Practical Chemical Thermodynamics for Geoscientists

By Bruce Fegley Jr.

Practical Chemical Thermodynamics for Geoscientists covers classical chemical thermodynamics and focuses on applications to practical problems in the geosciences, environmental sciences, and planetary sciences. This book will provide a strong theoretical foundation for students, while also proving beneficial for earth and planetary scientists seeking a review of thermodynamic principles and their application to a specific problem.

• Strong theoretical foundation and emphasis on applications
• Numerous worked examples in each chapter
• Brief historical summaries and biographies of key thermodynamicists—including their fundamental research and discoveries
• Extensive references to relevant literature

• Language: English
• Publisher: Academic Press
• Release date: Oct 22, 2012
• ISBN: 9780080918143

Book preview

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Preface

In this work, when it shall be found that much is omitted, let it not be forgotten that much likewise is performed…

—Samuel Johnson (1755)

If during the course of this book we help disclose to the student some of the beauty and simplicity of the thermodynamic method, if we convince a few practical chemists of the extreme practicality of the results of thermodynamic calculations, if we contribute in some measure toward making chemistry an exact science, our task is rewarded.

—Gilbert N. Lewis and Merle Randall (1923)

I decided to write this book because of my research and teaching experience, which involve the application of chemical thermodynamics to problems in the earth and planetary sciences. During the past 18 years, I have taught Thermodynamics and Phase Equilibria, a one-semester course for advanced undergraduates and incoming graduate students in the Department of Earth and Planetary Sciences at Washington University. Chemical thermodynamics is also extensively used in my one-semester courses Planetary Geochemistry and Earth System Science for advanced undergraduates and incoming graduate students.

I wrote this book for two groups of readers. The first group is the advanced undergraduates or graduate students in the earth and planetary sciences who are taking a course in chemical thermodynamics. The second group is the researchers in earth and planetary sciences who want a review of one or more topics in chemical thermodynamics or who need some thermodynamic data to help them solve problems related to their work. I hope that students and researchers with interests in the earth and planetary sciences but who are formally in related disciplines such as astronomy, chemistry, and physics, will also find this book a useful resource.

The organization and style of the book reflect my biases and opinions about the good and bad points of the many thermodynamic monographs and texts that I have used at one time or another over the past 30 years. Most chapters here begin with an overview of the subjects covered in each section of the chapter. All chapters except the first one have worked examples in the text, a number of original figures and/or tables, and about 20 problems. The references to articles and books cited in all chapters are collected together into one list. Some of the citations are to classic works by the pioneers of thermodynamics but the more recent literature is covered as well. The sidebars with brief biographies of scientists who developed classical thermodynamics and/or applied it in the earth and planetary sciences are intended to show a little of the human side of the development of thermodynamics.

I personally find worked examples extremely useful for learning and to illustrate the methods involved in solving the real-world problems that one encounters. Consequently, the book contains a large number of examples. A few examples are long because they deal with the complicated type of real-world problems that are seldom treated in most texts.

This book emphasizes the fundamental principles and their application to practical problems in calculations that students can do by hand or by using readily available computer programs such as spreadsheets and curve-fitting programs. However, the use of specialized programs for doing thermodynamic calculations is not covered. I believe it is a mistake to teach the use of computer programs in an introductory course. Students must first learn the fundamental principles and use them to solve problems themselves. Only after this firm foundation is established can students learn how to use specialized computer programs for thermodynamic calculations with critical judgment. Instead of training students to use computer programs (and many times to blindly trust the results because they do not understand how they were obtained), we should be training them to write the next generation of programs.

Many of the examples and problems involve non-SI units. Anyone reading the older scientific literature (i.e., pre-1970) inevitably encounters units such as calories, atmospheres, psi, mmHg, Amagats, and other officially disapproved units. It is important for students to have experience converting between these units and SI units so that they understand the older literature.

I dislike the qualitative figures schematically showing the alphabet variables A, B, and C in many thermodynamic texts. Thus, this book has many quantitative figures with numerical axes, and many of the figures here refer to concrete examples and to real compounds. For example, Figures 4-6 and 6-6 show the heat capacity, enthalpy, and entropy for corundum and molten alumina. Unfortunately, it is impossible and sometimes impractical to use quantitative figures throughout and to totally avoid the use of the dreaded A, B, and C.

The problems in this book have been tested in my courses and fall into three categories: (1) examples from everyday life, (2) research done by some of the scientists who made important contributions in thermodynamics, and (3) problems of current interest in the earth and planetary sciences. A solutions manual gives answers to all of the problems in the book.

It is very useful to have reliable thermodynamic data at hand for solving problems and examples in the book and for scientific research. For this purpose, the book contains tabular data throughout the text and two short appendices with thermodynamic data for some elements, inorganic compounds, minerals, and some organic compounds at 298.15 K and as a function of temperature. However, only a relatively small number of compounds can be included in the data tables without making the book unreasonably long. I selected compounds of broad interest in the geosciences. For example, Tables 3-3 and 3-4 give heat capacity equations for many common minerals in the Earth’s crust and for many important gases in planetary atmospheres. The thermodynamic data in the book focus on: (1) important gases for atmospheric chemistry in the terrestrial and other planetary atmospheres; (2) carbon, nitrogen, sulfur, and phosphorus compounds involved in the biogeochemical cycles of these elements; and (3) common terrestrial minerals and minerals found in lunar samples and meteorites.

A number of colleagues have assisted me in one way or another during preparation of this book and I am glad to acknowledge their help. My collaborator Rose Osborne made important contributions in several areas. She had the thankless task of drawing (and redrawing) many of the figures from my sketches and descriptions; she prepared a number of the biographies and was responsible for finding portraits to accompany them; she was responsible for many of the computations involved in preparation of the thermodynamic data appendices, and she provided solutions for a number of the problems. Laura Schaefer, who has worked with me during the past eight years, provided invaluable assistance on many aspects of the book. Many of the students in my classes have asked questions, found errors, and made comments that have helped me during the preparation of this book. In this regard, in particular I want to acknowledge the help of Brian Shiro. The Washington University library system provided invaluable assistance in finding and obtaining numerous articles and books, and I especially thank Nada Vaughn (former head of interlibrary loans), Rob McFarland (chemistry librarian), and Ben Woods (former chemistry library assistant) for their efforts. Frank Cynar, my original editor at Academic Press, and subsequent editors at Elsevier have shown the patience of Job in waiting for delivery of the book manuscript. Heather Tighe (project manager) and Katy Morrissey (editorial project manager) at Elsevier helped Rose and me to transform the manuscript into a book. And as always, Katharina Lodders, my wife and colleague, has provided valuable advice and support during the past years while I was finishing the manuscript.

