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    THE ROLE OF INTERNAL ENERGY AND LOST WORK IN UNDERSTANDING THERMODYNAMIC FUNCTIONS

     A number of recent articles in chemical education have expressed a need for less abstract and a more intuitive approach to the teaching and learning introductory thermodynamics. This article summarises these, and suggests additional strategies for making thermodynamics more user-friendly to the neophyte chemistry student. Gone are the days when our students were mathematical wizards at differential calculus, and dedicated to the rigour of that tedious chain of reasoning proving that entropy is a state Many of these students become graduates in a variety of professions, analytical chemistry to nutrition. They do not need to be experts but should have least an intuitive understanding of the basic thermodynamics of heat and work enthalpy and free energy, and of internal energy and entropy.

    KEEPING TRACK...

    "Thermodynamics is not difficult if you can just keep track of what it is you talking about" is a quote attributed to a Professor Giauque. It appears in opening lines of two articles on the difficulties of teaching and thermodynamics by Henry Bent in 1972 (1) and Gordon Barrow in 1988 (2). Equations 1-4 are a sample of thermodynamic equations that present a challenge to students. These are not necessarily an intellectual challenge but a challenge to know exactly what thermodynamic changes that each of these thermodynamic variables U, H, G, S, q, and w are referring to.
    ΔU = q + w (1)
    H = U + PV (2)
    G = U + PV - TS (3)
    dU = -PdV + TdS (4)
    ln order to make thermodynamics appear less abstract these authors argued the thermodynamic system is over-emphasised and that equal billing should given its surroundings. John P. Lowe (3) discloses how the role of the can easily be overlooked in discussions of spontaneous changes from randomness to positional order in arguments such as,

    "formation of the first replicating molecules - primordial DNA - could not have occurred through natural processes because it would involve spontaneous increase in order, violating the second law".

    Students should be alerted to the increase in the thermal energy in surroundings that automatically accompanies these changes (3).
    Henry Bent suggested a "σ-θ-wt" notation for "system-thermal surroundings- mechanical surroundings" (1). The First Law was then neatly summed up in eq 5, and the Second Law in eq 6.

    ΔUtotal = ΔUσ + ΔUθ + ΔUwt = 0 (5)
    ΔStotal or ΔSuniverse = ΔSθ + ΔSθ ≤ 0 (6)
    Gordon Barrow argued for the abolition of the terms "heat" and 'work" because they do not let us begin by knowing what it is we are talking about (2). While the functions U, G ,H ,P, S, and V have definite meanings as properties of a system, heat and work are "indefinite properties that seem to float between the system and the surroundings as a result of a temperature or pressure differences". They should be seen as changes occurring in the surroundings, not in the system. In 1985 Boyd Waite described thermodynamics as a complex theory whose physical explanations are often obscured by complicated mathematics (4). He set out to describe heat and work in terms of gas kinetic theory, and his conclusion was that "heat and work, which are often misrepresented or misinterpreted as being different forms of energy, are best considered as different mechanisms of energy transfer".

    The notion of "energy transfer" is deeply entrenched in our language in such phrases as "the flow of heat", "electricity supply", and "energy consumption". There is no point in trying to change this. But the notion of energy transfer has its difficulties, as it does not draw our attention to the mechanisms that have to exist for a system to act on its surroundings. There are other ways of describing thermodynamic changes without using the idea of energy transfer. A discussion of the work of a lever or a see-saw illustrates this. A heavy mass can raise the potential energy of a lighter mass with the help of a mechanism such as that provided by a see-saw. We can say that in this process the heavier mass acts on the lighter mass through the see-saw mechanism. Work is done by the heavier mass because the lighter mass gained energy while the heavier mass lost it. It seems inappropriate here to say that work or mechanical energy was transferred from the heavier mass to the lighter one. Nothing of any substance was transferred at all! The same argument should apply to the process of heating a kettle on a stove. The hot surface (the stove) acts on a cold surface (the kettle base) by a molecular mechanism which then raises the temperature of the kettle base, and ultimately the temperature of the water (4).

    Karen Sanchez and Robert Vergenz argued that the thermal meaning of entropy has been misunderstood and underused in chemical education (5). The reasoning traditionally used to develop it is long, circuitous, and based on experiences largely unfamiliar chemistry students. They developed instead the WISE equation (eq. 7) as a simplification in the logical development of thermodynamics.

    dSuniv = [(T - Tsur)/T]dSsur + [(P - Psur)/T]dV (7)

    According to the Second Law the incremental entropy change in the universe, dSuniv is zero for reversible processes but positive for all irreversible processes. A process done reversibly produces maximum work. The same process done irreversibly produces less than this maximum work, and the difference is called lost work (6). Irreversible changes are spontaneous changes that happen by virtue of a temperature, a pressure, height or other potential energy difference. The first term on the right hand side of eq 7 is lost Carnot cycle work divided by the system temperature T. The lost work in the second term, (P - Psur)dV, becomes the waste heat that could have been PV work if the process was performed reversibly. The positive change in the entropy of the universe that arises from irreversible processes is lost work divided by the system temperature.

