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# 16 Thermodynamics

## Chapter Outline

16.1 Spontaneity

16.2 Entropy

16.3 The Second and Third Laws of Thermodynamics

16.4 Free Energy

## Introduction

Among the many capabilities of chemistry is its ability to predict if a process will occur under specified conditions. Thermodynamics, the study of relationships between the energy and work associated with chemical and physical processes, provides this predictive ability. Previous chapters in this text have described various applications of thermochemistry, an important aspect of thermodynamics concerned with the heat flow accompanying chemical reactions and phase transitions. This chapter will introduce additional thermodynamic concepts, including those that enable the prediction of any chemical or physical changes under a given set of conditions.

## 16.1 Spontaneity

By the end of this section, you will be able to:

•  Distinguish between spontaneous and nonspontaneous processes
•  Describe the dispersal of matter and energy that accompanies certain spontaneous processes

In this section, consider the differences between two types of changes in a system: Those that occur spontaneously and those that occur by force. In doing so, we’ll gain an understanding as to why some systems are naturally inclined to change in one direction under certain conditions and how relatively quickly or slowly that natural change proceeds. We’ll also gain insight into how the spontaneity of a process affects the distribution of energy and matter within the system.

Spontaneous and Nonspontaneous Processes

Processes have a natural tendency to occur in one direction under a given set of conditions. Water will naturally flow downhill, but uphill flow requires outside intervention such as the use of a pump. Iron exposed to the earth’s atmosphere will corrode, but rust is not converted to iron without intentional chemical treatment. A spontaneous process is one that occurs naturally under certain conditions. A nonspontaneous process, on the other hand, will not take place unless it is “driven” by the continual input of energy from an external source. A process that is spontaneous in one direction under a particular set of conditions is nonspontaneous in the reverse direction. At room temperature and typical atmospheric pressure, for example, ice will spontaneously melt, but water will not spontaneously freeze.

The spontaneity of a process is not correlated to the speed of the process. A spontaneous change may be so rapid that it is essentially instantaneous or so slow that it cannot be observed over any practical period of time. To illustrate this concept, consider the decay of radioactive isotopes, a topic more thoroughly treated in the chapter on nuclear chemistry. Radioactive decay is by definition a spontaneous process in which the nuclei of unstable isotopes emit radiation as they are converted to more stable nuclei. All the decay processes occur spontaneously, but the rates at which different isotopes decay vary widely. Technetium-99m is a popular radioisotope for medical imaging studies that undergoes relatively rapid decay and exhibits a half-life of about six hours. Uranium-238 is the most abundant isotope of uranium, and its decay occurs much more slowly, exhibiting a half-life of more than four billion years (Figure 16.2).

As another example, consider the conversion of diamond into graphite (Figure 16.3).

The phase diagram for carbon indicates that graphite is the stable form of this element under ambient atmospheric pressure, while diamond is the stable allotrope at very high pressures, such as those present during its geologic formation. Thermodynamic calculations of the sort described in the last section of this chapter indicate that the conversion of diamond to graphite at ambient pressure occurs spontaneously, yet diamonds are observed to exist, and persist, under these conditions. Though the process is spontaneous under typical ambient conditions, its rate is extremely slow, and so for all practical purposes diamonds are indeed “forever.” Situations such as these emphasize the important distinction between the thermodynamic and the kinetic aspects of a process. In this particular case, diamonds are said to be thermodynamically unstable but kinetically stable under ambient conditions.

### Dispersal of Matter and Energy

As we extend our discussion of thermodynamic concepts toward the objective of predicting spontaneity, consider now an isolated system consisting of two flasks connected with a closed valve. Initially there is an ideal gas on the left and a vacuum on the right (Figure 16.4). When the valve is opened, the gas spontaneously expands to fill both flasks. Recalling the definition of pressure-volume work from the chapter on thermochemistry, note that no work has been done because the pressure in a vacuum is zero.

Note as well that since the system is isolated, no heat has been exchanged with the surroundings (q = 0). The first law of thermodynamics confirms that there has been no change in the system’s internal energy as a result of this process.

The spontaneity of this process is therefore not a consequence of any change in energy that accompanies the process. Instead, the driving force appears to be related to the greater, more uniform dispersal of matter that results when the gas is allowed to expand. Initially, the system was comprised of one flask containing matter and another flask containing nothing. After the spontaneous process took place, the matter was distributed both more widely (occupying twice its original volume) and more uniformly (present in equal amounts in each flask).

Now consider two objects at different temperatures: object X at temperature TX and object Y at temperature TY, with TX > TY (Figure 16.5). When these objects come into contact, heat spontaneously flows from the hotter object (X) to the colder one (Y). This corresponds to a loss of thermal energy by X and a gain of thermal energy by Y.

From the perspective of this two-object system, there was no net gain or loss of thermal energy, rather the available thermal energy was redistributed among the two objects. This spontaneous process resulted in a more uniform dispersal of energy.

As illustrated by the two processes described, an important factor in determining the spontaneity of a process is the extent to which it changes the dispersal or distribution of matter and/or energy. In each case, a spontaneous process took place that resulted in a more uniform distribution of matter or energy.

### Example 16.1Redistribution of Matter during a Spontaneous Process

Describe how matter is redistributed when the following spontaneous processes take place:

(a) A solid sublimes.

(b) A gas condenses.

(c) A drop of food coloring added to a glass of water forms a solution with uniform color.

Solution

(a) Sublimation is the conversion of a solid (relatively high density) to a gas (much lesser density). This process yields a much greater dispersal of matter, since the molecules will occupy a much greater volume after the solid-to-gas transition.

(b) Condensation is the conversion of a gas (relatively low density) to a liquid (much greater density). This process yields a much lesser dispersal of matter, since the molecules will occupy a much lesser volume after the solid-to-gas transition.

(c) The process in question is dilution. The food dye molecules initially occupy a much smaller volume (the drop of dye solution) than they occupy once the process is complete (in the full glass of water). The process therefore entails a greater dispersal of matter. The process may also yield a more uniform dispersal of matter, since the initial state of the system involves two regions of different dye concentrations (high in the drop, zero in the water), and the final state of the system contains a single dye concentration throughout.

Describe how matter and/or energy is redistributed when you empty a canister of compressed air into a room.

Answer: This is also a dilution process, analogous to example (c). It entails both a greater and more uniform dispersal of matter as the compressed air in the canister is permitted to expand into the lower-pressure air of the room.

## Exercises

### Question 16.1

16.1

1. What is a spontaneous reaction?

### Question 16.2

16.2

2. What is a nonspontaneous reaction?

### Question 16.3

16.3

3. Indicate whether the following processes are spontaneous or nonspontaneous. (a) Liquid water freezing at a temperature below its freezing point (b) Liquid water freezing at a temperature above its freezing point (c) The combustion of gasoline (d) A ball thrown into the air (e) A raindrop falling to the ground (f) Iron rusting in a moist atmosphere

### Question 16.4

16.4

4. A helium-filled balloon spontaneously deflates overnight as He atoms diffuse through the wall of the balloon. Describe the redistribution of matter and/or energy that accompanies this process.

### Question 16.5

16.5

5. Many plastic materials are organic polymers that contain carbon and hydrogen. The oxidation of these plastics in air to form carbon dioxide and water is a spontaneous process; however, plastic materials tend to persist in the environment. Explain.

## 16.2 Entropy

By the end of this section, you will be able to:

•  Define entropy
•  Explain the relationship between entropy and the number of microstates
•  Predict the sign of the entropy change for chemical and physical processes

In 1824, at the age of 28, Nicolas Léonard Sadi Carnot (Figure 16.7) published the results of an extensive study regarding the efficiency of steam heat engines. In a later review of Carnot’s findings, Rudolf Clausius introduced a new thermodynamic property that relates the spontaneous heat flow accompanying a process to the temperature at which the process takes place. This new property was expressed as the ratio of the reversible heat (qrev) and the kelvin temperature (T). The term reversible process refers to a process that takes place at such a slow rate that it is always at equilibrium and its direction can be changed (it can be “reversed”) by an infinitesimally small change is some condition. Note that the idea of a reversible process is a formalism required to support the development of various thermodynamic concepts; no real processes are truly reversible, rather they are classified as irreversible.

Similar to other thermodynamic properties, this new quantity is a state function, and so its change depends only upon the initial and final states of a system. In 1865, Clausius named this property entropy (S) and defined its change for any process as the following:

The entropy change for a real, irreversible process is then equal to that for the theoretical reversible process that involves the same initial and final states.

### Entropy and Microstates

Following the work of Carnot and Clausius, Ludwig Boltzmann developed a molecular-scale statistical model that related the entropy of a system to the number of microstates possible for the system. A microstate (W) is a specific configuration of the locations and energies of the atoms or molecules that comprise a system like the following:

Here k is the Boltzmann constant and has a value of 1.38 × 10−23 J/K.