Although I have tried to eliminate all errors, this is impossible to do in any book, especially in a thermodynamics book with a large number of equations and numerical values. I also aimed to include a wide range of examples and problems in the earth and planetary sciences, but I have undoubtedly slighted one discipline or another to some extent. Much effort went into the preparation of thermodynamic data tables throughout the book and the appendices, but still, some values may be undoubtedly incorrect because of sign errors, mistakes in transcription, or use of the wrong reference. I hope that readers will bear in mind Dr. Samuel Johnson’s often repeated quote and kindly advise me of the errors, miscalculations, mistakes, and omissions that they may find.

Chapter 1

Definition, Development, and Applications of Thermodynamics

A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression which classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown. …

—Albert Einstein (1949)

I Historical Development of Chemical Thermodynamics

Thermodynamics is the scientific study of the relationships between heat and other forms of energy. From 1790 to about 1850, experimental and theoretical work by Sadi Carnot, Emile Clapyron, Rudolf Clausius, Count Rumford, James Joule, Hermann Helmholtz, G. H. Hess, Lord Kelvin (William Thomson), and Robert Mayer led to a quantitative understanding of the transformation of heat into work and the maximum amount of work that can be obtained from an engine. These relationships form the first and second laws of thermodynamics.

Scientists began to use the word thermodynamics in scientific articles and books in the mid-19th century. For example, while he was developing the second law of thermodynamics (1849–1854), William Thomson (later Lord Kelvin) wrote several papers published in the Transactions of the Royal Society of Edinburgh about fundamental principles of general thermo-dynamics and about a perfect thermo-dynamic engine. The great English geologist Sir Roderick Impey Murchison (1792–1871) referred to the principles of thermo-dynamics in the 1867 edition of Siluria, which summarized his work on the Silurian system in Great Britain and other countries. By the early 1870s the hyphen was dropped and James Clerk Maxwell wrote about the First Law of Thermodynamics in his book The Theory of Heat.

From about 1850 to 1890, experimental studies by Marcellin Bertholet, Henri Le Chatelier, Robert Kirchhoff, Walther Nernst, Julius Thomsen, J. H. van’t Hoff, and other scientists developed many applications of thermodynamics to chemistry. During the same period, theoretical work originated by J. W. Gibbs and later developed by J. D. van der Waals and his colleagues, along with experimental studies by H. W. B. Roozeboom, led to an understanding of phase equilibria in single-component and multicomponent systems.

From about 1890 to 1920, experimental studies by Fritz Haber, G. N. Lewis, Walther Nernst, Wilhelm Ostwald, and others led to an understanding of the driving force of chemical reactions and finally to the prediction of chemical equilibria. The two landmark books in this era were Haber’s 1905 book, Thermodynamics of Technical Gas Reactions, and Thermodynamics by G. N. Lewis and M. Randall (1923). Around 1908 Haber synthesized ammonia from nitrogen and hydrogen. This breakthrough led to a Nobel Prize in Chemistry for Haber (1918). Carl Bosch, who developed Haber’s work into industrial NH3 synthesis at the German chemical company BASF, later shared the Nobel Prize in Chemistry in 1931.

During the first half of the 20th century, W. F. Giauque demonstrated the validity of the third law of thermodynamics (Nernst’s heat theorem) and developed magnetic cooling (adiabatic demagnetization). He was awarded the 1949 Nobel Prize in Chemistry for this work. During the same period, Sir Ralph Fowler, E. A. Guggenheim, Joseph Mayer, Maria Goeppert Mayer, and Richard C. Tolman developed statistical mechanics, which Gibbs originated around 1900. Statistical mechanics is used to calculate the thermodynamic properties of materials from the properties of their constituent atoms and molecules.

Three figures stand out in the era since World War II. The Norwegian chemist Lars Onsager (1903–1976) won the 1968 Nobel Prize in Chemistry for his development of the reciprocal relations that describe the mutual transport of mass and heat during irreversible processes, such as a sugar cube dissolving in a cup of coffee. Irreversible processes are ubiquitous in nature and are important not only for everyday events such as sugar dissolving in coffee or tea, but also for the operation of thermocouples (simultaneous conduction of electricity and heat), chemical weathering of minerals, metamorphic reactions, and gas-grain reactions in the solar nebula (some combination of mass, electrical charge, and heat transport in these three cases). The Russian chemist Ilya Prigogine (1917–2003), who did most of his work in Belgium, won the 1977 Nobel Prize in Chemistry for his research in nonequilibrium thermodynamics. Prigogine developed a nonlinear theory to describe systems (such as living organisms) that are far from equilibrium. His work shows how order can develop from chaos. Harold Urey (1893–1981), who won the 1934 Nobel Prize in Chemistry for his discovery of deuterium, pioneered the application of chemical thermodynamics to cosmochemistry and developed methods to compute equilibrium isotopic fractionations. An important application of the latter work is the use of isotopes, such as oxygen isotopes, as geothermometers.

II Pioneering Applications in the Geosciences

Going back to the 19th century, geoscientists became aware of the field of thermodynamics at an early stage of its development. As we discuss in Chapter 4, the German mineralogist and physicist Franz Neumann (1798–1895) developed a rule for calculating the heat capacities of chemical compounds that is known as the Neumann-Kopp rule (actually a good approximation under the right circumstances). During the mid-19th century, the French chemist and mineralogist Henri Sainte-Claire Deville (1818–1881) conducted pioneering experiments on the thermal dissociation of steam and other compounds.

