# ENTROPY AND THE SECOND LAW OF THERMODYNAMICS

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1 ENTROPY AND THE SECOND LAW OF THERMODYNAMICS Energy Reservoir The system consists of the red circles in the blue box. Energy and entropy fl ow out of the system. TIME Additional Energy is added to the system, The system decreases in entropy by DR. STEPHEN THOMPSON MR. JOE STALEY The contents of this module were developed under grant award # P116B from the Fund for the Improvement of Postsecondary Education (FIPSE), United States Department of Education. However, those contents do not necessarily represent the policy of FIPSE and the Department of Education, and you should not assume endorsement by the Federal government.

2 CONTENTS 2 Introduction To Entropy 2 Energy Disperses 3 Entropy 4 Enthalpy And Entropy 5 Thermal Entropy 6 Confi gurational Entropy 7 Confi gurational Entropy: Cellular Representation 8 Confi gurational Entropy: Combined Representation 9 Dispersible Energy 10 Diffusion 11 Liquid Crystal 12 Salt Dissolving In Water 13 The Pfeffer Tube 14 The Second Law Of Thermodynamics 15 Gibbs Free Energy 16 Gibbs Free Energy And Temperature 17 Gibbs Free Energy And Temperature 18 How Entropy Can Decrease (In A System) 19 Periodic Entropy Of The Elements

3 INTRODUCTION TO ENTROPY ENERGY DISPERSES Metal Time Styrofoam Time Time Time TIME Time Time In the experiments pictured above, the blue represents cooling, or loss of thermal energy. Is the evaporation of water exothermic or endothermic.? What is the evidence? If it is endothermic, how can it proceed spontaneously in the isolated system where the petri dish is placed on styrofoam? In the picture above the red ink represents energy. As time proceeds there is the same amount of ink (energy) but it spreads out, becomes less concentrated, disperses. Entropy is the measure of this dispersal The second law of thermodynamics says that the opposite change is impossible in an isolated system. Spontaneous endothermic reactions do occur and that means that there must be another factor than enthalpy involved. Scientists call this factor entropy. We have personal experience of entropy when we feel the coolness of evaporation. 2

4 ENTROPY Suppose three molecules have a total of three quanta of energy to share between them and that each molecule can occupy one of four energy states requiring zero, one, two or three quanta to occupy. E 3 E 2 E 1 E 0 o x o y o z Macrostate 1 has one possibility, that is, one microstate. E 3 E 2 E 1 E 0 o x o y oz o x o y o z o x E 3 E 2 E 1 E 0 o y o z o x o y o z o x o y o z o x o y o z o x o y o z o x o y Macrostate 2 has three possibilities, that is, three microstates. Macrostate 3 has six possibilities, six microstates. o z o x o y Suppose each microstate is as likely to be occupied as any other microstate. What is the most likely macrostate to be occupied? Suppose that the system shifts from one microstate to another at random times, what proportion of the time will the system be in macrostate 1? in macrostate 2? in macrostate 3? Assume the three quanta of energy are distributed among four molecules. How many macrostates will there be and how many microstates will there be for each macrostate? Suggestion: use drawings like the ones above to fi gure this out. Assume four quanta of energy are distributed among four molecules with four available energy states. How many macrostates will there be and how many microstates to each macrostate? o z In chemistry there are several different means by which energy can be dispersed and thus entropy created. These include: 1. The number of molecules among which the entropy can be shared. The rest of these examples refer to the same number of molecules: 2. The volume of space which the molecules can occupy. 3. The freedom with which the molecules can move about that space, e.g, the difference between a solid and a liquid. This would include the freedom to change location and, in the case of nonspherical molecules, the freedom to change oritentation or rotation. 4. The amount of energy available, which determines the range of energy states which the molecules can occupy. 5. The complexity of the molecules, which determines how many rotational and vibrational states they can have. A modern way to describe entropy is to say that entropy increases with the number of ways energy can be distributed in a system. More Particles Added More Particles Due to Chemical Reaction Larger Volume In each of the above sets of pictures, there is a change between the left hand side and the right hand side. Explain how the change would increse the number of ways energy can be distributed in the system.. We have described several sources of entropy. You describe several conditons that can restrain the growth of entropy or reduce it in a system. 3