As for other state functions, the change in entropy for a process is the difference between its final (Sf) and initial (Si) values:

For processes involving an increase in the number of microstates, Wf > Wi, the entropy of the system increases, ΔS > 0. Conversely, processes that reduce the number of microstates, Wf < Wi, yield a decrease in system entropy, ΔS < 0. This molecular-scale interpretation of entropy provides a link to the probability that a process will occur as illustrated in the next paragraphs.

Consider the general case of a system comprised of N particles distributed among n boxes. The number of microstates possible for such a system is n N . For example, distributing four particles among two boxes will result in 24 = 16 different microstates as illustrated in Figure 16.8. Microstates with equivalent particle arrangements (not considering individual particle identities) are grouped together and are called distributions. The probability that a system will exist with its components in a given distribution is proportional to the number of microstates within the distribution. Since entropy increases logarithmically with the number of microstates, the most probable distribution is therefore the one of greatest entropy.

For this system, the most probable configuration is one of the six microstates associated with distribution (c) where the particles are evenly distributed between the boxes, that is, a configuration of two particles in each box. The probability of finding the system in this configuration is or

The least probable configuration of the system is one in which all four particles are in one box, corresponding to distributions (a) and (d), each with a probability of

The probability of finding all particles in only one box (either the left box or right box) is then

As you add more particles to the system, the number of possible microstates increases exponentially (2N ). A macroscopic (laboratory-sized) system would typically consist of moles of particles (N ~ 1023), and the corresponding number of microstates would be staggeringly huge. Regardless of the number of particles in the system, however, the distributions in which roughly equal numbers of particles are found in each box are always the most probable configurations.

The previous description of an ideal gas expanding into a vacuum (Figure 16.4) is a macroscopic example of this particle-in-a-box model. For this system, the most probable distribution is confirmed to be the one in which the matter is most uniformly dispersed or distributed between the two flasks. The spontaneous process whereby the gas contained initially in one flask expands to fill both flasks equally therefore yields an increase in entropy for the system.

A similar approach may be used to describe the spontaneous flow of heat. Consider a system consisting of two objects, each containing two particles, and two units of energy (represented as “*”) in Figure 16.9. The hot object is comprised of particles A and B and initially contains both energy units. The cold object is comprised of particles C and D, which initially has no energy units. Distribution (a) shows the three microstates possible for the initial state of the system, with both units of energy contained within the hot object. If one of the two energy units is transferred, the result is distribution (b) consisting of four microstates. If both energy units are transferred, the result is distribution (c) consisting of three microstates. And so, we may describe this system by a total of ten microstates. The probability that the heat does not flow when the two objects are brought into contact, that is, that the system remains in distribution (a), is 3/10. More likely is the flow of heat to yield one of the other two distribution, the combined probability being 7/10. The most likely result is the flow of heat to yield the uniform dispersal of energy represented by distribution (b), the probability of this configuration being 4/10. As for the previous example of matter dispersal, extrapolating this treatment to macroscopic collections of particles dramatically increases the probability of the uniform distribution relative to the other distributions. This supports the common observation that placing hot and cold objects in contact results in spontaneous heat flow that ultimately equalizes the objects’ temperatures. And, again, this spontaneous process is also characterized by an increase in system entropy.

### Example 16.2Determination of ΔS

Consider the system shown here. What is the change in entropy for a process that converts the system from distribution (a) to (c)?

Solution

We are interested in the following change: The initial number of microstates is one, the final six:

The sign of this result is consistent with expectation; since there are more microstates possible for the final state than for the initial state, the change in entropy should be positive.

Consider the system shown in Figure 16.9. What is the change in entropy for the process where all the energy is transferred from the hot object (AB) to the cold object (CD)?

### Predicting the Sign of ΔS

The relationships between entropy, microstates, and matter/energy dispersal described previously allow us to make generalizations regarding the relative entropies of substances and to predict the sign of entropy changes for chemical and physical processes. Consider the phase changes illustrated in Figure 16.10. In the solid phase, the atoms or molecules are restricted to nearly fixed positions with respect to each other and are capable of only modest oscillations about these positions. With essentially fixed locations for the system’s component particles, the number of microstates is relatively small. In the liquid phase, the atoms or molecules are free to move over and around each other, though they remain in relatively close proximity to one another. This increased freedom of motion results in a greater variation in possible particle locations, so the number of microstates is correspondingly greater than for the solid. As a result, Sliquid > Ssolid and the process of converting a substance from solid to liquid (melting) is characterized by an increase in entropy, ΔS > 0. By the same logic, the reciprocal process (freezing) exhibits a decrease in entropy, ΔS < 0.

Now consider the vapor or gas phase. The atoms or molecules occupy a much greater volume than in the liquid phase; therefore each atom or molecule can be found in many more locations than in the liquid (or solid) phase. Consequently, for any substance, Sgas > Sliquid > Ssolid, and the processes of vaporization and sublimation likewise involve increases in entropy, ΔS > 0. Likewise, the reciprocal phase transitions, condensation and deposition, involve decreases in entropy, ΔS < 0.

According to kinetic-molecular theory, the temperature of a substance is proportional to the average kinetic energy of its particles. Raising the temperature of a substance will result in more extensive vibrations of the particles in solids and more rapid translations of the particles in liquids and gases. At higher temperatures, the distribution of kinetic energies among the atoms or molecules of the substance is also broader (more dispersed) than at lower temperatures. Thus, the entropy for any substance increases with temperature (Figure 16.11).

Try this simulator with interactive visualization of the dependence of particle location and freedom of motion on physical state and temperature.

The entropy of a substance is influenced by structure of the particles (atoms or molecules) that comprise the substance. With regard to atomic substances, heavier atoms possess greater entropy at a given temperature than lighter atoms, which is a consequence of the relation between a particle’s mass and the spacing of quantized translational energy levels (which is a topic beyond the scope of our treatment). For molecules, greater numbers of atoms (regardless of their masses) increase the ways in which the molecules can vibrate and thus the number of possible microstates and the system entropy.

Finally, variations in the types of particles affects the entropy of a system. Compared to a pure substance, in which all particles are identical, the entropy of a mixture of two or more different particle types is greater. This is because of the additional orientations and interactions that are possible in a system comprised of nonidentical components. For example, when a solid dissolves in a liquid, the particles of the solid experience both a greater freedom of motion and additional interactions with the solvent particles. This corresponds to a more uniform dispersal of matter and energy and a greater number of microstates. The process of dissolution therefore involves an increase in entropy, ΔS > 0.

Considering the various factors that affect entropy allows us to make informed predictions of the sign of ΔS for various chemical and physical processes as illustrated in Example 16.3.

### Example 16.3Predicting the Sign of ∆S

Predict the sign of the entropy change for the following processes. Indicate the reason for each of your predictions.

(a) One mole liquid water at room temperature

one mole liquid water at 50 °C

Solution

(a) positive, temperature increases

(b) negative, reduction in the number of ions (particles) in solution, decreased dispersal of matter

(c) negative, net decrease in the amount of gaseous species

(d) positive, phase transition from solid to liquid, net increase in dispersal of matter

Predict the sign of the enthalpy change for the following processes. Give a reason for your prediction.

Answer: (a) Positive; The solid dissolves to give an increase of mobile ions in solution.

(b) Negative; The liquid becomes a more ordered solid.

(c) Positive; The relatively ordered solid becomes a gas.

(d) Positive; There is a net production of one mole of gas.

## Exercies

### Question 16.6

16.6

6. In Figure all possible distributions and microstates are shown for four different particles shared between two boxes. Determine the entropy change, ΔS, if the particles are initially evenly distributed between the two boxes, but upon redistribution all end up in Box (b). Figure: The sixteen microstates associated with placing four particles in two boxes are shown. The microstates are collected into five distributions—(a), (b), (c), (d), and (e)—based on the numbers of particles in each box.

### Question 16.7

16.7

7. In Figure all of the possible distributions and microstates are shown for four different particles shared between two boxes. Determine the entropy change, ΔS, for the system when it is converted from distribution (b) to distribution (d).

Figure: The sixteen microstates associated with placing four particles in two boxes are shown. The microstates are collected into five distributions—(a), (b), (c), (d), and (e)—based on the numbers of particles in each box.

### Question 16.8

16.8

8. How does the process described in the previous item relate to the system shown in Figure?

Figure: An isolated system consists of an ideal gas in one flask that is connected by a closed valve to a second flask containing a vacuum. Once the valve is opened, the gas spontaneously becomes evenly distributed between the flasks.