At the end of the 19th century, thermodynamics began playing an increasingly important role in the geosciences. The German salt industry hired the Dutch physical chemist J. H. van’t Hoff to study the origin of the Stassfurt salt deposits in Saxony. These deposits date from the Zechstein period, about 250 million years ago, and were enormously valuable at the time for the production of halite, magnesium salts, potassium salts, and bromine. Van’t Hoff’s study of the phase equilibria during evaporation of seawater was a major advance in understanding the origin of these and other evaporate deposits. At about the same time, the American geologist and physicist Carl Barus (1856–1935) made measurements of the specific heat, volume change on melting, and change of the melting point with pressure for diabase. Barus also studied the pressure-volume-temperature (PVT) properties of gases and the use of different alloys for high-temperature thermocouples.

Two other major advances in the application of chemical thermodynamics to geosciences took place at about the same time. One was the foundation in 1905 of the Geophysical Laboratory of the Carnegie Institution in Washington, D.C. The personnel in the Geophysical Laboratory, in particular Norman L. Bowen (1887–1956), pioneered the experimental study of igneous phase equilibria. The other advance was V. M. Goldschmidt’s (1888–1947) study of metamorphic phase equilibria, notably the reaction

(1-1)

which takes place when a silica-bearing limestone is altered to wollastonite. Goldschmidt used Nernst’s heat theorem (1906) to determine the P-T boundary between calcite + quartz and CO2 + wollastonite from the meager experimental data available at the time (1912). As part of this work Goldschmidt developed the mineralogical phase rule, which states that the number of minerals in a rock is less than or equal to the number of components.

III Thermodynamics versus Kinetics

Today, chemical thermodynamics is an integral part of modern geosciences, with applications ranging from the geochemistry of the Earth’s interior to marine chemistry, biogeochemistry, air chemistry, and planetary science. Some examples of what thermodynamics can tell us include:

• The composition of a volcanic gas, such as those erupted at Kilauea on Hawaii.

• The crystallization sequence of minerals during cooling of an igneous magma or during evaporation of seawater.

• The condensation sequence of minerals in the solar nebula and expected in refractory inclusions in meteorites, such as the Ca,Al-rich inclusions in the Allende CV3 chondrite.

However, thermodynamics, although extremely useful, has its limitations. Two examples of what thermodynamics cannot and does not tell us are the rate of a chemical reaction and the mechanism by which a reaction proceeds. These two topics fall into the domain of chemical kinetics. The difference between kinetics and thermodynamics is easily illustrated by considering formation of corundum (Al2O3) grains in the solar nebula.

Corundum is found in some of the Ca,Al-rich inclusions in Allende and other meteorites. Chemical and mineralogical evidence indicates that the corundum formed via gas-solid reactions in the solar nebula, the cloud of gas and dust from which the solar system originated. Chemical thermodynamic models of nebular chemistry predict that corundum formed via the reaction

(1-2)

as the nebular gas cooled. Reaction (1-2) is a net thermochemical reaction that shows the overall change of reactants into products, and it predicts the temperature at which corundum forms (commonly called the condensation temperature) as a function of the total pressure. On the other hand, Eq. (1-2) does not express the actual course of events or tell us how fast things happen as atoms and molecules react to form corundum.

However, we can attempt to answer these questions using information from laboratory studies of metal oxidation and reactions in flames. This research gives information about the elementary reactions that actually take place between atoms and molecules. The overall result of a sequence of elementary reactions is simply the net thermochemical reaction. In the case of Eq. (1-2), a plausible set of elementary reactions is

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

Reactions (1-3) to (1-7) add up to Eq. (1-2), which is the net thermochemical reaction. The M in several reactions is a third body, which is any other gas. In the absence of kinetic data we cannot say anything about the rate of Eq. (1-2). Although it is possible to estimate the reaction rate, we will not do so here. Instead we will embark on our study of chemical thermodynamics and learn how to apply it to a wide variety of problems in the earth and planetary sciences.

Chapter 2

Important Concepts and Mathematical Methods

Mathematics is a language.

—Remarks attributed to Willard F. Gibbs at a Yale faculty meeting about the relative merits of mathematics and languages (Lewis and Randall, 1923)

In this chapter we lay the foundation for our study of thermodynamics. Several terms used throughout the book are introduced. Then, the nature of temperature and pressure and their measurement are explored. Pressure and temperature are used in almost every thermodynamics problem we solve, so it is important to have a practical understanding of these properties and how they are measured. Finally, we describe some basic concepts in calculus.

I Definitions

Before we can begin discussing the laws and applications of thermodynamics, we have to understand the language that we will be using. This section presents and defines some unfamiliar words, and some familiar words are given specific meanings to avoid ambiguity. Other definitions will be given throughout the book when we need to use them and are not given here.

System. A system encompasses everything being studied and can be defined in such a way as to make it easier to solve the problem at hand. The rest of the universe outside the system is the surroundings. Some examples of a system are the air + fuel mixture inside the cylinder of a car engine, a balloon filled with helium gas, the coffee in a coffee cup, an electric or magnetic field in an otherwise empty container, or a mineral assemblage in a rock. The boundary between a system and its surroundings can be real, like the cup holding the coffee, or it can be imaginary. The example of a mineral assemblage in a rock illustrates a system with an imaginary boundary because we chose only the assemblage as the system, even though it is an integral part of the rock. A system also need not contain matter, as in the example of an electric or a magnetic field; however, in this book we will generally consider systems that involve matter in some form (solid, liquid, or gas). A system can be closed or open; a closed system (e.g., a sealed bottle) does not exchange matter with its surroundings, but an open system (e.g., an active volcano erupting magma and volcanic gases) does. Both closed and open systems can exchange energy with their surroundings; an isolated system does not exchange anything with the surroundings.

Homogeneous system. Only one phase with uniform properties is present in a homogeneous system. For instance, the water in a glass and the air inside a balloon are two examples of homogeneous systems. Even though air is composed of a number of gases, it has a uniform composition and properties and acts as a homogeneous system.