5 ENTHALPY AND ENTROPY Consider this experiment: a drop of water is placed in a clean Petrie dish and the cover is put on. What happens and and what are the causes? The system is the Petri dish and its contents. The surroundings include the table and the air outside of the Petri dish. In the pictures below. each column shows the same state of the system, but from a different perspective. The fi rst column shows just the changes in molecular location. The second column shows changes in energy (temperature) and the third column shows changes in entropy. Temperature Increase Temperature Decrease Entropy Increase Entropy Decrease MOLECULES ENERGY ENTROPY TIME TIME Describe what is happening to the molecules. What do you think will happen later? Why are the gas phase molecules warmer than the liquid phase in the intermediate time. Why do they return to equal temperature? 4 In the energy column, the gas phase molecules return to their original temperature. Why doesn t the same hold true for entropy? Is entropy conserved?

6 THERMAL ENTROPY FUEL TO FUMES 5

7 CONFIGURATIONAL ENTROPY MIXING OF GASES UNLIKE LIKE SOLUTION 6

8 CONFIGURATIONAL ENTROPY: CELLULAR REPRESENTATION NUMBER OF MOLECULES NUMBER OF STATES Ω = 144 Ω =4 Ω = 144x143 Ω = 144! 72! Ω = 144x143x142 MOLECULAR DISSOCIATION Ω = 144x143 Ω = 144x143x142x141 7

9 CONFIGURATIONAL ENTROPY: COMBINED REPRESENTATION EXPANDING GAS Molecule Molecular Weight Ω = 1 Water 18 Dinitrogen Dioxygen Argon Carbon Dioxide 144! Ω = 72! 8

10 DISPERSIBLE ENERGY Enthalpy Entropy Universe Surroundings System In this pictorial representation, the system is shown qualitatively with an original enthalpy and entropy. In the surroundings - the rest of the universe - the original state is shown blank, since the actual amount of enthalpy and entropy in the universe is uncalculated and since it is the change which is relevant. H Surroundings = Η System If S System = 0, then S Universe = S Surroundings = ( H/T) System 9

11 DIFFUSION Enthalpy Entropy Universe Surroundings System 10

12 LIQUID CRYSTAL EXPERIMENT INTERPRETATION Universe Surroundings System Enthalpy Entropy The system is a horizontal rectangle of encapsulated liquid crystal (ELC). To begin with, the ELC is in thermal equilibrium with its surroundings. The surroundings include the surface upon the which ELC rests and the air above and around it. A drop of water is placed upon the surface of the ELC. Assume that the water is originally at the same temperature as the system and surroundings (the water is part of the surroundings). Experiment shows that the ELC cools beneath the drop as the drop evaporates and then that the cool region both spreads and diminishes in intensity. After the drop is completely evaporated the ELC eventually returns to its equilibrium temperature. The cooling is due to a warmer than average fraction of the water molecules escaping from the drop; although they lose energy to the work function of the water surface, they nevertheless retain enough energy to cool the drop. Since the ELC is cooled its entropy is decreased, unless there is an increase in some confi gurational entropy. The entropy of the water is confi gurationally increased by evaporation by the energy drawn from the ElC. And since the water is part of the surroundings, the entropy of the surroundings is thereby increased. Also, the thermal energy of the surroundings is increased. Eventually we see, as and/or after the water fi nishes evaporating, the cool region of the ELC spreads out, diminishing in intensity, and eventually disappears, from which we conclude that the ELC returns to thermal equilibrium with its surroundings. The entropy of the ELC also re-arises to its original level through absorption of heat from the surroundings. The surroundings will correspondingly return to its same energy level but will retain an increase in entropy; consider that the water which was once a liquid drop is now a gas.lorem quiscip umsan heniametum ipit, 11

13 SALT DISSOLVING IN WATER Ionic solvation in water has a dual entropy effect. The entropy is increased by the additonal space occupied by the salt ions, e.g., Na + and Cl and the entropy is decreased by the orientation of the water molecules about the ions