### Question 16.9

16.9

9. Consider a system similar to the one in Figure, except that it contains six particles instead of four. What is the probability of having all the particles in only one of the two boxes in the case? Compare this with the similar probability for the system of four particles that we have derived to be equal to 1/8. What does this comparison tell us about even larger systems?

Figure: The sixteen microstates associated with placing four particles in two boxes are shown. The microstates are collected into five distributions—(a), (b), (c), (d), and (e)—based on the numbers of particles in each box.

### Question 16.10

16.10

10. Consider the system shown in Figure. What is the change in entropy for the process where the energy is initially associated only with particle A, but in the final state the energy is distributed between two different particles?

Figure: This shows a microstate model describing the flow of heat from a hot object to a cold object. (a) Before the heat flow occurs, the object comprised of particles A and B contains both units of energy and as represented by a distribution of three microstates. (b) If the heat flow results in an even dispersal of energy (one energy unit transferred), a distribution of four microstates results. (c) If both energy units are transferred, the resulting distribution has three microstates.

### Question 16.11

16.11

11. Consider the system shown in Figure. What is the change in entropy for the process where the energy is initially associated with particles A and B, and the energy is distributed between two particles in different boxes (one in A-B, the other in C-D)?

Figure: This shows a microstate model describing the flow of heat from a hot object to a cold object. (a) Before the heat flow occurs, the object comprised of particles A and B contains both units of energy and as represented by a distribution of three microstates. (b) If the heat flow results in an even dispersal of energy (one energy unit transferred), a distribution of four microstates results. (c) If both energy units are transferred, the resulting distribution has three microstates.

### Question 16.12

16.12

12. Arrange the following sets of systems in order of increasing entropy. Assume one mole of each substance and the same temperature for each member of a set. (a) H$_{2}$(g), HBrO$_{4}$(g), HBr(g) (b) H$_{2}$O(l), H$_{2}$O(g), H$_{2}$O(s) (c) He(g), Cl$_{2}$ (g), P$_{4}$ (g)

### Question 16.13

16.13

13. At room temperature, the entropy of the halogens increases from I$_{2}$ to Br$_{2}$ to Cl$_{2}$. Explain.

### Question 16.14

16.14

14. Consider two processes: sublimation of I$_{2}$ (s) and melting of I$_{2}$ (s) (Note: the latter process can occur at the same temperature but somewhat higher pressure). I$_{2}$ (s)⟶I$_{2}$ (g) I$_{2}$ (s)⟶I$_{2}$ (l) Is ΔS positive or negative in these processes? In which of the processes will the magnitude of the entropy change be greater?

### Question 16.15

16.15

15. Indicate which substance in the given pairs has the higher entropy value. Explain your choices. (a) C$_{2}$H$_{5}$OH(l) or C$_{3}$H$_{7}$OH(l) (b) C$_{2}$H$_{5}$OH(l) or C$_{2}$H$_{5}$OH(g) (c) 2H(g) or H(g)

### Question 16.16

16.16

16. Predict the sign of the entropy change for the following processes. (a) An ice cube is warmed to near its melting point. (b) Exhaled breath forms fog on a cold morning. (c) Snow melts.

### Question 16.17

16.17

17. Predict the sign of the enthalpy change for the following processes. Give a reason for your prediction. (a) Pb$^{2+}$ (aq)+S$^{2-}$ (aq)⟶PbS(s) (b) 2Fe(s)+3O$_{2}$ (g)⟶Fe$_{2}$O$_{3}$ (s) (c) 2C$_{6}$H$_{14}$ (l)+19O$_{2}$ (g)⟶14H$_{2}$O(g)+12CO$_{2}$ (g)

### Question 16.18

16.18

18. Write the balanced chemical equation for the combustion of methane, CH$_{4}$ (g), to give carbon dioxide and water vapor. Explain why it is difficult to predict whether ΔS is positive or negative for this chemical reaction.

### Question 16.19

16.19

19. Write the balanced chemical equation for the combustion of benzene, C$_{6}$H$_{6}$ (l), to give carbon dioxide and water vapor. Would you expect ΔS to be positive or negative in this process?

## 16.3 The Second and Third Laws of Thermodynamics

By the end of this section, you will be able to:

•  State and explain the second and third laws of thermodynamics
•  Calculate entropy changes for phase transitions and chemical reactions under standard conditions

The Second Law of Thermodynamics

In the quest to identify a property that may reliably predict the spontaneity of a process, we have identified a very promising candidate: entropy. Processes that involve an increase in entropy of the systemS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, we may reach a significant conclusion regarding the relation between this property and spontaneity. In thermodynamic models, the system and surroundings comprise everything, that is, the universe, and so the following is true:

1.The objects are at different temperatures, and heat flows from the hotter to the cooler object. This is always observed to occur spontaneously. Designating the hotter object as the system and invoking the definition of entropy yields the following:

The arithmetic signs of qrev denote the loss of heat by the system and the gain of heat by the surroundings. Since Tsys > Tsurr in this scenario, the magnitude of the entropy change for the surroundings will be greater than that for the system, and so the sum of ΔSsys and ΔSsurr will yield a positive value for ΔSuniv. This process involves an increase in the entropy of the universe.

2. The objects are at different temperatures, and heat flows from the cooler to the hotter object. This is never observed to occur spontaneously. Again designating the hotter object as the system and invoking the definition of entropy yields the following:

The arithmetic signs of qrev denote the gain of heat by the system and the loss of heat by the surroundings. The magnitude of the entropy change for the surroundings will again be greater than that for the system, but in this case, the signs of the heat changes will yield a negative value for ΔSuniv. This process involves a decrease in the entropy of the universe.

3. The temperature difference between the objects is infinitesimally small, TsysTsurr, and so the heat flow is thermodynamically reversible. See the previous section’s discussion). In this case, the system and surroundings experience entropy changes that are equal in magnitude and therefore sum to yield a value of zero for ΔSuniv. This process involves no change in the entropy of the universe.

These results lead to a profound statement regarding the relation between entropy and spontaneity known as the second law of thermodynamics: all spontaneous changes cause an increase in the entropy of the universe. A summary of these three relations is provided in Table 16.1.

For many realistic applications, the surroundings are vast in comparison to the system. In such cases, the heat gained or lost by the surroundings as a result of some process represents a very small, nearly infinitesimal, fraction of its total thermal energy. For example, combustion of a fuel in air involves transfer of heat from a system (the fuel and oxygen molecules undergoing reaction) to surroundings that are infinitely more massive (the earth’s atmosphere). As a result, qsurr is a good approximation of qrev, and the second law may be stated as the following:

We may use this equation to predict the spontaneity of a process as illustrated in Example 16.4.

### Example 16.4Will Ice Spontaneously Melt?

The entropy change for the process

is 22.1 J/K and requires that the surroundings transfer 6.00 kJ of heat to the system. Is the process spontaneous at −10.00 °C? Is it spontaneous at +10.00 °C?

Solution

We can assess the spontaneity of the process by calculating the entropy change of the universe. If ΔSuniv is positive, then the process is spontaneous. At both temperatures, ΔSsys = 22.1 J/K and qsurr = −6.00 kJ.

At −10.00 °C (263.15 K), the following is true:

Suniv < 0, so melting is nonspontaneous (not spontaneous) at −10.0 °C.

Using this information, determine if liquid water will spontaneously freeze at the same temperatures. What can you say about the values of Suniv?

Answer: Entropy is a state function, and freezing is the opposite of melting. At −10.00 °C spontaneous, +0.7 J/K; at +10.00 °C nonspontaneous, −0.9 J/K.

### The Third Law of Thermodynamics

The previous section described the various contributions of matter and energy dispersal that contribute to the entropy of a system. With these contributions in mind, consider the entropy of a pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K). This system may be described by a single microstate, as its purity, perfect crystallinity and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.

This limiting condition for a system’s entropy represents the third law of thermodynamics: the entropy of a pure, perfect crystalline substance at 0 K is zero.

We can make careful calorimetric measurements to determine the temperature dependence of a substance’s entropy and to derive absolute entropy values under specific conditions. Standard entropies are given the label S°298  for values determined for one mole of substance at a pressure of 1 bar and a temperature of 298 K. The standard entropy change (ΔS°) for any process may be computed from the standard entropies of its reactant and product species like the following:

Here, ν represents stoichiometric coefficients in the balanced equation representing the process. For example, Δ for the following reaction at room temperature

is computed as the following:

Table 16.2 lists some standard entropies at 298.15 K. You can find additional standard entropies in Appendix G

### Example 16.5Determination of ΔS°

Calculate the standard entropy change for the following process:

Solution

The value of the standard entropy change at room temperature, ΔS°298  , is the difference between the standard entropy of the product, H2O(l), and the standard entropy of the reactant, H2O(g).

The value for Δ298  is negative, as expected for this phase transition (condensation), which the previous section discussed.