Heterogeneous system. Two or more phases are present in a heterogeneous system. For example, a soft-drink bottle containing soda and air is a heterogeneous system because two phases (gas and liquid) are present. Likewise, a glass of a soft drink containing air, the soda, and an ice cube is also a heterogeneous system because three phases (gas, liquid, solid) are present. Other examples of heterogeneous systems are crystals + melt in a cooling magma, rock + fluid in a hydrothermal system, or dust + gas in the solar nebula. We discuss heterogeneous systems and phase equilibria in more details in Chapters 7 and 10–12 Chapter 11 Chapter 12.

State. The properties of a system determine its state. For example, the temperature (T), pressure (P), and volume (V) determine the state of a gas. An equation of state mathematically relates the properties of a system. The ideal gas law (PV = nRT) is an equation of state that relates T, P, V, and n, the number of moles, using R, the ideal gas constant. For reference, a mole of a substance is Avogadro’s number (NA = 6.02214199 × 10²³ or 6.022 × 10²³ to three decimal places) of particles of that substance. For example, a mole of helium gas is 6.022 × 10²³ atoms of helium, and a mole of O2 gas is 6.022 × 10²³ oxygen molecules.

Extensive variable. An extensive variable depends on the size of a system or on the quantity of material in a system. For example, the volume of a gas is an extensive variable because it depends on the size of the container holding the gas. Likewise, the mass of a rock is an extensive variable. The kinetic energy (= ½mv²) of a system is also an extensive variable because it depends on the mass (m) as well as the velocity (v). In general, the energy of a system is an extensive variable.

Intensive variable. An intensive variable is characteristic of a system and does not depend on the size of the system. The temperature of a gas is an intensive variable because it does not depend on the size of the container holding the gas. Refractive index, viscosity, and surface tension are also intensive variables. Specifying the amount of material present (e.g., dividing by the mass) transforms extensive variables into intensive variables.

State variable. A state variable depends only on the difference between the initial and final states of the system. It does not depend on the path between the initial and final states. The schematic diagrams in Figure 2-1 illustrate this concept. These diagrams show the initial (PA, VA) and final (PB, VB) states of a system and four different paths between the two states. A path is the route from the initial to the final state of a system—for example, how the volume and pressure of the air + fuel mixture inside the cylinder of a car engine change as the piston moves. Pressure and volume are state variables. Thus, the volume change (ΔV = VB − VA) and the pressure change (ΔP = PB − PA) between the initial state A and the final state B are the same in all four cases shown in Figure 2-1, even though the paths between the initial and final states are different.

FIGURE 2-1 Volume plotted as a function of pressure. The change in volume Δ V = V B − V A and the change in pressure Δ P = P B − P A are the same in each case and are path independent.

All the thermodynamic functions that we use throughout this book are state variables. This is very important, because it means that changes in these functions do not depend on the way a mechanical process or chemical reaction occurs. However, as described in Chapter 3, heat and work are not state functions. The heat flow and the work done in a mechanical process, such as compression or expansion of the air + fuel mixture in the cylinder of a car engine, or during a chemical reaction such as iron rusting, depend on the path taken.

II Pressure and Temperature

Temperature and pressure are two variables of great importance in thermodynamics and are both intensive variables (i.e., their values do not depend on the size of the system in question). For example, if you have a drinking glass and a bucket sitting on top of your desk, the temperature and pressure in each is the same, even though the volumes and masses are different. Let’s take a moment to talk about temperature and pressure.

A What is temperature?

When we think of temperature in our everyday lives, it relates to the relative hotness or coldness of an object. However, what makes an object hot? The temperature of a body is related to the average kinetic energy of the atoms or molecules that make up the body by the equation

(2-1)

where K is the average kinetic energy and k is Boltzmann’s constant (k = 1.381×10−23 J K−1). Kinetic energy is the energy of motion and is related to the velocity of an object by the equation

(2-2)

where m is mass (kg) and v is velocity (m s−1). Equation (2-1) shows that by putting energy into a system, we can raise the temperature. Conversely, as energy dissipates, temperature falls. We can put energy into a system by heating it (e.g., heating a pan of water on top of a stove) or by other methods. Joule’s experiments, discussed in Chapter 3, showed that mechanical agitation of water would increase the water’s temperature. Water molecules gain kinetic energy as they move around, so water will be slightly warmer after it is stirred. Any process that increases the average velocity of particles in a system increases the temperature of the system.

What happens if two systems of different temperatures contact one another? Imagine putting a hot spoon that just came out of the dishwasher into a drawer on top of a cold spoon. The spoons are now touching each other. Since the molecules in the two spoons can collide with each other, those from the higher-temperature spoon will transfer some of their kinetic energy to the molecules in the lower-temperature spoon during collisions. In this way, the warmer spoon will cool off while the cooler spoon warms up. This process will continue until both spoons have the same temperature. Each spoon is a system, and when the temperatures of both spoons stop changing, the two systems will be at thermal equilibrium.

Imagine that a system A, spoon or otherwise, is in contact with a system B, and system B is in contact with system C. If B is in thermal equilibrium with both A and C, then A must be in thermal equilibrium with C as well. This is the zeroth law of thermodynamics, and it is the basis of temperature measurement. We discuss another important concept, the difference between heat and temperature, in Chapter 3 as part of our discussion of the first law.

B Temperature measurement

Generally, temperature is measured indirectly by measuring its effects. Most materials expand when heated, and each material has its own characteristic thermal expansion coefficient (α). Most familiar temperature-measuring devices are based on this principle. Table 2-1 lists average thermal expansion coefficients for some materials used in thermometers and for some common geological materials. Volumetric and linear thermal expansion coefficients express the fractional change in volume or length, respectively, with temperature.

Table 2-1. Some Typical Thermal Expansion Coefficients (α) for Different Materials a

aThe values listed are multiplied by 10⁵, that is, the thermal expansion coefficient for He gas is 3.661×10-3 and that for coesite is 6.9×10-6. The data are from Fei (1995), Skinner (1966), Smyth et al. (2000), Yang and Prewitt (2000), and Lide (2000).

bWater has a negative thermal expansion coefficient below 4°C, that is, it contracts when heated.

