In the Classroom
On the Importance of Ideality Rubin Battino* Department of Chemistry, Wright State University, Dayton, OH 45435;
[email protected] Scott E. Wood Department of Chemistry, Illinois Institute of Technology, Chicago, IL 60606 Arthur G. Williamson Department of Chemical and Process Engineering, University of Canterbury, Christchurch, New Zealand
The concept of an ideal phenomenon (such as ideal gas behavior or an ideal solution) is introduced early in the study of chemistry. In fact, some might think that most of the content of high school and general chemistry courses is “ideal” in the sense that simplifying generalizations are made that bear but a small connection to the more sophisticated and complete information found in graduate courses and monographs. It is generally considered acceptable to use these simplifications and generalizations—idealities—to introduce a subject, particularly to students who will not be learning any more chemistry. Idealizations are simplifying assumptions that represent limiting cases of real behavior. The discussion of “ideal” phenomena has a practical side in the ease of explaining those phenomena. But, ideality has an even more practical aspect for making calculations concerning real systems easier to carry out. In this essay we illustrate the utility of ideality in gaseous phenomena, solutions, and the thermodynamic concept of reversibility. So, the thesis of this paper is that idealists are practical. Some Basic Definitions The concept of reversibility is one of several kinds of limiting behavior, and it is central to understanding ideality in thermodynamics. Although ideal phenomena are “unreal”, the calculations of ideal changes of state set the limits for these changes of state and “limits” is the key word here. It is useful to know the maximum heat available from a particular process, the maximum efficiency of a heat engine operating between two temperature reservoirs, or the minimum amount of work required to compress a gas. Maxima and minima are connected to reversibility.
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A reversible change is one that can be “reversed” by an infinitesimal change in one of the variables (e.g., pressure, temperature, volume, or concentration) describing a particular state. In a sense, the driving and restoring forces approach zero. But before discussing the concept of reversibility in detail, it is first necessary to present some basic definitions. The definitions in this section are based heavily on Klotz and Rosenberg (1), Wood and Battino (2), and Battino and Wood (3). We define the system to be any region of matter that we wish to discuss and investigate. The surroundings consist of all other matter in the universe that can have an effect on or interact with the system. The universe (in thermodynamics) consists of the system plus its surroundings. There is always a boundary between the system and its surroundings; interactions between the two occur across this boundary, which may be real or hypothetical. The boundary has certain properties. It can be rigid or flexible or semipermeable, or adiabatic (a wall that permits no transfer of heat) or diathermic (a wall permitting heat transfer). An isolated system, which itself may be composed of many subsystems some of which may be considered as surroundings to a particular subsystem, is surrounded by a boundary that permits no interaction whatsoever between the entire isolated system and its surroundings beyond the boundary. A closed system is one in which no matter can be transferred between a system and its surroundings, whereas an open system permits such a transfer of matter. The state of the system is defined by ascribing definite values to a sufficient number of state variables (such as P, V, T, and n). A change of state involves changing the values of one or more of the state variables. We speak of the path when considering how these values change. That is, the path is the sequence of states between the initial and final states of the system. The
Journal of Chemical Education • Vol. 78 No. 10 October 2001 • JChemEd.chem.wisc.edu
In the Classroom
term process can then be used to describe how the system changes from one state to another. Consider an isothermal process—a change of state whose path is an isotherm; that is, the temperature is constant throughout the entire change of state. A change of state is isothermal when the initial and final states are at the same temperature. Thus, an isothermal change of state may or may not involve an isothermal path. The commonly discussed Joule expansion or experiment for an ideal gas involves the expansion of the gas into a vacuum. Two chambers that are separated by a valve are inside an adiabatic wall. An ideal gas is in one chamber, and the other chamber is evacuated. When the valve is opened, an adiabatic expansion occurs with attendant pressure and temperature fluctuations. In the case of the system being an ideal gas, the change of state is isothermal (same initial and final equilibrium temperatures), but the path between the two states, and hence the process, is not. There are many intermediate temperatures and pressures. Of course, a sufficient amount of time must elapse before equilibrium in this isolated system is attained. Many changes of state involve expansions and compressions of a gas. There are three ways we can interpret the word “compression” in changes of state: (i) increase in pressure at constant volume (corresponding to an isochoric or constantvolume increase in temperature); (ii) decrease in volume at constant pressure (corresponding to an isobaric or constantpressure decrease in temperature); and (iii) increase in pressure and decrease in volume at the same time (corresponding to an isothermal increase in pressure and decrease in volume or an adiabatic increase in temperature and pressure while the volume decreases). Although the processes in i and ii are not commonly thought of as compressions (process iii is the one most people think about), we can legitimately think of all three as compressions, and will do so in this paper. Expansions would simply be the reverse of these processes.