Calculate the standard entropy change for the following process:

Answer: −120.6 J mol−1 K −1

### Example 16.6Determination of ΔS°

Calculate the standard entropy change for the combustion of methanol, CH3OH:

Solution

The value of the standard entropy change is equal to the difference between the standard entropies of the products and the entropies of the reactants scaled by their stoichiometric coefficients.

Calculate the standard entropy change for the following reaction:

## Exercises

### Question 16.20

16.20

20. What is the difference between ΔS, ΔS$^{o}$, and ΔS$_{298}^{o}$ for a chemical change?

### Question 16.21

16.21

21. Calculate ΔS$_{298}^{o}$ for the following changes. (a) SnCl$_{4}$ (l)⟶SnCl$_{4}$ (g) (b) CS$_{2}$ (g)⟶CS$_{2}$ (l) (c) Cu(s)⟶Cu(g) (d) H$_{2}$O(l)⟶H$_{2}$O(g) (e) 2H$_{2}$ (g)+O$_{2}$ (g)⟶2H$_{2}$O(l) (f) 2HCl(g)+Pb(s)⟶PbCl$_{2}$ (s)+H$_{2}$ (g) (g) Zn(s)+CuSO$_{4}$ (s)⟶Cu(s)+ZnSO$_{4}$ (s) 16.2121. Calculate ΔS$_{298}^{o}$ for the following changes. (a) SnCl$_{4}$ (l)⟶SnCl$_{4}$ (g) (b) CS$_{2}$ (g)⟶CS$_{2}$ (l) (c) Cu(s)⟶Cu(g) (d) H$_{2}$O(l)⟶H$_{2}$O(g) (e) 2H$_{2}$ (g)+O$_{2}$ (g)⟶2H$_{2}$O(l) (f) 2HCl(g)+Pb(s)⟶PbCl$_{2}$ (s)+H$_{2}$ (g) (g) Zn(s)+CuSO$_{4}$ (s)⟶Cu(s)+ZnSO$_{4}$ (s)

### Question 16.22

16.22

22. Determine the entropy change for the combustion of liquid ethanol, C$_{2}$H$_{5}$OH, under standard state conditions to give gaseous carbon dioxide and liquid water.

### Question 16.23

16.23

23. Determine the entropy change for the combustion of gaseous propane, C$_{3}$H$_{8}$, under standard state conditions to give gaseous carbon dioxide and water.

### Question 16.24

16.24

24. “Thermite” reactions have been used for welding metal parts such as railway rails and in metal refining. One such thermite reaction is Fe$_{2}$O$_{3}$ (s)+2Al(s)⟶Al$_{2}$O$_{3}$ (s)+2Fe(s). Is the reaction spontaneous at room temperature under standard conditions? During the reaction, the surroundings absorb 851.8 kJ/mol of heat.

### Question 16.25

16.25

25. Using the relevant S$_{298}^{o}$ values listed in Appendix G, calculate S$_{298}^{o}$ for the following changes: (a) N$_{2}$ (g)+3H$_{2}$ (g)⟶2NH$_{3}$ (g) (b) N$_{2}$ (g)+ $\5\over 2$O$_{2}$ (g)⟶N$_{2}$O$_{5}$ (g)

### Question 16.26

16.26

26. From the following information, determine ΔS$_{298}^{o}$ for the following: N(g)+O(g)⟶NO(g) ΔS$_{298}^{o}$ =? N$_{2}$ (g)+O$_{2}$ (g)⟶2NO(g) ΔS$_{298}^{o}$ =24.8 J/K N$_{2}$ (g)⟶2N(g) ΔS$_{298}^{o}$ =115.0 J/K O$_{2}$ (g)⟶2O(g) ΔS$_{298}^{o}$ =117.0 J/K

### Question 16.27

16.27

27. By calculating $ΔS_{univ}$ at each temperature, determine if the melting of 1 mole of NaCl(s) is spontaneous at 500 °C and at 700 °C.

$S^∘_{NaCl(s)}$=72.11Jmol⋅KS∘NaCl(l)=95.06J/mol⋅K $ΔH^∘_{fusion}$=27.95 kJ/mol What assumptions are made about the thermodynamic information (entropy and enthalpy values) used to solve this problem?

### Question 16.28

16.28

28. Use the standard entropy data in Appendix G to determine the change in entropy for each of the reactions listed in Exercise 16.33. All are run under standard state conditions and 25 °C.

### Question 16.29

16.29

29. Use the standard entropy data in Appendix G to determine the change in entropy for each of the reactions listed in Exercise 16.34. All are run under standard state conditions and 25 °C.

## 16.4 Free Energy

By the end of this section, you will be able to:

•  Define Gibbs free energy, and describe its relation to spontaneity
•  Calculate free energy change for a process using free energies of formation for its reactants and products
•  Calculate free energy change for a process using enthalpies of formation and the entropies for its reactants and products
•  Explain how temperature affects the spontaneity of some processes
•  Relate standard free energy changes to equilibrium constants

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that we must determine the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs. This new property is called the Gibbs free energy change (G) (or simply the free energy), and it is defined in terms of a system’s enthalpy and entropy as the following:

Free energy is a state function, and at constant temperature and pressure, the standard free energy change (ΔG°) may be expressed as the following:

(For simplicity’s sake, the subscript “sys” will be omitted henceforth.) We can understand the relationship between this system property and the spontaneity of a process by recalling the previously derived second law expression:

The first law requires that qsurr = −qsys, and at constant pressure qsys = ΔH, and so this expression may be rewritten as the following:

ΔH is the enthalpy change of the system. Multiplying both sides of this equation by −T, and rearranging yields the following:

Comparing this equation to the previous one for free energy change shows the following relation:

The free energy change is therefore a reliable indicator of the spontaneity of a process, being directly related to the previously identified spontaneity indicator, ΔSuniv. Table 16.3 summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.

### Calculating Free Energy Change

Free energy is a state function, so its value depends only on the conditions of the initial and final states of the system that have undergone some change. A convenient and common approach to the calculation of free energy changes for physical and chemical reactions is by use of widely available compilations of standard state thermodynamic data. One method involves the use of standard enthalpies and entropies to compute standard free energy changes according to the following relation as demonstrated in Example 16.7.

### Example 16.7Evaluation of ΔG° Change from ΔH° and ΔS°

Use standard enthalpy and entropy data from Appendix G to calculate the standard free energy change for the vaporization of water at room temperature (298 K). What does the computed value for Δ say about the spontaneity of this process?

Solution

The process of interest is the following:

The standard change in free energy may be calculated using the following equation:

From Appendix G, here is the data:

Combining at 298 K:

Converting everything into kJ and combining at 298 K:

At 298 K (25 °C) ΔG°298 > 0, and so boiling is nonspontaneous (not spontaneous).

Use standard enthalpy and entropy data from Appendix G to calculate the standard free energy change for the reaction shown here (298 K). What does the computed value for ΔG° say about the spontaneity of this process?

Answer: ΔG°298  = 102.0 kJ/mol; the reaction is nonspontaneous (not spontaneous) at 25 °C.

Free energy changes may also use the standard free energy of formationf  ), for each of the reactants and products involved in the reaction. The standard free energy of formation is the free energy change that accompanies the formation of one mole of a substance from its elements in their standard states. Similar to the standard enthalpies of formation, Δf  is by definition zero for elemental substances under standard state conditions. The approach to computing the free energy change for a reaction using this approach is the same as that demonstrated previously for enthalpy and entropy changes. For the reaction

the standard free energy change at room temperature may be calculated as

### Example 16.8Calculation of ΔG298

Consider the decomposition of yellow mercury(II) oxide

Calculate the standard free energy change at room temperature, ΔG°298  , using (a) standard free energies of formation and (b) standard enthalpies of formation and standard entropies. Do the results indicate the reaction to be spontaneous or nonspontaneous under standard conditions?

Solution

The required data are available in Appendix G and are shown here.

(a) Using free energies of formation:

(b) Using enthalpies and entropies of formation:

Both ways to calculate the standard free energy change at 25 °C give the same numerical value (to three significant figures), and both predict that the process is nonspontaneous (not spontaneous) at room temperature.

Calculate ΔG° using (a) free energies of formation and (b) enthalpies of formation and entropies (Appendix G). Do the results indicate the reaction to be spontaneous or nonspontaneous at 25 °C?