14 ENTROPY AND THE SECOND LAW OF THERMODYNAMICS THE PFEFFER TUBE H 2O Semi-permeable membrane NaCl 13

15 THE SECOND LAW OF THERMODYNAMICS In Thermochemistry we have seen that reactions are infl uenced by the comparative enthalpies of reactants and products. Reactions tend to occur which lower the enthalpy. However, this is not the whole story; there is another factor involved, called entropy. Entropy has often been described as disorder, which is only partially correct. Here we will look at some types of entropy which are relevant to chemical reactions. In classical thermodynamics, e.g., before about 1900, entropy, S, was given by the equation S = Q/T where S is the entropy change in a system, Q is heat energy added to or taken from the system, and T is the temperature of the system. The units for entropy are Joules/Kelvin, except in chemistry we work with the quantity of a mole, so in chemistry the units of entropy are Joules/mole-Kelvin. Around 1900 Boltzmann found another basis for entropy as the number of ways a system can be in a given state (actually the logarithm of that number). For example, there are vastly more ways the air molecules in a room can be spread out all over the room than there are ways in which they would all be in one side of the room. Nature just does the most likely thing, when nothing prevents that. This is formally called the Second Law of Thermodynamics and can be stated as follows: For combined system and surroundings, entropy never decreases. Actually, it always increases. This is really what makes things happen. The fi rst law of thermodynamics, that energy is conserved, just ells us what can happen; it is the second law that makes things go. One of the early statements of the Second Law of Thermodynamics is that heat always fl ows downhill. More exactly, if two bodies are in thermal contact, heat energy will always fl ow from the warmer to the cooler one. In terms of heat energy, describe what happens when two bodies at the same temperature are brought into thermal contact? Does it depend upon the sizes of the bodies? Explain your answer. Another statement of the Second Law is that there is a state variable called entropy which never decreases and, in effect, always increases. Entropy Time In the box outlined above, the green dot represents the entropy at some starting time. Time passes as we go to the right. Draw a line or curve from the green dot to the right side of the box which represents a possible chart of the amount of entropy. Suppose you know that over a certain interval of time the entropy of a system decreased by the amount, A. What can you say about the entropy of the surroundings over that same interval of time? Compare and contrast the fl ow of heat energy according to the Second Law of Thermodynamics with the fl ow of water on earth. Describe some of the ways the world would be different if heat energy could fl ow from a cooler to a hotter body. Or what if that always happened? 14

16 GIBBS FREE ENERGY The enthalpy of a system is the energy of the system at constant temperature and pressure. However, not all of that energy is available for the system to do work or contribute to a chemical reaction. There is another factor, which we have introduced as entropy. In order to relate the entropy to the enthalpy we need to multiply the entropy by the temperature (in Kelvin). Temperature H T S Gibbs free energy, G is defi ned by G = H - TS where H is the enthalpy, T is the temperature (in Kelvins), and S is the entropy. In a chemical reaction, R P (R are reactants and P are products) at a constant temperature we have G = H T S. If G < 0 the reaction may proceed spontaneously to the right. If G = 0 the reaction is in equilibrium. If G 0 the reaction may proceed spontaneously to the left. The bar graph above shows H and T S for the same chemical reaction at different temperatures. At which temperature is the reaction in equilibrium? Which temperature will maximize the reactants? Which temperature will maximize the products? Since S (entropy) has units of kj mol 1 K 1 (kilojoules per mole-kelvin), when we multiply it by K (temperature in Kelvin) we get units of kj mol 1 (kilojoules per mole), which are the same units as energy. Entropy times temperature is not actually an energy but it controls the availability of energy to do work, such as making chemical reactions happen. These four ChemLogs show four possible sign combinations for Gibb s Free Energy: G = H T S REACTION TYPE ONE + 0 REACTION TYPE TWO + 0 G H REACTION TYPE THREE + 0 H T S T S G T S T S T S T S T S T S G H H G REACTION TYPE FOUR + 0 Which of the four reaction types above would be thermodynamically spontaneous? Why? Tell which reaction type each of the following reactions would fi t into and explain why. 1. H 2(g) + O 2(g) H 2 O (g) 2. H 2(g) + O 2(g) H 2 O (l) 3. H 2 O (l) H 2 O (g) 15

