In the Classroom
The Microscopic Statement of the Second Law of Thermodynamics Igor Novak Department of Chemistry, National University of Singapore, Singapore 117543, Singapore;
[email protected] One source of problems in teaching and learning chemistry has been identified as the threefold representation of matter—macroscopic, microscopic, and symbolic (1). Chemical education at the university level encompasses all three representations. The students thus have to be comfortable with analyzing a particular phenomenon in any representation and be able to make conceptual shifts from one representation to another. Thermodynamics is one of the fields within the undergraduate curriculum where students encounter many learning difficulties. The difficulties are related to the nature of thermodynamics and the cumbersome ways in which it is sometimes presented. Understanding thermodynamics requires familiarity with macroscopic and microscopic concepts of matter simultaneously, unlike, for example, in quantum mechanics where the microscopic aspect prevails. Macroscopic versus Microscopic Representation The teaching of thermodynamics usually begins from the macroscopic aspect (classical thermodynamics) in the first year of study, followed by the introduction of the microscopic picture via statistical mechanics in later years. Freshmen are confronted with a variety of processes (isothermal, adiabatic, reversible, irreversible), functions (heat, work, enthalpy, entropy), and technological devices (heat engines, chemically driven systems, etc.) on an ad hoc, phenomenological basis. These topics are introduced early in the syllabus without a coherent conceptual framework and justification. In fact, the distinction between work and heat, which is one of the first topics described, is presented on a phenomenological level only. The notions of entropy and the second law of thermodynamics are also introduced from the macroscopic viewpoint. Their full understanding can, however, only be achieved at the microscopic level with statistical mechanics acting as a “bridge” between macroscopic and microscopic representations. Statistical mechanics uses the concept of partition functions and ensembles, which are too difficult for freshmen. The macroscopic approach to teaching thermodynamics is well established in practice, but has no pedagogical justification as it hinders the building of rational concept networks. One should proceed by first teaching simple concepts before proceeding to more complex concepts and systems—not the other way around. The reasons why the macroscopic approach appeared in the first place are partly historical (this is how thermodynamics as a discipline evolved) and partly mathematical (a more sophisticated mathematical reasoning is required in the statistical approach). Because the notions of heat, work, entropy, et cetera are so important, yet elusive, false analogies and imprecise statements abound in lectures and textbooks. The first inaccuracy is the association of entropy with disorder. The second 1428
inaccuracy is that heat and work are forms of energy. We have surveyed a few recently published, popular physical chemistry textbooks and found that definitions and ideas are still presented with confusing nonuniformity. In one of the best known, physical chemistry textbooks (2), the entropy is described as “a measure of the molecular disorder of a system”. Even though the wording “molecular disorder” was carefully chosen so as not to imply false analogies, such description is still likely to be visualized by analogy with the macroscopic disorder; a view that had been decisively refuted (3). On the other hand, the same textbook (2) states correctly that heat and work are modes of energy transfer. Another textbook (4) goes even further and uses the term “order–chaos change” together with the statement, “When thermal energy flows from one system to another, it is called heat …”. The third textbook (5) introduces similar unsatisfactory notions of heat flowing from one body to another, but adds a very useful and correct caveat, “However, order and disorder are subjective concepts, whereas probability is a precise quantitative concept. It is therefore preferable to relate S to probability rather than to disorder.” This brief perusal of the latest textbooks is sufficient to highlight problems related to the teaching of thermodynamics. We propose a complementary introduction to thermodynamics by using microscopic concepts only, followed by an explanation of how certain macroscopic results arise naturally from the microscopic picture. Some elements of the microscopic approach have been mentioned previously regarding the first law of thermodynamics (6, 7), but have not yet been adopted by standard textbooks of physical chemistry. Discussion We begin teaching thermodynamics by describing the system at the microscopic level, where only total energies of individual particles (atoms, molecules, ions), εi, and their number, ni, are well defined. There is no pedagogical difficulty because students are familiar with atomic and molecular orbitals, energy levels, and the conservation of energy. The notion of the bulk material containing a multitude of particles with different energies can be linked to the kinetic molecular theory of gases (and its experimental justification), which appears in the first-year chemistry syllabus. The key point is for students to realize that variables such as pressure, entropy, enthalpy, Gibbs free energy, heat capacity, et cetera describe huge ensembles of particles and have no meaning at the level of individual particles. The pedagogical advantage of microscopic approach lies in concept integration. This approach enables students to rationalize and predict a plethora of phenomena by using two fundamental ideas (quantization of matter and the probability-driven changes between the states of the system) and their corresponding quantitative expressions (first and second law of thermodynamics).