### Temperature Dependence of Spontaneity

As was previously demonstrated in this chapter’s section on entropy, the spontaneity of a process may depend upon the temperature of the system. Phase transitions, for example, will proceed spontaneously in one direction or the other depending upon the temperature of the substance in question. Likewise, some chemical reactions can also exhibit temperature dependent spontaneities. To illustrate this concept, the equation relating free energy change to the enthalpy and entropy changes for the process is considered:

The spontaneity of a process, as reflected in the arithmetic sign of its free energy change, is then determined by the signs of the enthalpy and entropy changes and, in some cases, the absolute temperature. Since T is the absolute (kelvin) temperature, it can only have positive values. Four possibilities therefore exist with regard to the signs of the enthalpy and entropy changes:

1. Both ΔH and ΔS are positive. This condition describes an endothermic process that involves an increase in system entropy. In this case, ΔG will be negative if the magnitude of the TΔS term is greater than ΔH. If the TΔS term is less than ΔH, the free energy change will be positive. Such a process is spontaneous at high temperatures and nonspontaneous at low temperatures.

2. Both ΔH and ΔS are negative. This condition describes an exothermic process that involves a decrease in system entropy. In this case, ΔG will be negative if the magnitude of the TΔS term is less than ΔH. If the TΔS term’s magnitude is greater than ΔH, the free energy change will be positive. Such a process is spontaneous at low temperatures and nonspontaneous at high temperatures.

3. ΔH is positive and ΔS is negative. This condition describes an endothermic process that involves a decrease in system entropy. In this case, ΔG will be positive regardless of the temperature. Such a process is nonspontaneous at all temperatures.

4. ΔH is negative and ΔS is positive. This condition describes an exothermic process that involves an increase in system entropy. In this case, ΔG will be negative regardless of the temperature. Such a process is spontaneous at all temperatures.

These four scenarios are summarized in Figure 16.12.

### Example 16.9Predicting the Temperature Dependence of Spontaneity

The incomplete combustion of carbon is described by the following equation:

How does the spontaneity of this process depend upon temperature?

Solution

Combustion processes are exothermic (ΔH < 0). This particular reaction involves an increase in entropy due to the accompanying increase in the amount of gaseous species (net gain of one mole of gas, ΔS > 0). The reaction is therefore spontaneous (ΔG < 0) at all temperatures.

Popular chemical hand warmers generate heat by the air-oxidation of iron:

How does the spontaneity of this process depend upon temperature?

Answer: ΔH and ΔS are negative; the reaction is spontaneous at low temperatures.

When considering the conclusions drawn regarding the temperature dependence of spontaneity, it is important to keep in mind what the terms “high” and “low” mean. Since these terms are adjectives, the temperatures in question are deemed high or low relative to some reference temperature. A process that is nonspontaneous at one temperature but spontaneous at another will necessarily undergo a change in “spontaneity” (as reflected by its ΔG) as temperature varies. This is clearly illustrated by a graphical presentation of the free energy change equation, in which ΔG is plotted on the y axis versus T on the x axis:

Such a plot is shown in Figure 16.13. A process whose enthalpy and entropy changes are of the same arithmetic sign will exhibit a temperature-dependent spontaneity as depicted by the two yellow lines in the plot. Each line crosses from one spontaneity domain (positive or negative ΔG) to the other at a temperature that is characteristic of the process in question. This temperature is represented by the x-intercept of the line, that is, the value of T for which ΔG is zero:

And so, saying a process is spontaneous at “high” or “low” temperatures means the temperature is above or below, respectively, that temperature at which ΔG for the process is zero. As noted earlier, this condition describes a system at equilibrium.

### Example 16.10Equilibrium Temperature for a Phase Transition

As defined in the chapter on liquids and solids, the boiling point of a liquid is the temperature at which its solid and liquid phases are in equilibrium (that is, when vaporization and condensation occur at equal rates). Use the information in Appendix G to estimate the boiling point of water.

Solution

The process of interest is the following phase change:

When this process is at equilibrium, ΔG = 0, so the following is true:

Using the standard thermodynamic data from Appendix G,

The accepted value for water’s normal boiling point is 373.2 K (100.0 °C), and so this calculation is in reasonable agreement. Note that the values for enthalpy and entropy changes data used were derived from standard data at 298 K (Appendix G). If desired, you could obtain more accurate results by using enthalpy and entropy changes determined at (or at least closer to) the actual boiling point.

Use the information in Appendix G to estimate the boiling point of CS2.

Answer: 313 K (accepted value 319 K)

### Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a finite driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔG is zero, the forward and reverse driving forces are equal, and so the process occurs in both directions at the same rate (the system is at equilibrium).

In the chapter on equilibrium the reaction quotient, Q, was introduced as a convenient measure of the status of an equilibrium system. Recall that Q is the numerical value of the mass action expression for the system, and that you may use its value to identify the direction in which a reaction will proceed in order to achieve equilibrium. When Q is lesser than the equilibrium constant, K, the reaction will proceed in the forward direction until equilibrium is reached and Q = K. Conversely, if Q < K, the process will proceed in the reverse direction until equilibrium is achieved.

The free energy change for a process taking place with reactants and products present under nonstandard conditions, ΔG, is related to the standard free energy change, Δ, according to this equation:

R is the gas constant (8.314 J/K mol), T is the kelvin or absolute temperature, and Q is the reaction quotient. We may use this equation to predict the spontaneity for a process under any given set of conditions as illustrated in Example 16.11.

### Example 16.11Calculating ΔG under Nonstandard Conditions

What is the free energy change for the process shown here under the specified conditions?

T = 25 °C, PN2 = 0.870 atm, PH2 = 0.250 atm, and PNH3 = 12.9 atm

Solution

The equation relating free energy change to standard free energy change and reaction quotient may be used directly:

Since the computed value for ΔG is positive, the reaction is nonspontaneous under these conditions.

Calculate the free energy change for this same reaction at 875 °C in a 5.00 L mixture containing 0.100 mol of each gas. Is the reaction spontaneous under these conditions?

Answer: ΔG = −136 kJ; yes

For a system at equilibrium, Q = K and ΔG = 0, and the previous equation may be written as

This form of the equation provides a useful link between these two essential thermodynamic properties, and it can be used to derive equilibrium constants from standard free energy changes and vice versa. The relations between standard free energy changes and equilibrium constants are summarized in Table 16.4.

### Example 16.12Calculating an Equilibrium Constant using Standard Free Energy Change

Given that the standard free energies of formation of Ag+ (aq), Cl (aq), and AgCl(s) are 77.1 kJ/mol, −131.2 kJ/mol, and −109.8 kJ/mol, respectively, calculate the solubility product, Ksp, for AgCl.

Solution

The reaction of interest is the following:

The standard free energy change for this reaction is first computed using standard free energies of formation for its reactants and products:

The equilibrium constant for the reaction may then be derived from its standard free energy change:

This result is in reasonable agreement with the value provided in Appendix J.

Use the thermodynamic data provided in Appendix G to calculate the equilibrium constant for the dissociation of dinitrogen tetroxide at 25 °C.

To further illustrate the relation between these two essential thermodynamic concepts, consider the observation that reactions spontaneously proceed in a direction that ultimately establishes equilibrium. As may be shown by plotting the free energy change versus the extent of the reaction (for example, as reflected in the value of Q), equilibrium is established when the system’s free energy is minimized (Figure 16.14). If a system is present with reactants and products present in nonequilibrium amounts (Q ≠ K), the reaction will proceed spontaneously in the direction necessary to establish equilibrium.

## Exercies

### Question 16.30

16.30

30. What is the difference between ΔG, ΔG°, and $ΔG^∘_{298}$ for a chemical change?

### Question 16.31

16.31

31. A reactions has $ΔH^∘_{298}$ = 100 kJ/mol and $ΔS^∘_{298}=250$ J/mol⋅K. Is the reaction spontaneous at room temperature? If not, under what temperature conditions will it become spontaneous?

### Question 16.32

16.32

32. Explain what happens as a reaction starts with ΔG < 0 (negative) and reaches the point where ΔG = 0.

### Question 16.33

16.33

33. Use the standard free energy of formation data in Appendix G to determine the free energy change for each of the following reactions, which are run under standard state conditions and 25 °C. Identify each as either spontaneous or nonspontaneous at these conditions. (a) $MnO_2(s)⟶Mn(s)+O_2(g)$ (b) $H2(g)+Br_2(l)⟶2HBr(g)$ (c) $Cu(s)+S(g)⟶CuS(s)$ (d) $2LiOH(s)+CO_2(g)⟶Li_2CO_3(s)+H_2O(g)$ (e) $CH4(g)+O_2(g)⟶C(s,graphite)+2H_2O(g)$ (f) $CS_2(g)+3Cl_2(g)⟶CCl_4(g)+S_2Cl_2(g)$

### Question 16.34

16.34

34. Use the standard free energy data in Appendix G to determine the free energy change for each of the following reactions, which are run under standard state conditions and 25 °C. Identify each as either spontaneous or nonspontaneous at these conditions. (a) $C(s, graphite)+O_2(g)⟶CO_2(g)$ (b) $O_2(g)+N_2(g)⟶2NO(g)$ (c) $2Cu(s)+S(g)⟶Cu_2S(s)$ (d) $CaO(s)+H_2O(l)⟶Ca(OH)_2(s)$ (e) $Fe_2O_3(s)+3CO(g)⟶2Fe(s)+3CO_2(g)$ (f) $CaSO_4⋅2H_2O(s)⟶CaSO_4(s)+2H_2O(g)$

### Question 16.35

16.35

35. Given:

$P_4(s)+5O_2(g)⟶P_4O_{10}(s)\ \ \ \ \ \ ΔG^∘_{298}=−2697.0\ kJ/mol$

$2H_2(g)+O_2(g)⟶2H_2O(g)\ \ \ \ \ ΔG^∘_{298}=−457.18\ kJ/mol$

$6H_2O(g)+P_4O_{10}(g)⟶4H_3PO_4(l)\ \ \ \ \ ΔG^∘_{298}=−428.66\ kJ/mol$ (a) Determine the standard free energy of formation, $ΔG^∘_f,$ for phosphoric acid. (b) How does your calculated result compare to the value in Appendix G? Explain.