Figure 2-2 shows a simplified diagram of a liquid-in-glass thermometer, which is an evacuated tube filled partly with alcohol or mercury and sealed at both ends. The liquid in the tube expands on heating and fills more of the tube. The tube has marks that relate liquid volume (proportional to liquid column length) with temperature via the thermometer equation,

(2-3)

FIGURE 2-2 Liquid expanding inside a mercury-in-glass thermometer.

The calibration of a mercury-in-glass thermometer illustrates the use of Eq. (2-3). The thermometer is immersed in a mixture of ice + water at the melting point of ice (0°C), and the distance from the bottom of the bulb to the top of the mercury column is measured. This is the distance l0. A similar measurement is made when the thermometer is immersed in boiling water at 1 atmosphere pressure. This is the distance l100. Then the temperature (t°C) corresponding to distance l can be calculated and the temperature scale can be etched on the thermometer. Note that when l = l0, Eq. (2-3) gives t = 0°C, and that when l = l100, Eq. (2-3) gives t = 100°C.

Another familiar device is made of two strips of metal with different coefficients of thermal expansion. Figure 2-3 is a schematic diagram of this device, which is used in household thermostats. Two thin strips of metal are fused together in a coil. The metal with the larger thermal expansion coefficient is on the outside of the coil. As this system warms up, the outside layer expands more than the inside layer and forces the coil to tighten. A pointer attached to the end of the coil moves in the direction of the curve (to the right in our picture) as the coil gets hotter. Again, there are marks that correlate a given amount of curvature in the coil with a given temperature.

FIGURE 2-3 The bimetallic metal coil winding inside a thermostat.

Thermal expansion is not the only temperature-dependent property that is measured. The frequency of the light given off by a hot body varies with temperature and is measured with an optical pyrometer. Temperature is calculated by comparison to the frequency of the radiation that would be given off by an ideal black body. A black body is a theoretically perfect absorber and emitter of radiation. The radiation emitted through a pinhole in a hot cavity (e.g., a metal sphere) gives a good approximation to an ideal black body.

As you might expect, temperature affects the electrical properties of materials as well. The temperature-dependent electrical resistance of platinum metal wire in a platinum resistance thermometer or of a semiconducting metal oxide bead in a thermistor is used to measure temperature. Thermocouples determine temperature by measuring the temperature-dependent electromotive force (EMF) in volts developed between two different metals. A different EMF is generated at each temperature by each different pair of metals. The use of thermocouples is widespread in industry and research laboratories.

Figure 2-4 shows a schematic diagram of a thermocouple. The thermocouple is made of two wires that have different compositions and are welded together at one end. Figure 2-5 shows such a weld. The wires are inside an electrically insulating tube (e.g., plastic or ceramic) so that they do not touch each other at any point other than the welded spot. This part is exposed to the system of which the temperature is being measured. The other ends of the wires connect to copper leads at the cold junction, which is a reference point. The cold junction is inside an ice bath at 0°C or is left at room temperature, in which case a correction must be made. The copper leads connect to an instrument such as a potentiometer or a digital voltmeter that measures EMF. Each EMF corresponds to a particular temperature. Commercial thermocouple meters typically store emf-T data for different types of thermocouples in memory. The emf-T data for the type of thermocouple connected to the meter are selected by flipping a switch. These meters also automatically correct for the nonzero temperature at the cold junction and convert the EMF readings into temperatures. Table 2-2 lists common thermocouple types and their temperature ranges. Figure 2-6 shows a plot of T versus EMF for K- and S-type thermocouples.

FIGURE 2-4 A sketch of a thermocouple.

FIGURE 2-5 Photograph of the welded bead at the end of a thermocouple.

Table 2-2. Some Common Types of Thermocouples

Data from The Temperature Handbook, Omega Engineering, Inc., Stamford, CT, 1995, and the NIST ITS-90 thermocouple Web pages (http://srdata.nist.gov/its90/main/).

FIGURE 2-6 Electromotive force plotted versus temperature for K- and S-type thermocouples.

As you can see, thermocouples are useful because they can measure a wide range of temperatures. They also have a small mass, which means that they reach thermal equilibrium relatively quickly and are used to follow a change in temperature.

The low mass of a temperature-measuring device is important for another reason: If you were to put a large, cold thermometer into a small, hot sample, you would not get a very accurate reading. This is because the thermometer would effectively lower the temperature of the sample! You can see that this might be a problem during the measurement of minute temperature changes or of the temperatures of very low-mass systems.

C Development of the international temperature scale

Another difficulty in temperature measurement has been deciding on a scale to use. Scientists want to be able to compare their results, so it is necessary to have one standard. What should that standard be based on? Historically, there has been little agreement, so a number of different temperature scales (and thermometers) have been used (see Partington, 1949). In late 16th- and early 17th- century Italy, Galileo used a thermoscope, a constant-volume gas thermometer. This device consisted of an air-filled bulb with a vertical tube below it dipping into a vessel of water or wine. As the bulb’s temperature varied, the fluid column either rose or fell; however, the readings also depended on atmospheric pressure because the bottom vessel was open to air. Sealed alcohol in glass thermometers with etched scales were used at the Florentine Academy in Italy and by Ferdinand II, Duke of Tuscany, in the 1640s. The latter is credited with their invention. The fixed points for these thermometers were those of a mixture of ice + salt and blood heat. In the 1660s, Robert Hooke (1635–1703) and Robert Boyle (1627–1691) in England used thermometers with the melting point of ice, 0°C (Hooke), or the freezing point of oil of anise, about 17°C (Boyle), as the lower fixed points.

DANIEL GABRIEL FAHRENHEIT (1689–1736)

Fahrenheit began to take an interest in making scientific instruments in 1701, after the premature deaths of his parents, when he had to take up a trade. Beginning in 1707, he traveled extensively to meet and observe other instrument makers. He set up his own business in Amsterdam in 1717. In about 1708 he began making the thermometers that have made his name famous. He believed them to be graduated on Ole Roemer’s scale; however, there was a misunderstanding about the upper fixed point, so Fahrenheit’s scale was in fact unique. (Roemer had not published a description of his scale, but Fahrenheit watched him graduating thermometers during a visit.)

Fahrenheit made all sorts of equipment, including a thermometer that could be used to measure atmospheric pressure. He knew that the boiling points of liquids varied with atmospheric pressure, so he made a device that allowed a person to read the atmospheric pressure directly from a measurement of the boiling point of water.