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
In the Classroom
The notion of total individual particle energy εi is valid for systems in any phase or of any composition. The macroscopic total energy (internal energy; E ) of the system can be expressed as the sum over i particles (6, 7) E = dE =
∑ i ni εi ∑ i dni εi
∑ i ni dεi
+
(1)
εi k ln
ni n0
(2)
where n0 is the number of particles in the lowest energy level, ε0 = 0, and k is the Boltzmann constant. Equation 2 is simply a rearranged Boltzmann distribution equation. The introduction of Boltzmann distribution at this point does not present great teaching difficulties, because the distribution can be derived without recourse to statistical mechanics (8, 9). The expression for dT, derived from eq 2 is:
d dT = k
ni εi n0
ni n ln i n0 n0
2
1
Changes of internal energy (dE ) are usually induced by energy transfer between the system and surroundings. The energy transfer can be realized in various ways and this variety leads to the existence of many functions and variables in classical thermodynamics. For reversible processes, changes in level populations, dni, or changes in level energies, dεi, are small. Another important macroscopic variable is temperature, T, which can also be expressed in terms of ni and εi , T = −
E
2
−
dεi n k ln i n0
(3)
Equation 3 suggests that changes in the system’s temperature can be induced by the changes in energy levels, dεi , level populations, dni, or both. This fact dispels students’ misconception that only the energy exchanged as heat leads to the temperature change in the system. Although eq 2 can be readily obtained, it does not appear in the textbooks where only a macroscopic definition of temperature (in terms of thermal equilibrium between two systems separated by a diathermic wall) is given. We discuss various thermodynamic processes by analyzing how the microscopic variables in eqs 1 and 2 can vary. The trivial case, dni = 0, dεi = 0, dE = 0, dT = 0, when nothing changes in the system, is excluded. At this stage, it is important for the students to appreciate that eqs 1 and 2 represent molecular forms of the first and second law of thermodynamics, respectively. Equation 1 has been introduced previously (6, 7) as the molecular form of the first law of thermodynamics, but without pedagogically insightful justification, which can be made as follows. The students are familiar from pre-university education with the principle of the conservation of energy, which they may have even verified experimentally. The notion of energy conservation leads directly to eq 1; that is,
0
Figure 1. The energy level diagram for process 1 (adapted from ref 6). The isochoric process, energy transfer as heat, dni ≠ 0, dεi = 0, dE = 2, is a macroscopic example of microscopic process 1.