### Question 16.36

16.36

36. Is the formation of ozone $(O_3 (g))$ from oxygen $(O_2(g))$ spontaneous at room temperature under standard state conditions?

### Question 16.37

16.37

37. Consider the decomposition of red mercury(II) oxide under standard state conditions.

$2HgO(s,red)⟶2Hg(l)+O_2(g)$ (a) Is the decomposition spontaneous under standard state conditions? (b) Above what temperature does the reaction become spontaneous?

### Question 16.38

16.38

38. Among other things, an ideal fuel for the control thrusters of a space vehicle should decompose in a spontaneous exothermic reaction when exposed to the appropriate catalyst. Evaluate the following substances under standard state conditions as suitable candidates for fuels. (a) $Ammonia: 2NH_3(g)⟶N_2(g)+3H_2(g)$ (b) $Diborane: B_2H_6(g)⟶2B(g)+3H_2(g)$ (c) $Hydrazine: N_2H_4(g)⟶N_2(g)+2H_2(g)$ (d) $Hydrogen peroxide: H_2O_2(l)⟶H_2O(g)+1/2O_2(g)$

### Question 16.39

16.39

39. Calculate ΔG° for each of the following reactions from the equilibrium constant at the temperature given. (a) $N_2(g)+O_2(g)⟶2NO(g)\ \ \ \ \ T=2000°C\ \ \ \ \ K_p=4.1×10^{−4}$ (b) $H_2(g)+I_2(g)⟶2HI(g)\ \ \ \ \ T=400°C\ \ \ \ \ K_p=50.0$ (c) $CO_2(g)+H_2(g)⟶CO(g)+H_2O(g)\ \ \ \ \ T=980°C\ \ \ \ \ K_p=1.67$ (d) $CaCO_3(s)⟶CaO(s)+CO_2(g)\ \ \ \ \ T=900°C \ \ \ \ \ K_p=1.04$ (e) $HF(aq)+H_2O(l)⟶H3O+(aq)+F−(aq)\ \ \ \ \ T=25°C\ \ \ \ \ K_p=7.2×10^{−4}$ (f) $AgBr(s)⟶Ag+(aq)+Br^−(aq)\ \ \ \ \ T=25°C\ \ \ \ \ K_p=3.3×10^{−13}$

### Question 16.40

16.40

40. Calculate ΔG° for each of the following reactions from the equilibrium constant at the temperature given. (a) $Cl_2(g)+Br_2(g)⟶2BrCl(g)\ \ \ \ \ T=25°C\ \ \ \ \ K_p=4.7×10^{−2}$ (b) $2SO_2(g)+O_2(g)⇌2SO_3(g)\ \ \ \ \ T=500°C\ \ \ \ \ K_p=48.2$ (c) $H_2O(l)⇌H_2O(g)\ \ \ \ \ T=60°C\ \ \ \ \ K_p=0.196 atm$ (d) $CoO(s)+CO(g)⇌Co(s)+CO_2(g)\ \ \ \ \ T=550°C\ \ \ \ \ K_p=4.90×10^2$ (e) $CH_3NH_2(aq)+H_2O(l)⟶CH_3NH_3+(aq)+OH^−(aq)\ \ \ \ \ T=25°C\ \ \ \ \ Kp=4.4×10^{−4}$ (f) $PbI_2(s)⟶Pb_2+(aq)+2I^−(aq)\ \ \ \ \ T=25°CKp=8.7×10^{−9}$

### Question 16.41

16.41

41. Calculate the equilibrium constant at 25 °C for each of the following reactions from the value of ΔG° given. (a) $O_2(g)+2F_2(g)⟶2OF_2(g)\ \ \ \ \ ΔG°=−9.2 kJ$ (b) $I_2(s)+Br_2(l)⟶2IBr(g)\ \ \ \ \ ΔG°=7.3 kJ$ (c) $2LiOH(s)+CO_2(g)⟶Li_2CO_3(s)+H2O(g)\ \ \ \ \ ΔG°=−79 kJ$ (d) $N_2O_3(g)⟶NO(g)+NO2(g)\ \ \ \ \ ΔG°=−1.6 kJ$ (e) $SnCl_4(l)⟶SnCl_4(l)\ \ \ \ \ ΔG°=8.0 kJ$

### Question 16.42

16.42

42. Calculate the equilibrium constant at 25 °C for each of the following reactions from the value of ΔG° given. (a) $I_2(s)+Cl_2(g)⟶2ICl(g)\ \ \ \ \ ΔG°=−10.88 kJ$ (b) $H_2(g)+I_2(s)⟶2HI(g)\ \ \ \ \ ΔG°=3.4 kJ$ (c) $CS_2(g)+3Cl_2(g)⟶CCl_4(g)+S_2Cl_2(g)\ \ \ \ \ ΔG°=−39 kJ$ (d) $2SO_2(g)+O_2(g)⟶2SO_3(g)\ \ \ \ \ ΔG°=−141.82 kJ$ (e) $CS_2(g)⟶CS_2(l)\ \ \ \ \ ΔG°=−1.88 kJ$

### Question 16.43

16.43

43. Calculate the equilibrium constant at the temperature given. (a) $O_2(g)+2F_2(g)⟶2F_2O(g)(T=100°C)$ (b) $I_2(s)+Br_2(l)⟶2IBr(g)(T=0.0°C)$ (c) [math]2LiOH(s)+CO2(g)⟶Li2CO3(s)+H2O(g)\ \ \ \ \ (T=575°C) (d) [math]N2O3(g)⟶NO(g)+NO_2(g)\ \ \ \ \ (T=−10.0°C) (e) $SnCl_4(l)⟶SnCl_4(g)\ \ \ \ \ (T=200°C)$

### Question 16.44

16.44

44. Calculate the equilibrium constant at the temperature given. (a) $I_2(s)+Cl_2(g)⟶2ICl(g)(T=100°C)$ (b) $H_2(g)+I_2(s)⟶2HI(g)(T=0.0°C)$ (c) $CS_2(g)+3Cl_2(g)⟶CCl_4(g)+S_2Cl_2(g)(T=125°C)$ (d) $2SO_2(g)+O_2(g)⟶2SO_3(g)(T=675°C)$ (e) $CS_2(g)⟶CS_2(l)(T=90°C)$

### Question 16.45

16.45

45. Consider the following reaction at 298 K:

$N_2O_4(g)⇌2NO_2(g)\ \ \ \ \ KP=0.142$ What is the standard free energy change at this temperature? Describe what happens to the initial system, where the reactants and products are in standard states, as it approaches equilibrium.

### Question 16.46

16.46

46. Determine the normal boiling point (in kelvin) of dichloroethane, $CH_2Cl_2.$ Find the actual boiling point using the Internet or some other source, and calculate the percent error in the temperature. Explain the differences, if any, between the two values.

### Question 16.47

16.47

47. $Under what conditions is N_2O_3(g)⟶NO(g)+NO_2(g) spontaneous?$

### Question 16.48

16.48

48. At room temperature, the equilibrium constant $(K_w)$ for the self-ionization of water is $1.00 × 10^{-14}.$ Using this information, calculate the standard free energy change for the aqueous reaction of hydrogen ion with hydroxide ion to produce water. (Hint: The reaction is the reverse of the self-ionization reaction.)

### Question 16.49

16.49

49. Hydrogen sulfide is a pollutant found in natural gas. Following its removal, it is converted to sulfur by the reaction $2H_2S(g)+SO_2(g)⇌3_8S_8(s,rhombic)+2H_2O(l).$ What is the equilibrium constant for this reaction? Is the reaction endothermic or exothermic?

### Question 16.50

16.50

50. Consider the decomposition of $CaCO_3$(s) into CaO(s) and $CO_2$(g). What is the equilibrium partial pressure of $CO_2$at room temperature?