Fahrenheit became a member of the Royal Society in 1724, though he had no formal scientific training and published very little. This was common at the time. Instrument makers and scientists worked closely together because the design and quality of the instruments were critical for allowing the researchers to carry out their work.

WILLIAM JOHN MACQUORN RANKINE (1820–1872)

William John Macquorn Rankine was a Scottish engineer. During his youth, poor health meant that he had to be educated at home by his father and by private tutors. He later attended college but did not complete a degree. In spite of this fact, he became professor of civil engineering and mechanics at the University of Glasgow in 1855. He was also one of the founders and first president of the Institution of Engineers in Scotland and a Fellow of the Royal Society.

Like many of his time who contributed to the field of thermodynamics, Rankine worked for a railroad. He developed a technique for laying out circular curves of track and investigated such topics as the best shape for wheels and how to prevent axles from breaking. He also penned a popular series of engineering textbooks and worked on plans for the water supply of the city of Glasgow.

Rankine developed his own absolute temperature scale that was based on the Fahrenheit scale because he felt reluctant to give up traditional British units of measurement in favor of the Celsius scale and the metric system. He even wrote a song about the issue of the encroaching metric system, titled The Three-Foot Rule. Its lyrics went, in part, A party of astronomers went measuring of the Earth, And forty million mètres they took to be its girth; Five hundred million inches, though, go through from pole to pole; So let’s stick to inches, feet, and yards and the good old three-foot rule.

Modern temperature scales date to the early 18th century. In 1702, Ole Christian Roemer (1644–1710), the Danish astronomer, adopted a temperature scale based on two fixed points: the temperature of ice or snow and the boiling point of water. In 1714, Daniel Gabriel Fahrenheit, whose life is briefly described in the sidebar, constructed the first useful sealed mercury thermometer and took 0°F as the temperature of a mixture of solid NH4Cl, ice, and water (−18°C). He arbitrarily divided the interval between this temperature and that of the melting point of ice into 32 degrees. Fahrenheit measured the boiling point of water to be 212°F. The Fahrenheit scale is still used in daily life in the United States. The Rankine temperature scale, named after the Scottish engineer William J. M. Rankine (see his biography in the sidebar), is the absolute Fahrenheit scale. The Rankine scale is related to the Fahrenheit scale by the equation T (°R) = T (°F) + 459.67°, and it is still used today by engineers.

The Centigrade scale, which is based on the freezing (0°C) and boiling (100°C) points of air-saturated water, dates to the 1740s, when centigrade scales (= 100 degrees) were proposed by Anders Celsius (see sidebar) and Carl Linnaeus (1707–1778). The temperatures of the freezing and boiling points are arbitrarily chosen; in fact, Celsius proposed 100° for the freezing point and 0° for the boiling point of water. Linnaeus proposed the reverse, and his Centigrade scale was the one used. In 1954, Linneaus’s Centigrade scale was replaced by a new scale named after Celsius, which is not a centigrade scale but is based on the triple point of water.

ANDERS CELSIUS (1701–1744)

Anders Celsius was a Swedish astronomer who became a professor of mathematics and astronomy at the University of Uppsala, where his father and grandfather had also taught astronomy. After being appointed professor in 1730, Celsius went abroad to round out his education. He joined a French group in planning and executing a trip to Lapland to measure meridian lines. Their findings confirmed Isaac Newton’s hypothesis that Earth is somewhat flattened at the poles. This was a daring undertaking for several reasons, not least of which was at that time in Sweden and elsewhere, the Copernican view of the solar system was sacrilegious.

Celsius had an ability to win friends and garner support. After his travels, he was able to obtain permission and funding for an observatory at the university. His other achievements in the field of astronomy included measuring the magnitudes of the stars in Aries and writing a discourse on observations by himself and others of the aurora borealis.

Celsius’s name is best known for the 100° temperature scale. Celsius was not the first to propose such a scale, but his observation that two constant degrees, or fixed points, were enough to define a temperature scale led to the general acceptance of the 100° scheme.

Before describing this revision, we introduce the Kelvin, or absolute, temperature scale used in thermodynamics. William Thomson (Lord Kelvin; see sidebar) proposed this scale in 1848 and discussed it in several later papers. The Kelvin scale is related to the Celsius (and Centigrade) scales by the equation T (°C) = T (K) − 273.15, where the temperature of the freezing point of air-saturated water is 273.15 K. The Kelvin scale is based on the second law of thermodynamics and, like the Centigrade scale, was originally defined to have a 100° interval between the freezing and boiling points of water.

WILLIAM THOMSON, LORD KELVIN (1824–1907)

Lord Kelvin was born William Thomson in Belfast, Ireland, though he was of Scottish decent. He entered the University of Glasgow at the age of 10 and the University of Cambridge at the age of 17. He became professor of natural philosophy at the University of Glasgow in 1846 and taught there until his retirement in 1895. At that time, he enrolled himself as a research student and continued his association with the university until his death.

Kelvin’s long friendship with James Joule began in 1847 at an historic meeting of the British Association. Kelvin and Joule discovered the principle of Joule-Kelvin cooling of gases, which is important for refrigeration. In the early 1850s Kelvin developed the thermodynamic (Kelvin) temperature scale and formulated the second law of thermodynamics contemporaneously with Rudolf Clausius. Kelvin was the leading British physicist throughout his lifetime. Queen Victoria knighted him in 1866 for his work on the first transatlantic telegraph cable, and she made him a peer in 1892. Kelvin is buried in Westminster Abbey, next to Sir Isaac Newton.

In 1854, Kelvin pointed out that this was a clumsy definition and instead proposed to define the temperature of a single fixed point and the size of the degree. His suggestion was not adopted at the time. The Nobel Prize–winning chemist William F. Giauque (1895–1982) revived Kelvin’s proposal in 1939. By then, it was clear that it was experimentally very difficult to reproduce the freezing point of air-saturated water within a few hundredths of a degree. Different laboratories reported values ranging from 273.13 K to 273.17 K for the freezing point (0°C). These difficulties led to problems at low temperatures, where an uncertainty of 0.01 degree gave larger and larger errors as lower temperatures were achieved.