Calculations can then be done in terms of system properties, and in particular, in terms of the system equation of state. That is, Pext dV(reversible) = Pgas dV = (RT/V )dV. A common illustration used to help students understand reversible processes involves a piston-and-cylinder arrangement that is immersed in a constant-temperature environment (thermostat). At equilibrium, the piston has a small mound of sand on it. A grain of sand may be added to the top of the piston by sliding the grain from an adjacent shelf at the same height as the piston, or a grain of sand may be removed in a similar fashion. So, the process may be reversed by the addition or removal of a single grain of sand. After equilibrium is once again attained, another grain of sand is added (compression of the confined gas) or removed (expansion of the gas). In the limit, one can imagine that the grains of sand become infinitesimally small. The movement of finite-sized grains of sand imparts some momentum to the piston, which is assumed to have mass. If the piston rises or falls in a gravitational field, then there will also be a potential energy change related to position in this field. The movement of the piston is assumed to be frictionless. This illustration has some difficulties, but is readily understood by students. How the grains of sand are moved is not considered. An illustration of how four (isothermal, isobaric, isochoric, adiabatic) reversible processes may occur is shown in Figures 1a to 1d. There are still some difficulties with this apparatus, but conceptually, it has a number of advantages. In all cases, we use a horizontal piston and cylinder arrangement.
a
Reversible and Irreversible Processes A reversible process is an idealized process that proceeds through an infinite number of equilibrium states, any one of which may be “reversed” by an infinitesimal change. At each and every equilibrium state all of the forces within the system and between it and its surroundings are in balance. These sequential equilibrium states that make up the reversible process will differ, for example, by a δP or a δT or a δF or a δᏱ or a δni ; that is, infinitesimal changes in pressure (P) or temperature (T ) or force (F ) or EMF (Ᏹ) or amount of substance (ni) (number of moles). Since a reversible process is an ideal process, it cannot be realized in nature. Its value in thermodynamics as a thought experiment is that reversible processes are useful for establishing the limits of real processes, and that real-world processes would be easier to calculate. Also, the changes in the thermodynamic functions entropy, Gibbs energy, and Helmholtz energy can only be evaluated along these idealized reversible paths. Reversible processes may be approximated by some real processes. One aspect of the reversible process, which is used but not explicitly recognized, is that the work exchange between the system and surroundings (e.g., Pext dV ) can, in the limiting case of reversibility when Pext = Psys, be written Psys dV.
b
c
d
Figure 1. a: Isothermal reversible compression or expansion. b: Adiabatic reversible compression or expansion. c: Isochoric temperature change. d: Isobaric temperature change.