### Question 16.51

16.51

51. In the laboratory, hydrogen chloride (HCl(g)) and ammonia $(NH_3(g))$ often escape from bottles of their solutions and react to form the ammonium chloride $(NH_4Cl(s)),$ the white glaze often seen on glassware. Assuming that the number of moles of each gas that escapes into the room is the same, what is the maximum partial pressure of HCl and $NH_3$ in the laboratory at room temperature? (Hint: The partial pressures will be equal and are at their maximum value when at equilibrium.)

### Question 16.52

16.52

52. Benzene can be prepared from acetylene. $3C_2H_2(g)⇌C_6H_6(g).$ Determine the equilibrium constant at 25 °C and at 850 °C. Is the reaction spontaneous at either of these temperatures? Why is all acetylene not found as benzene?

### Question 16.53

16.53

53. Carbon dioxide decomposes into CO and O2 at elevated temperatures. What is the equilibrium partial pressure of oxygen in a sample at 1000 °C for which the initial pressure of $CO_2$was 1.15 atm?

### Question 16.54

16.54

54. Carbon tetrachloride, an important industrial solvent, is prepared by the chlorination of methane at 850 K.

$CH_4(g)+4Cl_2(g)⟶CCl_4(g)+4HCl(g)$ What is the equilibrium constant for the reaction at 850 K? Would the reaction vessel need to be heated or cooled to keep the temperature of the reaction constant?

### Question 16.55

16.55

55. Acetic acid, $CH3CO2H,$ can form a dimer, (CH3CO2H)_2, in the gas phase.

$2CH_3CO_2H(g)⟶(CH_3CO_2H)2(g)$ The dimer is held together by two hydrogen bonds with a total strength of 66.5 kJ per mole of dimer. At 25 °C, the equilibrium constant for the dimerization is $1.3 × 10^3$(pressure in atm). What is ΔS° for the reaction?

### Question 16.56

16.56

56. Nitric acid, $HNO_3,$ can be prepared by the following sequence of reactions:

$4NH_3(g)+5O_2(g)⟶4NO(g)+6H_2O(g)$

$2NO(g)+O_2(g)⟶2NO_2(g)$

$3NO_2(g)+H_2O(l)⟶2HNO_3(l)+NO(g)$ How much heat is evolved when 1 mol of $NH3(g)$ is converted to $HNO3$(l)? Assume standard states at 25 °C

### Question 16.57

16.57

57. Determine ΔG for the following reactions. (a) Antimony pentachloride decomposes at 448 °C. The reaction is:

$SbCl_5(g)⟶SbCl_3(g)+Cl_2(g)$ An equilibrium mixture in a 5.00 L flask at 448 °C contains 3.85 g of $SbCl_5,$ 9.14 g of $SbCl3,$ and 2.84 g of $Cl_2.$ (b) Chlorine molecules dissociate according to this reaction:

$Cl_2(g)⟶2Cl(g)$ 1.00% of $Cl2$ molecules dissociate at 975 K and a pressure of 1.00 atm.

### Question 16.58

16.58

58. Given that the $ΔG^∘_f$ for Pb^{2+} (aq) and $Cl^−(aq)$ is −24.3 kJ/mole and −131.2 kJ/mole respectively, determine the solubility product, $K_sp,$ for $PbCl_2(s).$

### Question 16.59

16.59

59. Determine the standard free energy change, $ΔG^∘_f$ , for the formation of $S^{2−}(aq)$ given that the $ΔG^∘_f$ for Ag+(aq) and $Ag_2S(s)$ are 77.1 k/mole and −39.5 kJ/mole respectively, and the solubility product for $Ag_2S(s)$ is $8 × 10^{−51}.$

### Question 16.60

16.60

60. Determine the standard enthalpy change, entropy change, and free energy change for the conversion of diamond to graphite. Discuss the spontaneity of the conversion with respect to the enthalpy and entropy changes. Explain why diamond spontaneously changing into graphite is not observed.

### Question 16.61

16.61

61. The evaporation of one mole of water at 298 K has a standard free energy change of 8.58 kJ.

$H_2O(l)⇌H_2O(g)ΔG^∘298=8.58 kJ$ (a) Is the evaporation of water under standard thermodynamic conditions spontaneous? (b) Determine the equilibrium constant, $K_P,$ for this physical process. (c) By calculating ∆G, determine if the evaporation of water at 298 K is spontaneous when the partial pressure of water, $P_{H_2O,}$ is 0.011 atm. (d) If the evaporation of water were always nonspontaneous at room temperature, wet laundry would never dry when placed outside. In order for laundry to dry, what must be the value of $P_{H_2O}$ in the air?

### Question 16.62

16.62

62. In glycolysis, the reaction of glucose (Glu) to form glucose-6-phosphate (G6P) requires ATP to be present as described by the following equation:

$Glu+ATP⟶G6P+ADPΔ\ \ \ \ \ G∘298=−17 kJ$ In this process, ATP becomes ADP summarized by the following equation:

$ATP⟶ADP\ \ \ \ \ ΔG^∘298=−30 kJ$ Determine the standard free energy change for the following reaction, and explain why ATP is necessary to drive this process:

$Glu⟶G6P\ \ \ \ \ ΔG^∘298=?$

### Question 16.63

16.63

63. One of the important reactions in the biochemical pathway glycolysis is the reaction of glucose-6-phosphate (G6P) to form fructose-6-phosphate (F6P):

$G6P⇌F6P\ \ \ \ \ ΔG^∘298=1.7 kJ$ (a) Is the reaction spontaneous or nonspontaneous under standard thermodynamic conditions? (b) Standard thermodynamic conditions imply the concentrations of G6P and F6P to be 1 M, however, in a typical cell, they are not even close to these values. Calculate ΔG when the concentrations of G6P and F6P are 120 μM and 28 μM respectively, and discuss the spontaneity of the forward reaction under these conditions. Assume the temperature is 37 °C.

### Question 16.64

16.64

64. Without doing a numerical calculation, determine which of the following will reduce the free energy change for the reaction, that is, make it less positive or more negative, when the temperature is increased. Explain. (a) $N_2(g)+3H_2(g)⟶2NH_3(g)$ (b) $HCl(g)+NH_3(g)⟶NH_4Cl(s)$ (c) $(NH4)2Cr_2O_7(s)⟶Cr_2O_3(s)+4H_2O(g)+N_2(g)$ (d) $2Fe(s)+3O_2(g)⟶Fe_2O_3(s)$

### Question 16.65

16.65

65. When ammonium chloride is added to water and stirred, it dissolves spontaneously and the resulting solution feels cold. Without doing any calculations, deduce the signs of ΔG, ΔH, and ΔS for this process, and justify your choices.

### Question 16.66

16.66

66. An important source of copper is from the copper ore, chalcocite, a form of copper(I) sulfide. When heated, the $Cu_2S$ decomposes to form copper and sulfur described by the following equation:

$Cu_2S(s)⟶Cu(s)+S(s)$ (a) Determine $ΔG^∘_{298}$for the decomposition of Cu2S(s). (b) The reaction of sulfur with oxygen yields sulfur dioxide as the only product. Write an equation that describes this reaction, and determine $ΔG^∘_{298}$for the process. (c) The production of copper from chalcocite is performed by roasting the $Cu_2S$ in air to produce the Cu. By combining the equations from Parts (a) and (b), write the equation that describes the roasting of the chalcocite, and explain why coupling these reactions together makes for a more efficient process for the production of the copper.