the conservation of energy on the macroscopic scale is a direct consequence of additivity of particle energies on the microscopic scale. Thus energy additivity and energy conservation represent two sides of the same coin! Equations 2 and 3 embody the second law of thermodynamics, because the Boltzmann distribution is derived from it. Consequently eqs 2 and 3 also include notions of the most probable configuration and maximum entropy. After restating the laws in the language of microscopic descriptors we proceed with the analysis of some typical thermodynamic processes. Process 1: dni ≠ 0, dεεi = 0 In this process only the number of particles having particular energy is changed; that is, the particles are redistributed among energy levels after the energy exchange between the system and the surroundings took place (Figure 1). Equation 3 requires that in such process dT ≠ 0; that is, the temperature of the system must change as a consequence. A macroscopic example would be the isochoric process where the heating or cooling of the fixed quantity of gas in a container of constant volume takes place. One can prove that it is possible to change only the level populations by an example from spectroscopy. Heating the sample causes changes in the spectral band intensities, but not in the band positions (level energies). Process 2: dni = 0, dεεi ≠ 0 In this case the energy levels change, but not their populations (Figure 2). On the macroscopic scale this corresponds to energy transfer in the form of work that takes place between the system and the surroundings (6, 7). The work can be of mechanical (pV work), electrical, magnetic, or chemical type, but the process description is valid for all of them. This is an example of concept integration in teaching, which is our main objective. The explanation of how, for example, the volume change for gas leads to dεi ≠ 0 is available in undergraduate textbooks on thermodynamics (6, 7). Consideration of eq 3 reveals that also in this case dT ≠ 0, even though the energy exchange between the system and surroundings was in the form of work only. This is an important insight because students tend to think that temperature change occurs as a result of energy transfer in the form of heat only. A relevant macroscopic example for process 2 is an adiabatic
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In the Classroom
E
E
2
2
1
1
0.5
0.5
0
0
Figure 2. The energy level diagram for process 2 (adapted from ref 6). The adiabatic process, energy transfer as work, dni = 0, dεi ≠ 0, dE = 2, is a macroscopic example of microscopic process 2.
Figure 3. The energy level diagram for process 3 (adapted from ref 6). The energy transfer as heat and work, dni ≠ 0, dεi ≠ 0, dE = 3, is a macroscopic example of microscopic process 3.
expansion or compression of a fixed quantity of gas, which as our analysis makes clear, leads to a change in the system’s temperature. This is not surprising since it is known that real gases are cooled or heated by adiabatic expansion or compression, respectively.
applicability of the second law of thermodynamics. The third statement tempts the students to associate entropy with disorder and may give rise to the incorrect notion about the second law as “the disorder enforcer”. We propose an alternative statement of the second law, which avoids heat engines, Carnot cycles, entropy and disorder, but relies solely on the microscopic descriptors of matter.
Process 3: dni ≠ 0, dεεi ≠ 0 In this case the energy is transferred in both work and heat forms. Equations 1 and 3 suggest that there are two ways in which process 3 can be achieved. The first possibility is dT = 0 and dE = 0, where the temperature and internal energy of the system are unchanged. An illustrative example is an isothermal expansion or compression of gas. The isothermal process is possible if the first and second terms in eqs 1 and 3 have the same magnitudes, but opposite signs. However, this equality of terms is a restriction (a special case), so most processes fall in the second category where dT ≠ 0 and dE ≠ 0 (Figure 3). Change in Energy Levels versus Change in Population The important question arising at this stage is whether the energy levels and populations can be varied completely independently of each other in a closed system. In a closed system energy exchange in any form is allowed, but not the exchange of matter between the system and surroundings. We shall see that the second law of thermodynamics introduces some restrictions on level and population changes. The second law has been expressed through several equivalent statements: No process is possible in which the sole result is the absorption of heat from the reservoir and its complete conversion into work (Kelvin statement) (10). No process is possible in which the sole result is the transfer of energy from the cooler to a hotter body (Clausius statement) (10). When left to themselves systems tend to alter in such a way that the entropy of the system increases (11).
The first two statements refer to specific macroscopic processes and devices (heat engines) thus obscuring the general
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It is impossible in a closed system, to change energy levels available to the particles without at the same time redistributing the particles among the levels. The converse need not apply.