The Tenth General Conference on Weights and Measures adopted the Kelvin-Giauque proposal in 1954. The Kelvin and Centigrade (renamed Celsius) temperature scales are based on a single fixed point, the triple point of water. This is the equilibrium temperature for ice + water + water vapor. It was set at 0.01°C (273.16 K). The size of the degree was set as one Kelvin (1 K), which is 1/273.16 of the thermodynamic (Kelvin) temperature of the triple point of water. One degree Celsius has the same size as one Kelvin.

Although these definitions unambiguously define the Kelvin and Celsius temperature scales, it is not possible for every laboratory to maintain a triple point cell as a temperature standard. An international temperature scale that allows users all over the world to calibrate their thermometric devices to various fixed points is needed.

The Seventh General Conference of Weights and Measures adopted the first international temperature scale in 1927. This specified the temperatures of fixed points, such as the freezing point of gold, and the instruments and methods for making temperature measurements with as close an approximation to thermodynamic temperatures as could be done at that time. The freezing point and the melting point are the same at equilibrium, but the temperature scale is specified in terms of the one that can be determined more accurately. The international temperature scale was revised in 1948, 1968, 1976, and most recently when the Eighteenth General Conference of Weights and Measures adopted the International Temperature Scale of 1990 (ITS-90). Table 2-3 lists temperatures of fixed points on ITS-90. These temperatures are as close as possible to the actual thermodynamic values. As mentioned earlier, the Kelvin scale is based on the second law of thermodynamics. It does not depend on the physical properties (e.g., thermal expansion coefficient, electrical properties) of any given material. We return to these issues in Chapter 6 when we discuss the second law of thermodynamics, entropy, and heat engines. Table 2-4 lists conversion factors between common temperature scales. Further information about the history of thermometry, the development of temperature scales, and modern methods of temperature measurement are given in volume 1 of Partington (1949), by Quinn (1983), and in Zemansky (1957).

Table 2-3. Defining Fixed Points of the ITS-90

a The light isotope ³He is used from 0.65 K to 3.2 K, and the heavier, more abundant isotope ⁴He is used from 1.25 K to 2.18 K and 2.18 K to 5.0 K.

b The triple point is the temperature and pressure at which the gas, liquid, and solid phases all exist in equilibrium.

Table 2-4. Temperature Conversion Factors

D What is pressure?

Pressure is defined as force per unit area, and force is given by mass times acceleration. Thus, we can calculate the pressure (P) exerted by this book or another object lying on a table by multiplying the mass (m) of the object by the average acceleration of gravity (g = 9.81 m s−2) and dividing by the surface area (A) of the bottom of the object.

Example 2-1. The dictionary in Figure 2-7 has the dimensions 26 by 22 by 13 centimeters and mass 4 kg. The average acceleration due to gravity (g) is 9.81 m s−2. The pressure exerted is

(2-4)

When the book is lying on its side (a), the area of the bottom of the book is 0.0572 m². When it is standing on end (b), the area is only 0.0286 m², or half of what it was before. Therefore,

(2-5)

(2-6)

However, in the two cases the force (about 39.2 N) is the same. The SI unit of pressure is the pascal (Pa). A pascal is one newton per meter squared (N m−2). A newton is the SI unit of force and is equal to one kilogram meter per second squared (kg m s−2).

FIGURE 2-7 A dictionary standing on its side (a) and standing upright (b).

In thermodynamics, we consider the pressure exerted by a gas more often than that exerted by a solid mass. Gas pressure is a statistical property because it depends on averaging values such as velocity over a very large number of bodies. Gases exert pressure in a way that is similar to solid bodies in that the pressure can still be described as force per unit area. A collection of gas particles in a balloon is moving around in all directions. The force of a great many collisions of gas particles against the sides of the balloon is what keeps the balloon inflated. The pressure inside the balloon is equal to the average amount of force per area exerted against the sides of the balloon, which will depend on the average velocity of the particles, their mass, and on how many collisions occur. The average number of collisions in turn depends on the gas density or number of particles (N) per volume (V). The kinetic theory of gases shows us that

(2-7)

for an ideal gas, where m is the average mass of the particles, N is the number of particles present, v is their average velocity, and V is the volume of their container. We discuss the relationship between P and V of a gas when we describe Boyle’s law later in this chapter.

Example 2-2. At 300 K, N2 molecules have an average velocity of about 517 m s−1. What pressure would 0.04 moles of N2 gas at 300 K exert inside a 1 liter cylinder? The atomic weight of N is 14.00674 g mol−1, so the molecular weight of N2 is 28.01348 g mol−1.

(2-7)

E Pressure measurement

One of the first devices used to measure pressure was the manometer, a U-shaped tube partially filled with a fluid such as water, oil, or mercury, as pictured in Figure 2-8. One end of the tube connects to the system under observation. The other end is either open to the air, in which case the pressure measured is relative to the ambient pressure, or is a sealed vacuum, in which case an absolute pressure measurement is achieved (as in a barometer). A change in the pressure of the system changes the height of the mercury in the tube. The changes in height are measured visually or more precisely using a small telescope on a calibrated stand (a cathetometer). The change in height (h) is related to a change in pressure by

(2-8)

where ρ is the density of the fluid in the column, g is the acceleration due to gravity, and Pref is the atmospheric pressure (in the case of an open-ended manometer) or zero (for an absolute manometer that has a sealed vacuum at one end).

FIGURE 2-8 Pressure measurement with a manometer.

Example 2-3. In the late 19th century, the French physicist Émile Amagat (1841–1915) used very long, open-ended mercury manometers to measure pressures of compressed gases in his experiments. What is the gas pressure corresponding to a mercury column height of 200 meters (Δh = 200 m), assuming constant density (ρ) for Hg and constant acceleration due to gravity throughout the column and a reference pressure of one bar? At room temperature and ambient pressure ρ (Hg) = 13.534 g mL−1 = 13.534 kg L−1. Substituting into Eq. (2-8) and converting to consistent units we find

Figure 2-9 shows a Bourdon gauge. This device is used on compressed gas tanks in laboratories. The Bourdon gauge contains a curved tube, which uncoils as the pressure in it increases. The end of the tube connects to a pointer that indicates the pressure on a scale.

