M13_PETR4521_10_SE_C13.QXD 9/17/15 4:31 PM Page 579 Spontaneous Change: Entropy and Gibbs Energy 13 CONTENTS 13-1 Entropy: Boltzmann’s View 13-5 Gibbs Energy of a System of Variable ¢ ¢ 13-2 Entropy Changes: Clausius’s View Composition: rG° and rG ¢ LEARNING OBJECTIVES 13-3 Combining Boltzmann’s and 13-6 rG° and K as Functions of Clausius’s Ideas: Absolute Entropies Temperature 13.1 Describe the concepts of microstate 13-4 Criterion for Spontaneous Change: 13-7 Coupled Reactions and entropy, and discuss how they are The Second Law of 13-8 Chemical Potential and Thermodynamics related. Identify situations in which entropy Thermodynamics of Spontaneous Chemical Change generally increases, and describe them in terms of the microstates involved. 13.2 Describe how Clausius’s equation can be used to obtain equations for calculating entropy changes for simple physical changes, including phase changes, constant pressure heating/cooling, and isothermal expansion/compression. Apply the resulting equations to calculate entropy changes. 13.3 Explain how the standard molar entropy of a substance is obtained. Use the standard molar entropies of reactants and products to determine the entropy change for a chemical reaction. 13.4 State the second law of thermodynamics, and identify the relationship between Gibbs energy, enthalpy, and entropy. 13.5 Predict the direction of spontaneous chemical change by using Kenneth William Caleno/Shutterstock values of the standard Gibbs energy of reaction (¢ G° ) and the thermodynamic Thermodynamics originated in the early nineteenth century with attempts to r reaction quotient (Q). improve the efficiency of steam engines. However, the laws of thermodynamics are widely useful throughout the field of chemistry and in biology and physics, as we 13.6 Use the van’t Hoff equation to discover in this chapter. calculate the equilibrium constant as a function of temperature. ur everyday experiences have conditioned us to accept that certain 13.7 Describe how the coupling of things happen naturally in one direction only. For example, a chemical reactions may make a nonspontaneous process become a bouncing ball eventually comes to rest on the floor, but a ball at rest O spontaneous one. will not begin to bounce. An ice cube placed in hot water eventually melts, but a glass of water will not produce an ice cube and hot water. A shiny iron 13.8 Discuss the relationships among nail rusts in air, but a rusty nail will not naturally shed its rusty exterior to chemical potential activity and the Gibbs produce a shiny nail. In this chapter, we explore concepts needed to under- energy of a mixture. stand why change happens naturally in one direction only. At the end of Chapter 7, we noted some chemical and physical processes that proceeded in a certain direction without external influence, that is, 579 M13_PETR4521_10_SE_C13.QXD 9/17/15 4:31 PM Page 580 580 Chapter 13 Spontaneous Change: Entropy and Gibbs Energy spontaneously (Section 7-10). Among those examples, we saw situations in which the internal energy, U, of the system increased, decreased, or stayed the same. Clearly, the internal energy change, ¢U, is not a reliable criterion for deciding whether or not a particular change will occur spontaneously. In 1850, Clausius introduced the concept of entropy to explain the direction of spontaneous change. Twenty-seven years later Ludwig Boltzmann pro- posed an alternative view of entropy based on probability theory. Not surpris- Sebastian Duda/Fotolia ingly, Clausius’s and Boltzmann’s definitions of entropy were eventually ▲ The melting of an ice cube shown to be equivalent. So what is entropy and why is it important? Simply occurs spontaneously at stated, entropy measures the dispersal of energy. It is an important concept temperatures above 0 °C. because a great deal of experimental evidence supports the notion that energy spontaneously “spreads out” or “disperses” if it is not hindered from doing so. Entropy is the yardstick for measuring the dispersal of energy. In this chapter, we will continue to interpret observations about macro- scopic systems by using a microscopic point of view. We will develop a con- ceptual model for understanding entropy and learn how to evaluate entropy changes for a variety of physical and chemical processes. Most importantly, we will define the criterion for spontaneous change and discover that it con- siders not only the entropy change for the system but also that of the sur- roundings. Finally, we will also learn about another important thermody- namic quantity, called Gibbs energy, which can also be used for understanding the direction of spontaneous change. 