    Reversible work is a process that involves a succession of equilibrium states, so that reversible work can take forever to happen (7). But the role of lost work get things moving. This principle can be illustrated in the above example of the see-saw. The heavier weight loses potential energy while the lighter weight gains potential energy. But the heavier weight loses more energy than the lighter weight gains, because some of the potential energy lost by the heavier weight becomes the kinetic energy that gets the system moving. This same kinetic energy becomes lost work in the form of heat and noise, when the see-saw hits the ground. If you want to minimise this lost work, you make the heavier weight only slightly heavier than the lighter weight. The smaller the difference between the weights the slower is the movement of the see-saw. If there is no difference in the potential energies of the two weights the see-saw will be at equilibrium and wont do any work at all. In electrochemistry the same principle is involved in the concept of overpotential - the deviation of the potential at an electrode from its equilibrium value. An overpotential is lost work that must be applied in order to induce a net flow of current (8). It is work that becomes thermal energy in the surroundings.

    THREE ENERGY COMPARTMENTS IN THE SURROUNDINGS

    Henry Bent's eqs 5 and 6 are still rather abstract (1). The point of the First Law of thermodynamics (eq 5) is to allow us to calculate, from the measured energy changes in its surroundings, the corresponding internal energy changes (ΔU) in a system. We can do this more explicitly if we use Ew, Gw, and q (Figures 1 and 2) as symbols of the measurements that we make in the surroundings. Figure 1 is designed to be a diagram that is easy to imagine and reproduce. It has a circle representing the system that is surrounded by a triangle. The triangle represents the three compartments in the surroundings that a system can act upon. These compartments are the thermal reservoir, mechanical reservoir, and other coupled systems.

     Each compartment contains a particular mechanism that allows a system to act on or react to its surroundings. Thermal energy or heat, q, is an energy change that is measured in the thermal reservoir. The thermal reservoir and the system respond to each other through physical contact of the molecules at their respective boundaries, when there is a temperature difference between them (4). Expansion work or mechanical work, Ew, is the energy change in the mechanical reservoir. An example of a mechanism for measuring energy changes in the mechanical reservoir is a piston attached to a weight that moves upwards when the volume of the system increases. When a system causes a chemical change within another coupled system, Gw or Gibbs work is the measure of the energy change. A 12 Volt battery recharging a flat 9 Volt battery is an example of two coupled systems. The symbol Gw in this case represents the energy change in the 9 Volt battery (the coupled system) that was driven by the 12 Volt battery (the system). When we measure the electrical potential of an electrochemical system the coupled system is the potentiometer, where the electrodes and electronic circuit provide the mechanism that enables the system to act on the potentiometer.

     BALANCING THE THERMODYNAMIC ENERGY ACCOUNT

    Figure 2 provides a convenient way of balancing the thermodynamic energy account when a system undergoes a chemical or physical change and acts on its surroundings. It is a pictorial expression of eqs 1-3, that makes the relationships between Δu, ΔH, ΔG, PΔV and T.ΔS much clearer. We can see immediately, what is difficult to see from eqs 2, 3, and 5 that ΔU is the sum of all three measured energy changes - Ew, Gw, and q as in eq 8. By identifying these three energy measurements it is easier to appreciate that thermodynamic values are not just obtained from tables but are built on laboratory measurements in the surroundings
    ΔU = q + Ew + Gw (8)

    Another property of a system is the enthalpy change, ΔH,

    ΔH = q + Gw (9)

    An energy term not included in Figure 2 is the Helmholtz energy, ΔA, which is the sum of Gw and Ew. For a chemical change carried out at constant volume, Gw and PΔV are replaced by Hw (Helmholtz work), and ΔA replaces ΔG as the maximum work that can be done on a coupled system. The enthalpy change, ΔH, would be removed altogether because it requires constant pressure.

    LOST GIBB'S WORK AND ENTROPY CHANGE

    Entropy changes are usually seen in terms of changes in the thermal surroundings, or in terms of statistical changes in systems. But it may also be seen as changes in the capacity of the internal energy of a system to produce work. The expression TΔS represents that part of ΔU that cannot produce work when the change is
    carried out reversibly and q' is additional lost work when the change is carried out irreversibly. The symbol q' is here called "lost Gibbs work" or "irreversible heat", because q' is energy that is subtracted from ΔG, and observed and measured in the thermal reservoir. The value of q' has the same sign as its ΔG. When q' = 0 there is an equilibrium between the system and its coupled system compartment. For a spontaneous chemical change in the system Gw = ΔG - q', and q = TΔS + q'. The second Law (eq 6) requires that spontaneous processes increase the entropy of the universe. This is the effect of q' in eq 10.