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The piston may have mass, it can be adiabatic or diathermal, it is rigid, and it moves in a frictionless manner. There are two pressure reservoirs connected to the right-most chamber (RMC): one contains a gas at a pressure higher than any needed for the experiment, and the other is at a pressure lower than any that may be required. Each pressure reservoir is connected to the RMC via a three-way adjustable microflow valve so that at the smallest flow the pressure may be increased or decreased via an infinite number of equilibrium states. For example, the flows could conceivably be at the rate of one molecule per minute. Processes can obviously be reversed by turning the valve to reverse the flow of gas into or out of the RMC. Let us first consider an isothermal reversible expansion (or compression) using the apparatus in Figure 1a. The system and surroundings are initially at equilibrium at the temperature of the thermostat. The cylinder walls are diathermal. The microflow of gas into or out of the RMC will result in an isothermal compression or expansion of the ideal gas system. If this apparatus were modified so that slugs of gas were added to or removed from the RMC, then this would impart some momentum to the piston and also result in temperature and pressure fluctuations in the system, which would slowly damp out until equilibrium was once again reached before the addition of the next slug of gas. The compression (or expansion) would then be a quasi-static one; that is, it would go through a finite number of equilibrium states. The limit of the quasi-static finite stepwise process is the reversible one described above. Thus, a quasi-static process is irreversible because all forces are not balanced at each and every finite step. The essence of a reversible process is that all forces are balanced at each and every infinitesimal step. Two other examples of reversible processes are described briefly to extend the concept. In an electrochemical cell at equilibrium, the voltage supplied by a potentiometer (or a variable constant-voltage source) opposing and balancing the cell EMF may be infinitesimally increased or decreased to change the direction of the cell reaction from its equilibrium state. In a chemically reactive open system at equilibrium, the point of equilibrium in the reaction may be changed by infinitesimal changes in pressure (certain reactions only) or temperature, or by the addition or removal of an infinitesimal amount of reactant or product. Reversible processes result in changes of state via ideal processes in which forces (of all kinds) acting at the boundary between the system and its surroundings are balanced. They may be approximated, but only approximated, by quasi-static processes. An adiabatic reversible expansion (or compression) may be carried out using the apparatus in Figure 1b. The walls and the piston are adiabatic. Infinitesimal gas flows into or out of the RMC will result in adiabatic changes in the system. An isochoric reversible temperature increase or decrease utilizes the apparatus in Figure 1c. The system is separated from the middle chamber by a rigid diathermal wall. All the other walls are adiabatic. The micro addition of gas to the RMC will compress the gas in the middle chamber, raising its temperature and thus raising the temperature of the system. Infinitesimal gas flows can be reversed at any time to bring about temperature increases or decreases at constant volume. The apparatus for an isobaric reversible change in temperature is a bit more complicated, as indicated in Figure 1d. The 1366
temperature on the right side is changed infinitesimally as for the isochoric changes shown in Figure 1c. To maintain constant pressure, the gas in the left-hand chamber is actually a barostat that maintains constant pressure, and the adiabatic piston serves simply to keep the system pressure constant. As indicated, there is one diathermal wall; the rest are adiabatic. We have briefly mentioned irreversibility. An irreversible process is one involving unbalanced forces within the system or between it and its surroundings. Consider an isolated system that is left to itself. We know from experience that such a system will eventually change toward some final equilibrium state. We also know that the direction of this change cannot be reversed without the use of some other system external to the original isolated system. All experience shows that this characteristic of an isolated system progressing toward an equilibrium state is universal; such a change is called an irreversible (natural) process. Since a reversible process is an idealized limiting one, all natural processes (in isolated systems) must be irreversible. For example, a cube of ice in a glass of 20 °C water at atmospheric pressure will naturally melt. The addition of an external refrigerating system can stop the ice from melting by sufficiently cooling the water. The melting of ice at 20 °C in water is irreversible. Let us next consider the reversible and irreversible transfer of heat. First, the transfer of heat between a system and its surroundings requires a difference in their temperatures; for example, the system is at T and the surroundings are at T + ∆T. The heat effect, Q, is calculated from Q = C (T2 – T1) = C∆T, where C is the heat capacity. The transfer of an infinitesimal amount of heat, δQ, would require an infinitesimal temperature difference of δT between the two bodies, or δQ = C δT, and δT is the change in temperature of the system. But the reversible transfer of heat between the system and its surroundings requires that their temperatures be the same! (This is not a problem as long as one has the concept of a limiting process.) The isothermal reversible compression or expansion of a gas (see Fig. 1a) involves the transfer of heat between the system and the surrounding thermostat through the diathermal walls. The infinitesimal compression of the system increases its temperature an infinitesimal δT. It is this infinitesimal temperature difference that results in a flow of δQ from the system to the thermal reservoir. At the next equilibrium, the temperature of the system is the same as that of the thermal reservoir. In the limit, the reversible transfer of heat requires δT to become vanishingly small. Time, per se, is not a factor, since a sufficient amount of it is always allowed to attain equilibrium states. How can you reversibly add or remove a quantity of heat over a range of temperature? For example, isobaric and isochoric changes of state for an ideal gas involve significant changes of temperature and the transfer of significant amounts of heat. A way to conceptualize reversible isobaric and isochoric changes of the temperature of a system were illustrated in the discussion related to Figures 1c and 1d. In effect, these mechanisms expose the system via the diathermal wall to a series of thermal reservoirs with infinitesimally increasing or decreasing temperatures. So, rather than postulating an infinite series of thermal reservoirs, Figures 1c and 1d show how they can be realized, if only in an ideal manner. The quasi-static transfer of heat over a temperature range would involve the addition or removal of slugs of gas from the RMC.