17 GIBB S FREE ENERGY AND TEMPERATURE 500 kj mol 1 T S vs Temperature for Diatomic Gases 400 kj mol 1 Cl CO 2(g) 2(g) O 2(g) F N 2(g) 2(g) 300 kj mol 1 T S H 2(g) 200 kj mol kj mol 1 0 kj mol 1 0 K 400 K 800 K 1200 K 1600 K 2000 K T (temperature) Using the chart above, describe the relationship, if any, between entropy and molecular weights. EVAPORATION OF WATER H 2 O (l) H 2 O (g) H f º= 44 kj/k mol at K S º = 119 J/K mol at K G f º = H f o T Sº If we make the reasonable approximation that H and S do not (signifi cantly) vary between T = 273 K and T = 373 K, then we can produce the following chart: 15 J K 1 mol J K 1 mol 1 G 0 J K 1 mol K 298 K T 373 K 16

18 GIBB S FREE ENERGY AND TEMPERATURE We know that when G < 0 a reaction is spontaneous and when G 0 a reaction is nonspontaneous. However, G is composed of two terms, an enthalpy term and an entropy term. When both terms pull G in the same direction, then situation is clear, but what can we say ingeneral about situations where the enthalpy and entropy terms are of opposite effect? Because the entropy term, T S, is the entropy multiplied by the temperature, we would expect temperature to be an important contributing factor and we are right. The Effect of Temperature on Spontaneity 1. At high temperatures the entropy factor, T S, predominates 2. At low temperatures the enthalpy factor, H, predominates. The chart below shows the separate terms, H and T S, which combine to give Gibb s free energy. Reactions below the dashed line are spontaneous, those above it are nonspontaneous J mol J mol kJ mol 1 Reactions at K 2400 A H 2(g) + 1 2O 2(g) H 2 O (g) 2000 B 4Fe (s) + 3O 2(g) 2Fe 2 O 3(s) 1600 C C 6 H 12 O 6(s) + 6O 2(g) 6CO 2(g) + 6H 2 O (g) 1200 D Si (s) Si (g) G G 0 D D D Reactions at 1000 K A H 2(g) + 1 2O 2(g) H 2 O (g) H 0 B 4Fe (s) + 3O 2(g) 2Fe 2 O 3(s) AAA G G < C C 6 H 12 O 6(s) + 6O 2(g) 6CO 2(g) + 6H 2 O (g) D Si (s) Si (g) B B Reaction at 4000K D Si (s) Si (g) 2400 T S C C 2800 kj mol -1 You can see that the transition from solid silicon to gaseous silicon (reaction D) moves to the right on the table as the temperature increases. For what values of S will this be true? The reaction Br 2(l) Br 2(g) has Η = 3 kj mol 1 and S = 93 J K 1 mol 1. Mark its location on the graph. 17

19 HOW ENTROPY CAN DECREASE (IN A SYSTEM) One way of stating the second law of thermodynamics is to say that in any (nonreversible, i.e., real) process the entropy of the system plus the entropy of the surroundings must always increase. If energy disperses and entropy increases how is it possible that some systems, such as living beings, can maintain their energy and not be quickly disolved by entropy? There are even systems in which entropy decreases; for example, water can be frozen into ice. This can happen if energy fl ows out of the system, carrying entropy with it. Energy Reservoir The system consists of the red circles in the blue box. Energy and entropy fl ow out of the system. TIME Additional Energy is added to the system, The system decreases in entropy Name some systems and processes where entropy decreases in the system. Carefully distinguish between the system and the surroundings and describe the energy and entropy changes which occur when entropy decreses in the system. As the universe expands it s temperature decreses. It is now about 2.7 K. And yet the second law of thermodynamics says that the entropy of the universe always increases. How can these facts be reconciled? 18

20 PERIODIC ENTROPY OF THE ELEMENTS ENTROPY OF THE ELEMENTS ENTROPY J K 1 mol 1 ATOMIC NUMBER In the ENTROPY OF THE ELEMENTS CHART you can see that several of the elements have much higher entropy than the rest. Using the atomic numbers, determine and list what elments these are. Describe what they have in common which results in their high entropies. ENTROPY INCREASING EVENTS The following events either always or ordinarily involve an increase in entropy, either in the system or the surroundings or both. Heating any substance. Phase change from sold to liquid and from liquid to gas. Any reaction that increases the number of moles of gas molecules. Mixing two different liquids or two different gases. Dissolving solids in liquids. 19

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