In other words, if the energy levels in the system change, dεi ≠ 0, the necessary consequence is dni ≠ 0! On the other hand, dni ≠ 0 does not require dεi ≠ 0 as the discussion regarding process 1 makes clear. The change in energy levels may come about via, for example, chemical reaction or phase change. Our statement of the second law of thermodynamics is particularly useful for understanding chemical reactions. As a result of chemical reaction the energy levels of the system will change, because products and reactants have different structures (geometries, etc.). This level change, dεi ≠ 0, necessitates the change in level populations and temperature, that is, dni ≠ 0 and dT ≠ 0 according to eq 3. The change in T versus surroundings will make the reaction exothermic (dT > 0) or endothermic (dT < 0) and allow energy transfer as heat when the reaction system comes in contact with its environment. The absolute sign of dT will be determined by the individual signs and magnitudes of dni and dεi according to eq 3. The explanation of our second law of thermodynamics statement is based on the assumption that particles interact or collide among themselves and are thus able to exchange energy. The particle rearrangement always proceeds towards the most probable distribution, which is concomitant with the available internal energy, that is, the Boltzmann distribution. Once that distribution is achieved, the internal equilibrium is reached and the average values of ni and εi become time independent. This explanation does not present a pedagogical difficulty, because students learn early on in physical chemistry that particles within the gas or liquid have a wide range of energies (Maxwell–Boltzmann distribution of gas velocities) and collide frequently exchanging energy. Particles
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
In the Classroom E 4
2 1 0
Figure 4. The energy level diagram showing possible processes in the heat engine. Left: useful work, E = 8 to E = 16. Right: heat dissipation, E = 8 to E = 16.
in the solid also interact via strong interparticle forces, which again facilitates energy exchanges. Does our statement of the second law of thermodynamics make process 2 impossible? It does not! Our statement of the second law refers to a closed system and the system in process 2 is adiabatic (energy exchange is restricted to work form only) and thus not closed. Heat Engines Let us now consider how our statement of the second law of thermodynamics is related to heat engines. In such engines, the working fluid (system) is heated or cooled (dT ≠ 0). Inspection of eq 3 reveals that the change in T can be achieved by changing εi, ni, or both. Since there are no physical reasons for placing restrictions on variables εi and ni, eq 3 suggests that both ni and εi will change (Figure 4). The macroscopic implication (i.e., in practice) is that some of the energy transferred as heat will lead to useful work, dεi ≠ 0, while the rest will induce level population changes, dni ≠ 0, and lead to heat dissipation. This analysis explains the crux of Clausius and Kelvin statements (which sound puzzling to the students) as to why it is not possible to convert all energy transferred as heat into mechanical work; that is, why the maximum efficiencies of steam or internal combustion engines are well below 100%. Our explanation is scientifically more accurate than the subjective notions of energy being “degraded” from “the higher and more useful form” of work into “the lower and less useful form” of heat! If we transfer energy as work, dεi ≠ 0, eq 2 suggests that both T and ni will change when no variable restrictions are imposed. The conclusion is that a complete conversion of work into heat takes place as is indeed observed in practice (in for example a refrigerator).
Conclusion We have presented an integrated framework for discussing thermodynamics from the microscopic point of view. This nontraditional approach is logical because it starts from elementary, particle concepts and shows how it leads to specific processes at bulk level. We have expressed the second law of thermodynamics by using only microscopic descriptors. This article aims to help physical chemistry students establish a conceptual framework for their study of thermodynamics and enhance their understanding of the topic. This article presents insights from which physical chemistry students may benefit and paves the way for subsequent in-depth discussions of statistical thermodynamics. Our approach is challenging because the students must be willing to go beyond the typical introductory approach to thermodynamics. This approach addresses general concepts with a minimum of mathematics, avoids formal derivation or manipulation of quantities, and is not tied to any specific processes (e.g., gas expansion or compression or Carnot cycles). Finally this approach may be necessary to correct fundamental errors regarding thermodynamics that appear in textbooks. An example (12) of such an erroneous statement is: Solar cells have the advantage that no pollution (or very little) is associated with their use. Because they directly convert light into electricity, they are not constrained by the fundamental limitations of the second law of thermodynamics as are heat engines.
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