Journal of Chemical Education • Vol. 78 No. 10 October 2001 • JChemEd.chem.wisc.edu
In the Classroom
The entropy function is defined by the equation dS = DQrev /T That is, the entropy function is defined in terms of the reversible transfer of heat. (The D is used to indicate that the heat effect is an inexact differential.) Entropy is a state function, and changes in entropy are solely dependent on the initial and final states for any given change of state. But these entropy changes can only be evaluated along reversible paths. Since knowledge of entropy changes is important for understanding many phenomena and for many practical calculations, the seeming fiction of the idealized reversible process is of great practical value. The two practical thermodynamic functions, Gibbs energy (G = H – TS ) and Helmholtz energy (A = U – TS ), are defined in terms of the entropy function. This means that ∆G and ∆A must also be evaluated along reversible paths via reversible processes. Ignoring this fundamental requirement for the evaluation of ∆S, ∆A, and ∆G has led many an unwary person to grief. On the other hand, the calculation of changes in the energy (∆U ) and the enthalpy (∆H ), as well as heat and work effects, do not require the condition of reversibility. Reversible Heat and Work Effects The first law may be represented in differential form as dU = DQ + DW where d denotes an exact differential and D denotes an inexact differential. Upon integration, ∫ dU = ∫ DQ + ∫ DW or
∆U = Q + W
Energy changes as represented by ∆U are state functions whose values depend only on the initial and final states. However, the heat effect Q and the work effect W result from the integration of inexact differentials along specified paths, and their values are path dependent. (See Kivelson and Oppenheim [4 ] for a good description of work in irreversible processes.) The distinction between effects like heat and work, whose values are path dependent, and thermodynamic state functions like energy and entropy, which are not, cannot be overemphasized. The path must be specified to calculate values of Q and W for a given change of state. (Of course, if either Q = 0 or W = 0 for a given change of state, then the value of the other is given by ∆U.) Ideal and Real Gases The ideal gas is defined by the two relations PV = nRT and
(∂U/∂V )T,n = (∂U/∂P)T,n = 0
or U is a function of T and n only. The first is the familiar ideal gas equation (IGE), an equation of state (EOS) that expresses the relationship between the pressure, temperature, volume, and the amount of substance. R is the universal gas
constant. The second equation expresses the condition that for an ideal gas the energy is not dependent on the volume or the pressure, and is a function of the temperature alone. At room temperature and atmospheric pressure the behavior of gases like helium, hydrogen, neon, and argon is reasonably well described by the IGE. However, at high temperatures, pressures, and gas densities (related to V /n, the molar volume) this simple equation is no longer accurate. There are a great many EOSs that use additional terms to accurately describe real gas behavior over a wide range of conditions. A necessary characteristic of these EOSs is that they must reduce to PV = nRT in the limits of zero pressure or zero density. Implicit in the IGE are that the gas molecules occupy zero volume and there are no forces of interaction between the gas molecules. As zero pressure or density is approached, the size of a gas molecule and interactional forces become insignificant. Remember that the IGE is a bulk description. It has no direct molecular interpretations. Conversely, a molecular model with noninteracting point particles can lead to the IGE. We briefly illustrate EOSs with two examples. The van der Waals EOS introduces two constants in addition to R. The constant b is related to molecular size (repulsive forces) and a is related to interactional (attractive) forces. Notice that the following equation reduces to PV = nRT as V → ∞. (P + an2/V 2)(V – nb) = nRT In the first factor an2/V 2 becomes negligible with respect to P; in the second factor nb becomes negligible with respect to V. There are tables of experimentally determined a and b values for many gases. The virial EOS is in two forms: a power series in the reciprocal of the molar volume (V /n) or in the pressure. PV = nRT [1 + B(n/V ) + C(n/V )2 + D(n/V )3 + … ] PV = nRT [1 + βP + γP 2 + δP 3 + … ] Again, notice that both forms reduce to the IGE as V → ∞ or P → 0. Generally, two virial coefficients (B and C, or β and γ) are all that is needed for practical calculations. The form of the virial EOS permits greater accuracy by the addition of more terms. The virial coefficients are calculable from various theories. Equations of state are necessarily based on the IGE. Real gas calculations can be considered to be perturbations of the IGE—that is, changes that are relatively easy to evaluate— but are based on the IGE. Heat engines that operate between two thermal reservoirs can be modeled by the ideal Carnot cycle. When an ideal gas is used as the fluid in a Carnot cycle, the derivation of the efficiency yields e = (T2 – T1)/T2 This equation sets the limit of efficiency for such heat engines— a useful practical number. Since in practice T1 (the lowertemperature reservoir, typically a nearby river) could not be lowered, the only way to increase efficiency was to raise T2. In steam-powered plants this meant going to superheated steam. These efficiency limits are useful for the analysis of the heat engines that are invented yearly that violate the first or second law of thermodynamics.
