaneous, the reverse reaction is nonspontaneous. • All spontaneous reactions release heat. • The boiling of water at 100o C and 1 atm pressure is a spontaneous process. • If a process increases the disorder of the particles of a system, the entropy of the system increases. • The energy of the universe is a constant. The entropy of the universe just grows and grows. • Both the entropy change of the system and the entropy change of the surroundings are zero at equilibrium. • A spontaneous process has a positive change in entropy. 4.170 Free Energy Change List five things that ∆G is. 4.171 Enthalpy Change What is the enthalpy change? 4.172 Entropy What is entropy? 4.173 Boiling Point Assuming that ∆H o and ∆S o are independent of temperature, find the boiling point of CCl4 (`). 4.174 Boiling Point The normal boiling point of trimethylphosphine is 38.4o C. Its vapor pressure at -45.2o C is 0.017 atm. Find ∆H o , ∆S o , and the vapor pressure at 15o C. Chm 118 Exercise Set 4 1.184 21 4.175 Enthalpy Change Find the enthalpy change for the reaction BaCO3 (s) = BaO(s) + CO2 (g). 4.176 Enthalpy Change Find the enthalpy change for the reaction BaSO4 (s) = BaO(s) + SO3 (g) 4.177 Enthalpy of Combustion The enthalpy of combustion of acetone, CH3 C(O)CH3 , is -1821.4 kJ/mole and that of propanal, CH3 CH2 C(O)H, is -1845 kJ/mole. Which compound is more stable? By how much? 4.178 Spontaneity and Concentration For the reaction HNO2 (aq) = H+ (aq) + NO–2 (aq) determine if the process is spontaneous to the right when the concentrations are: [HNO2 ] = 0.010; [H + ] = 0.00010; [NO–2 ] = 0.0150. 4.179 Spontaneity and Concentration For the reaction HNO2 (aq) = H+ (aq) + NO–2 (aq) determine if the process is spontaneous to the right when the concentrations are: [HNO2 ] = 0.010; [H + ] = 0.0010; [NO–2 ] = 0.00150. 4.180 Spontaneity and Concentration For the reaction HNO2 (aq) = H+ (aq) + NO–2 (aq) determine if the process is spontaneous to the right when the concentrations are: [HNO2 ] = 0.0010; [H + ] = 0.0010; [NO–2 ] = 0.0150. 4.181 Mixed Equilibrium Constants Imagine you have a chemical reaction with a ∆Go 1 and an associated K1 . You also have a second chemical reaction with a a ∆Go 2 and an associated K2 . Show, using the additive property of ∆Go , that the equilibrium constant for the reaction that is the sum of chemical reactions 1 and 2 is K = K1 K2 . 4.182 Determining ∆H o and ∆S o from Equilibrium Constants Use equation 5 and the relationship between ∆Go and lnK to show if you determine K at various temperatures and then plot ln[K] versus 1/T that you can obtain ∆H o and ∆S o . 4.183 Review of Qualitative Microstate Reasoning Figure 3 show two sets of energy levels. In set A the long (black if you are viewing in color) lines represent the quantum levels of a reactant and the short (red) lines those of a product of a reaction. Is this reaction endo- or exothermic? Will it occur at higher temperatures to produce more product than reactant? Explain your reasoning. Chm 118 Exercise Set 4 1.188 22 Figure 3: Figure for Problems 183 and 184. The energy level in arbitrary units is listed on the left. 4.184 Review of Qualitative Microstate Reasoning There is a second set of energy levels in Figure 3, set B. The long (black if you are viewing in color) lines represent the quantum levels of a reactant and the short (red) lines those of a product of a reaction. Is this reaction endo- or exothermic? Will it occur at higher temperatures to produce more product than reactant? Why? Try to give an example of a system for which these quantum levels might occur. 4.185 Units What are the “units” of kb T? 4.186 Quantitative Application in Figure 3B In the following table, for several values of kb T, in the arbitrary units used for the spacings of the energy levels, are given the fractions of the total particles in “reactants” (long, black lines) and “products” (short red lines) quantum levels illustrated in Figure 3B. How, in general, did I get these data? Show using these values that the equilibrium constant for reactants going to products are consistent with your qualitative observations from the last problem. kb T 0.2 0.5 1.0 1.4 f(A) 0.9998 0.9196 0.5267 0.3610 f(B) 0.0001 0.0874 0.4733 0.6388 4.187 Estimations Using Thermochemical Data Make an estimate (why is it an estimate?) of the temperature at which SO3 and a mixture of SO2 and O2 are present in equilibrium with all pressures equal to one atmosphere? Chm 118 Exercise Set 4 1.197 23 4.188 Balancing Oxidation-Reduction Equations Write a stoichiometrically balanced reaction for the process Cl2 O7 + H2 O2 = ClO–2 + O2 in basic solution. 4.189 Balancing Oxidation-Reduction Equations Write a stoichiometrically balanced reaction for the process Cr3+ + MnO2 (s) = Mn2+ + CrO2– 4 in basic solution. 4.190 Balancing Oxidation-Reduction Equations Write a stoichiometrically balanced reaction for the process H2 SeO3 + ClO–3 = HSeO–4 + Cl2 in acidic solution. 4.191 Potential of Electrochemical Cell Find the potential for the cell Pt(s) | Fe3+ (aq),Fe2+ (aq) k Ag+ (aq) | Ag(s) if all concentrations are 1.00 M. 4.192 Potential of Electrochemical Cell Find the potential for the cell Pt(s) | Fe3+ (aq),Fe2+ (aq) k Ag+ (aq) | Ag(s) if all concentrations are 1.00 M except [Ag+ ] = 0.01M. 4.193 Potential of Electrochemical Cell Find the potential for the cell Ag(s) | AgI(s) | Ag+ (aq, 0.025M) k Cl– (aq, 0.67M) | AgCl(s) | Ag. 4.194 Concentration Cell Explain what is behind the workings of a “concentration” cell. 4.195 Concentration Cells Will the cell Ag | Ag+ (aq, 0.0010 M) k Ag+ (aq, 1.00 M) | Ag develop a voltage? What is it? Why does it develop a voltage? 4.196 Concentration Cells Assume that in a nerve cell that the concentration of potassium ion inside the cell is 25 times as great as that found outside. What potential is developed across the cell wall? 4.197 Electrochemical Cells and Entropy Assume for this problem that ∆H o and ∆S o are independent of temperature. Show that if you measure the potential of a cell at temperatures T1 and T2 (and get E1o and E2o , that nF (E2o −E1o ) . you can evaluate the value of ∆S o as (T2 −T 1) Chm 118 Exercise Set 4