FIGURE 2-9 Photographs showing a Bourdon pressure gauge.

There are several different units for pressure. We will usually talk about bars or atmospheres. One bar is equal to 10⁵ Pa or 10² kPa. One atmosphere is equal to 1.01325 bars. As mentioned earlier, the pascal is the SI unit of pressure and is used in most countries in the world. For example, TV weather reports in Canada give atmospheric pressure in kilopascal (kPa). Other common units for measuring pressure are millimeters mercury (mmHg), used in the older scientific literature, or inches of mercury, used for TV weather reports in the United States. You are probably already familiar with the unit pounds per square inch (psi), used for measuring the air pressure in tires. As shown in Figure 2-9, the gauges on compressed gas tanks give pressure per square inch gauge (psig), which is the pressure (in psi) above ambient pressure (about 14.70 psi). Table 2-5 lists conversion factors for pressure measurement units. Throughout the rest of this book, pressure and P refer to pressure in bar unless otherwise stated. Absolute and relative presssure measurements, including in high vacuum systems, are given in Le Neindre and Vodar (1975) and by Tilford (1992).

Table 2-5. Pressure Conversion Factors

The value of g used is 9.80665 m s-2.

Here psi denotes pounds per square inch absolute.

III Boyle’s Law

Robert Boyle published the results of his experiments using the newly invented mercury barometer and vacuum pump in the early 1660s. He found that at constant temperature the volume of a fixed mass of gas is inversely proportional to the pressure

(2-9)

where P is pressure, V is volume, and C is a constant. An equivalent equation is

(2-10)

where P1 and V1 are the initial pressure and volume and P2 and V2 are the final pressure and volume of the gas, or

(2-11)

The PV product has units of energy, which are the same units used for work. Pressure is force per unit area (F/A), so PV = (F/A)(V) = (F)(V/A) = force times length = work (or energy). A table listing conversion factors for different units used for energy and work is in Chapter 3.

Table 2-6 gives the molar PV product for air at several pressures at constant temperature. This is simply the product of the molar volume times the pressure. The molar volume Vm = V/n, where n is the number of moles. Under the conditions listed in Table 2-6, the PV product for air is constant within 0.03% of the one bar value. The deviations from Boyle’s law occur because air is not an ideal gas. In fact, no gas behaves ideally under all conditions, but several gases (e.g., air, H2, He, Ne, Ar, O2, N2) approach ideality at normal temperatures as pressure is reduced to lower and lower values. Table 2-7 lists extrapolated values of molar PV products. The extrapolated values are almost the same for many gases. The mean molar PV product, 22.4140 atm liter mol−1, gives the molar volume of an ideal gas at one atmosphere pressure and 273.15 K. The molar volume of an ideal gas at one bar pressure and 273.15 K is slightly larger because one bar is a slightly lower pressure than one atmosphere and is 22.7110 liter-bar mol−1.

Table 2-6. PV Product for Air at 350 K

Source: Thermodynamic Properties of Air by Sychev et al. (1987).

Table 2-7. PV Product (atm L mol −1 ) for Selected Gases at One Atmosphere and Extrapolated to Zero Pressure, 273.15 K (0°C)

a(1+λ)=(PV)0/(PV)1ratio.

Sources: Coplen (2001), Din (1962), Partington (1949), Pickering (1928).

The variation of pressure and molar volume at 273.15 K for an ideal gas and for dry air at 300 K, a nearly ideal gas, are shown in Figure 2-10. This type of curve is an isotherm. An isotherm shows the correlated variation of the volume and pressure of a constant amount of gas at constant temperature. We will come back to isothermal curves later in this chapter when we discuss the PVT surface for an ideal gas.

FIGURE 2-10 Isotherms for air (300 K) and an ideal gas (273.15 K).

Example 2-4. A piston in a glass cylinder contains one liter of air at one bar pressure (see Figure 2-11). A pressure gauge is attached to the cylinder to measure the air pressure, and the volume of the cylinder can be read from markings on the side. The cylinder and piston are in a water bath kept at constant temperature of 25°C (298.15 K). The piston is lowered until the pressure of the air in the cylinder is 2 bars. What is the final volume of the air? Using Eq. (2-9) we find

Thus, the volume is halved when the pressure is doubled.

FIGURE 2-11 A cartoon of a piston compressing gas inside a cylinder.

IV Charles’s or Gay-Lussac’s Law

The knowledge that air expands when heated dates back to the ancient Greeks and Romans. Hero of Alexandria (AD 50) used heated air to open the doors of a temple by kindling a fire on an altar containing a concealed air reservoir. The heated air then drove water into buckets, which pulled down cords to open the doors (Partington, 1949).

In 1787, Jacques-Alexandre-César Charles (1746–1823) discovered that the volume of a gas is proportional to temperature. Later, in 1802–1808, Joseph Gay-Lussac (1778–1850) measured the variation of volumes of permanent gases (N2, O2, H2) with temperature at constant pressure. His results are given by the equation

(2-12)

where V0 is gas volume at 0°C (273.15 K), t is temperature in degrees Celsius, α0 is the coefficient of thermal expansion, and V is gas volume at another temperature t. Gay-Lussac found α0 varied from 37.40 × 10−4 °C−1 to 37.57 ×10−4 °C−1 for various gases with an average value

(2-13)

Subsequent work by the French chemist Henri Victor Regnault (1810–1878) in 1847, and later by many other workers, improved on these results. For example, Regnault found

for air, which is close to the modern value of

This value is based on determinations of the thermal expansion of gases as a function of pressure and temperature. An example of measurements for He, compiled by Partington (1949), is in Table 2-8 and shows how the value of α0 is computed by linear extrapolation of measurements at several low pressures to the limiting value of zero pressure.

Table 2-8. α 0 Values for Helium

Example 2-5. You are blowing up balloons for a New Year’s party and fill a balloon with one liter of air indoors at 25°C (t1). If you hang the balloon outside the front door, where it is only 10°C (t2), what will its volume