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Ideal and Real Solutions For a two-component system an ideal solution is defined as one that is described by Raoult’s law: P1 = x1 P1° and P2 = x2 P2° where x1 and x2 are mole fractions, P1 and P2 are the partial pressures of components 1 and 2, and P1° and P2° are the saturated vapor pressures of the pure solvents at the temperature of the measurement. (This implies that the solution is ideal and that the vapors behave as an ideal gas.) This means that each partial pressure is directly proportional via the molecular concentration (mole fraction) to the pure solvent vapor pressure. Another way of characterizing an ideal solution is via the fugacities, where f1 = x1 f1° and f 2 = x 2 f 2° where f1° and f 2° are the fugacities of the pure liquids. In fact, fugacity is an artifice to maintain the simplicity of the ideal equations for real systems. (See a physical chemistry or thermodynamics text for more information on fugacities and activities.) A consequence of the definition of an ideal solution is that there is no heat or volume effect on mixing. That is, in the formation of an ideal solution where both components are at the same initial temperature, the final temperature is identical to the initial temperature. For volume, the volume of an ideal solution (constant-temperature process) is strictly the sum of the volumes of each component. Real solutions show a heat effect and a nonadditive volume change on mixing. The mixing can be endothermic or exothermic and can show an expansion or a contraction. Based on the definition of an ideal solution, it is possible to calculate the ideal solubility of one substance in another. This turns out to be purely a function of solvent properties; it is independent of the nature of the solute. These ideal solubilities permit the calculation of a two-component phase diagram
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and the composition of the eutectic mixture. Surprisingly, these “ideal” calculations are frequently close to reality and are good rule-of-thumb estimates. Colligative properties (boiling point elevation, freezing point depression, osmotic pressure) can be estimated from ideal solution calculations. For an ideal solution the solubility product is written in terms of molar concentrations. For real solutions the solubility product equation must be written in terms of activities (“effective” concentrations). For accurate calculations the activities must be used, but the “ideal” solubility product is quite useful, particularly at lower concentrations. Ideal solutions are the model and base for real-solution calculations. Summary It is our hope that the discussion in this paper will help the reader understand the practicality of ideality and the importance of limits. Ideal phenomena provide the bedrock for real phenomena. Acknowledgments RB acknowledges a guest professorship at the Institute of Physical Chemistry of the University of Vienna, an Erskine Fellowship at Canterbury University, and helpful discussions with E. Wilhelm. The comments of the reviewers of an earlier version of this paper are appreciated. Literature Cited 1. Klotz, I. M.; Rosenberg, R. M. Chemical Thermodynamics, 6th ed.; Benjamin/Cummings: Menlo Park, CA, 2000. 2. Wood, S. E.; Battino, R. Thermodynamics of Chemical Systems; Cambridge University Press: Cambridge, 1990. 3. Battino, R.; Wood, S. E. Thermodynamics: An Introduction; Academic: New York, 1968. 4. Kivelson, D.; Oppenheim, I. J. Chem. Educ. 1966, 43, 233.
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