    ΔSuniv = -q' / T (10)

    Whatever the value of q' the addition of Gw to q eliminates q' and produces the Gibbs-Helmholtz equation, eq 11.

    ΔH = ΔG + TΔS (11)

     AN EXAMPLE

    Figure 3 is an example thermodynamic energy account for a mountain climber who is consuming glucose as her food energy source. Glucose is being burned into carbon dioxide and water as in eq 12.
    C6H12O6 + 6O2 → 6CO2(g) + 6H2O(l) (12)

    The value of the irreversible heat q' depends on how vigorously the climber climbs the mountain. If she is very very slow, the respiration reaction would tend to cool her down, because q' will be very small and near zero, and q = 56 kJ/mole approx. If she climbs the mountain vigorously, q' will be 70% approx of ΔG (i.e. -2004 kJ/mole) and q = 56 - 2004 = -1952 kJ/mole. She will experience an increase in body heat at the rate of 1952 kJ/mole of glucose digested and the need for the surroundings to cool her down. The remaining 30% of ΔG will be observed in the increase in the potential energy of the weight of her body as she climbs. The value of the expansion work of -2 kJ/mole is the PV work expended in exhaling into the atmosphere the CO2 and the water vapour products of eq 12. If the glucose is consumed by a person who does no exercise at all, Gw may be stored as fatty tissue.

    INTERNAL ENERGY AND CALORIMETRY

    When 2 mole of ozone (Figure 4) decompose inside a constant volume calorimeter at a constant temperature of 25°C the entire internal energy change ΔU is measured in the thermal surroundings as q = qV = -287.1 kJ. When the calorimeter allows the expansion against constant pressure of 1 atmosphere for this reaction the expansion work Ew amounts to -2.5 kJ. This expansion work is taken from the internal energy change of -287.1 kJ/mole, so that q = qp = - 284.6 kJ/mole as a result. This measurement of q at constant pressure is called the "enthalpy change ΔH" where ΔH = ΔU - Ew. If the reaction was also harnessed to produce Gibb's work, Gw, at constant pressure this work would also be subtracted from the internal energy and q would equal ΔU - Gw - Ew.

    CONCLUSION

    For every physical or chemical change in ideal systems at constant temperature, whether it be at constant volume or constant pressure, there is a fixed internal energy change, ΔU. All of the other energy changes are derived from ΔU. The central status of ΔU, and the fact that we get the value of it experimentally by measuring energy changes in the surroundings, should be obvious in the way we introduce thermodynamics. Figures 1-4 are a visual method of doing this by showing how measurements of energy changes in the surroundings - Gw, Ew, and q - and the system state functions - ΔH, ΔG, TΔS, ΔA, PΔV - relate to the internal energy change. The functions ΔG and ΔA represent that part of the internal energy change that is available for the work that can be done on other systems. The values for ΔG and ΔA may also include a contribution from the surroundings when the value of TΔS is positive as in Figure 3. The value for lost work, q', is that part of ΔG or ΔA that is required to get the system to undergo an irreversible change. The energy in q' shows up as energy in the thermal reservoir. The Second Law that requires an increase in the entropy of the universe for all spontaneous irreversible physical and chemical changes (eq 6) is too abstract for most new students. But the concept of ΔSuniv in eq 10 as a function of lost work divided by the system temperature helps students to gain a more intuitive if incomplete understanding of the Second Law. Though entropy change is defined in terms of "reversible heat transfer" (7,8), it may also be seen in terms of the capacity of a system to produce work from a change in its internal energy. An internal energy change of one Joule at 1000°C does not produce the same entropy change as one Joule at 100°C because at the higher temperature a higher proportion of one Joule can be harnessed to produce work than can one Joule at the lower temperature. Figures 1-4 allow students to develop the confidence that they understand where each of the above energy changes and state functions come from, and provide a secure basis for advancing their knowledge of thermodynamics in both ideal and non-ideal systems.

    REFERENCES

    1. Bent, H. A., J. Chem. Educ.,1972, 49, 44-46.
    2. Barrow, G. M., J. Chem. Educ.,1988,65, 122-125
    3. Lowe, J. P., J. Chem. Educ.,1988,65,403-406
    4. Waite, B. A., J. Chem. Educ.,1985, 62,224-222
    5. Sanchez, K. S.; Vergenz, R. A., J. Chem. Educ. 1994,71,562-566
    6. Van Wylen, G. J.; Sonntag, R. E., Fundamentals of Classical Thermodynamics, 3rd ed.; Wiley: New York, 1986; p 205
    7. Moore, W. J., Basic Physical Chemistry, Prentice-Hall: New Jersey, 1983; p 95-96.
    8. Atkins, P. W., Physical Chemistry, Oxford University Press: Oxford, 1978; p 974.
    9. Nuffield Advanced Science, Book of Data, Harrison, R. D. ed., Penguin Books Ltd: Ringwood, Australia, 1972; p 78.

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