13-1 Entropy: Boltzmann’s View We will soon see that the criterion for spontaneous change can be expressed in terms of a thermodynamic quantity called entropy. Let’s first focus our atten- ▲ Spontaneous: “proceeding tion on developing a conceptual model for understanding entropy. Then, we from natural feeling or native will be able to use entropy, more specifically entropy changes, to explain why tendency without external certain processes are spontaneous and others are not. constraintÁ ; developing without apparent external Microstates influence, force, cause, or treatment” (Merriam-Webster’s The modern interpretation of entropy is firmly rooted in the idea that a macro- 23 Collegiate Dictionary, online, scopic system is made up of many particles (often 10 or more). Consider, for 2000). example, a fixed amount, n, of an ideal gas at temperature T in a container of volume V. The pressure of the gas is P = nRT/V. At the macroscopic level, the state of the gas is easily characterized by giving the values of n, T, V, and P. The state of the gas won’t change without some external influence (e.g., by adding more gas, increasing the temperature, compressing the gas). However, on the microscopic level, the state of the system is not so easily characterized: The molecules are in continuous random motion, experiencing collisions with each other or the walls of the container. The positions, velocities, and energies of individual molecules change from one instant to the next. The main point is that for a given macroscopic state, characterized by n, T, P, and V, there are ▲ A simple model for an many possible microscopic configurations (or microstates), each of which ideal gas is obtained by treat- might be characterized by giving the position, velocity, and energy of every ing the gas as a collection of molecule in the gas. Stated another way, the macroscopic properties of the gas, noninteracting particles con- such as its temperature, pressure, and volume, could be described by any one fined to a three-dimensional of a very large number of microscopic configurations. box. As we saw in Chapter 8, In this discussion, we suggested that the microstate of an ideal gas could be for a particle confined to a described by giving the position, velocity, and energy of every molecule in the box, the kinetic energy (trans- gas. However, such a description is not consistent with quantum mechanics lational energy) is quantized. Each microstate corresponds because, according to the Heisenberg uncertainty principle (page 323), exact to a particular way of distrib- values for the position and velocity (or momentum) of a particle cannot be uting the molecules among simultaneously specified. To be consistent with quantum mechanics, a the available translational microstate is characterized by specifying the quantum state (quantum numbers energy levels. and energy) of every particle or by specifying how the particles are distributed among the quantized energy levels. Thus, according to quantum mechanics, a M13_PETR4521_10_SE_C13.QXD 9/17/15 4:31 PM Page 581 13-1 Entropy: Boltzmann’s View 581 Relative energy E 2 = 4 E 1 = 1 L L U = 5, W = 1 U = 8, W = 5 (a) (b) E 4 = 4 Relative energy E = 9 3 4 E 2 = 1 E = 1 1 4 E 4 = 4 E = 9 3 4 E 2 = 1 E = 1 1 4 2L U = 5, W = 6 (c) ▲ FIGURE 13-1 Enumeration of microstates The distribution of five different particles among among the particle-in-a-box energy levels. The energies are expressed as multiples of h2/8mL2. The height of the pink shaded region represents the total internal energy of the system. (a) For a box of length L, there is only one possible microstate when U = 5 * (h2/8mL2). (b) When the internal energy is increased to 8 * (h2/8mL2), the number of microstates increases to five because more energy levels are accessible. (c) When length of the box is increased to 2L, the number of microstates for U = 5 * (h2/8mL2) increases to six. More energy levels are accessible because the energy levels are shifted to lower values and are more closely spaced. microstate is a specific microscopic configuration describing how the particles of a system are distributed among the available energy levels. Let’s explore the concept of a microstate by considering a system of five particles confined to a one-dimensional box of length L. (We discussed the model of a particle in a box on page 326.) To start, we use the energy level M13_PETR4521_10_SE_C13.QXD 9/17/15 4:31 PM Page 582 582 Chapter 13 Spontaneous Change: Entropy and Gibbs Energy expression E = n2h2 8mL2 to calculate a few energy levels. Representative n > energy levels, expressed as multiples of h 2/8mL2, are shown in the diagrams in Figure 13-1. Let’s place the five particles among these energy levels with the constraint that the total energy, U, of the system must be 5 * (h2/8mL2). The only possible arrangement (Fig. 13-1a) has all the particles in the n = 1 level. The total energy of this microstate is obtained by adding the particle energies: U = (1 + 1 + 1 + 1 + 1) * (h2/8mL2) = 5 * (h2 8mL2) > Notice that, for the situation just discussed, the state of the system can be described in two ways.