### Question 16.67

16.67

67. What happens to $ΔG^∘_{298}$ (becomes more negative or more positive) for the following chemical reactions when the partial pressure of oxygen is increased? (a) $S(s)+O_2(g)⟶SO_2(g)$ (b) $2SO_2(g)+O_2(g)⟶SO_3(g)$ (c) $HgO(s)⟶Hg(l)+O_2(g)$

## Key Terms

entropy (S) state function that is a measure of the matter and/or energy dispersal within a system, determined by the number of system microstates often described as a measure of the disorder of the system

Gibbs free energy change (G) thermodynamic property defined in terms of system enthalpy and entropy; all spontaneous processes involve a decrease in G

microstate (W) possible configuration or arrangement of matter and energy within a system

nonspontaneous process process that requires continual input of energy from an external source

reversible process process that takes place so slowly as to be capable of reversing direction in response to an infinitesimally small change in conditions; hypothetical construct that can only be approximated by real processes removed

second law of thermodynamics entropy of the universe increases for a spontaneous process

spontaneous change process that takes place without a continuous input of energy from an external source

standard entropy (S°) entropy for a substance at 1 bar pressure; tabulated values are usually determined at 298.15 K and denoted S°298

standard entropy change (ΔS°) change in entropy for a reaction calculated using the standard entropies, usually at room temperature and denoted ΔS°298

standard free energy change (ΔG°) change in free energy for a process occurring under standard conditions (1 bar pressure for gases, 1 M concentration for solutions)

standard free energy of formation (ΔG°fchange in free energy accompanying the formation of one mole of substance from its elements in their standard states

third law of thermodynamics entropy of a perfect crystal at absolute zero (0 K) is zero

## Key Equations

• ΔS = qrev/T

• S = k ln W

• ΔS = k ln Wf /Wi

• ΔS° = ΔS°298  = ∑ νS°298  (products) − ∑ νS°298  (reactants)

• ΔS = qrev/T

• ΔSuniv = ΔSsys + ΔSsurr

• ΔSuniv = ΔSsys + ΔSsurr = ΔSsys + qsurr /T

• ΔG = ΔH − TΔS

• ΔG = ΔG° + RT ln Q

• ΔG° = −RT ln K

## Summary

16.1 Spontaneity

Chemical and physical processes have a natural tendency to occur in one direction under certain conditions. A spontaneous process occurs without the need for a continual input of energy from some external source, while a nonspontaneous process requires such. Systems undergoing a spontaneous process may or may not experience a gain or loss of energy, but they will experience a change in the way matter and/or energy is distributed within the system.

16.2 Entropy

Entropy (S) is a state function that can be related to the number of microstates for a system (the number of ways the system can be arranged) and to the ratio of reversible heat to kelvin temperature. It may be interpreted as a measure of the dispersal or distribution of matter and/or energy in a system, and it is often described as representing the “disorder” of the system. For a given substance, Ssolid < Sliquid < Sgas in a given physical state at a given temperature, entropy is typically greater for heavier atoms or more complex molecules. Entropy increases when a system is heated and when solutions form. Using these guidelines, the sign of entropy changes for some chemical reactions may be reliably predicted.

16.3 The Second and Third Laws of Thermodynamics

The second law of thermodynamics states that a spontaneous process increases the entropy of the universe, Suniv > 0. If ΔSuniv < 0, the process is nonspontaneous, and if ΔSuniv = 0, the system is at equilibrium. The third law of thermodynamics establishes the zero for entropy as that of a perfect, pure crystalline solid at 0 K. With only one possible microstate, the entropy is zero. We may compute the standard entropy change for a process by using standard entropy values for the reactants and products involved in the process.

16.4 Free Energy

Gibbs free energy (G) is a state function defined with regard to system quantities only and may be used to predict the spontaneity of a process. A negative value for ΔG indicates a spontaneous process; a positive ΔG indicates a nonspontaneous process; and a ΔG of zero indicates that the system is at equilibrium. A number of approaches to the computation of free energy changes are possible.

A reaction has a natural tendency to occur and takes place without the continual input of energy from an external source.

(a) spontaneous; (b) nonspontaneous; (c) spontaneous; (d) nonspontaneous; (e) spontaneous; (f) spontaneous

Although the oxidation of plastics is spontaneous, the rate of oxidation is very slow. Plastics are therefore kinetically stable and do not decompose appreciably even over relatively long periods of time.

There are four initial microstates and four final microstates.

The probability for all the particles to be on one side is 1 32. This probability is noticeably lower than the 1 8 result for the four-particle system. The conclusion we can make is that the probability for all the particles to stay in only one part of the system will decrease rapidly as the number of particles increases, and, for instance, the probability for all molecules of gas to gather in only one side of a room at room temperature and pressure is negligible since the number of gas molecules in the room is very large.

There is only one initial state. For the final state, the energy can be contained in pairs A-C, A-D, B-C, or B-D. Thus, there are four final possible states.

The masses of these molecules would suggest the opposite trend in their entropies. The observed trend is a result of the more significant variation of entropy with a physical state. At room temperature, I2 is a solid, Br2 is a liquid, and Cl2 is a gas.

(a) C3H7OH(l) as it is a larger molecule (more complex and more massive), and so more microstates describing its motions are available at any given temperature. (b) C2H5OH(g) as it is in the gaseous state. (c) 2H(g), since entropy is an extensive property, and so two H atoms (or two moles of H atoms) possess twice as much entropy as one atom (or one mole of atoms).

(a) Negative. The relatively ordered solid precipitating decreases the number of mobile ions in solution. (b) Negative. There is a net loss of three moles of gas from reactants to products. (c) Positive. There is a net increase of seven moles of gas from reactants to products.

C6 H6 (l) + 7.5O2 (g) ⟶3H2 O(g) + 6CO2 (g) There are 7.5 moles of gas initially, and 3 + 6 = 9 moles of gas in the end. Therefore, it is likely that the entropy increases as a result of this reaction, and ΔS is positive.

(a) 107 J/K; (b) −86.4 J/K; (c) 133.2 J/K; (d) 118.8 J/K; (e) −326.6 J/K; (f) −171.9 J/K; (g) −7.2 J/K

100.6 J/K

(a) −198.1 J/K; (b) −348.9 J/K 1354 Answer Key This content is available for free at https://cnx.org/content/col11760/1.9

As ΔSuniv < 0 at each of these temperatures, melting is not spontaneous at either of them. The given values for entropy and enthalpy are for NaCl at 298 K. It is assumed that these do not change significantly at the higher temperatures used in the problem.

(a) 2.86 J/K; (b) 24.8 J/K; (c) −113.2 J/K; (d) −24.7 J/K; (e) 15.5 J/K; (f) 290.0 J/K

The reaction is nonspontaneous at room temperature. Above 400 K, ΔG will become negative, and the reaction will become spontaneous.

(a) 465.1 kJ nonspontaneous; (b) −106.86 kJ spontaneous; (c) −53.6 kJ spontaneous; (d) −83.4 kJ spontaneous; (e) −406.7 kJ spontaneous; (f) −30.0 kJ spontaneous

(a) −1124.3 kJ/mol for the standard free energy change. (b) The calculation agrees with the value in Appendix G because free energy is a state function (just like the enthalpy and entropy), so its change depends only on the initial and final states, not the path between them.

(a) The reaction is nonspontaneous; (b) Above 566 °C the process is spontaneous.

(a) 1.5 × 102 kJ; (b) 21.9 kJ; (c) −5.34 kJ; (d) −0.383 kJ; (e) 18 kJ; (f) 71 kJ

(a) K = 41; (b) K = 0.053; (c) K = 6.9 × 1013; (d) K = 1.9; (e) K = 0.04

In each of the following, the value of ΔG is not given at the temperature of the reaction. Therefore, we must calculate ΔG from the values ΔH° and ΔS and then calculate ΔG from the relation ΔG = ΔH° − TΔS°.

(a) K = 1.29;

(b) K = 2.51 × 10−3;

(c) K = 4.83 × 103 ;

(d) K = 0.219;

(e) K = 16.1

The standard free energy change is ΔG°298  = −RT ln K = 4.84 kJ/mol. When reactants and products are in their standard states (1 bar or 1 atm), Q = 1. As the reaction proceeds toward equilibrium, the reaction shifts left (the amount of products drops while the amount of reactants increases): Q < 1, and ΔG298 becomes less positive as it approaches zero. At equilibrium, Q = K, and ΔG = 0.

The reaction will be spontaneous at temperatures greater than 287 K.

K = 5.35 × 1015 The process is exothermic.

1.0 × 10−8 atm. This is the maximum pressure of the gases under the stated conditions.

x = 1.29 × 10−5 atm = PO2

−0.16 kJ

(a) −22.1 kJ; (b) 61.6 kJ/mol

90 kJ/mol

(a) Under standard thermodynamic conditions, the evaporation is nonspontaneous; (b) Kp = 0.031; (c) The evaporation of water is spontaneous; (d) PH2 O must always be less than Kp or less than 0.031 atm. 0.031 atm represents air saturated with water vapor at 25 °C, or 100% humidity.

(a) Nonspontaneous as ΔG298 ° > 0; (b) ΔG298 ° = −RT ln K, ΔG = 1.7 × 103 + (8.314 × 335 × ln 28/ 128) = −2.5 kJ. The forward reaction to produce F6P is spontaneous under these conditions.

ΔG is negative as the process is spontaneous. ΔH is positive as with the solution becoming cold, the dissolving must be endothermic. ΔS must be positive as this drives the process, and it is expected for the dissolution of any soluble ionic compound.

(a) Increasing PO2 will shift the equilibrium toward the products, which increases the value of K. ΔG°298 therefore becomes more negative.

(b) Increasing PO2 will shift the equilibrium toward the products, which increases the value of K. ΔG°298 therefore becomes more negative.

(c) Increasing PO2 will shift the equilibrium the reactants, which decreases the value of K.

ΔG°298 therefore becomes more positive.