A Comparison of Local and Global Formulations of Thermodynamics

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A Comparison of Local and Global Formulations of Thermodynamics Howard DeVoe* Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, United States S Supporting Information *

ABSTRACT: Several educators have advocated teaching thermodynamics using a “global” approach in place of the conventional “local” approach. This article uses four examples of experiments to illustrate the two formulations and the definitions of heat and work associated with them. Advantages and disadvantages of both approaches are discussed. The article concludes that either formulation can be used for conceptual understanding of the first and second laws of thermodynamics, that the local formulation is usually less complicated, and that the choice of a formulation for evaluating thermodynamic quantities from experimental measurements depends on the experiment.

KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Problem Solving/Decision Making, Thermodynamics, Calorimetry/Thermochemistry, Heat Capacity

S

everal educators, in this Journal and elsewhere, have advocated teaching thermodynamics using a “global” formulation rather than the conventional “local” approach. The global formulation focuses on changes in the surroundings of the system during a thermodynamic process. This approach seems to have been presented first in a 1962 article1 by Henry Bent, who followed up with a book2 and a second article.3 The formulation was developed further in articles4,5 and a book6 by Norman Craig, and in articles by Eric Gislason and Craig.7,8 Bent, Craig, and Gislason, parallel to their development of the global formulation, described “surroundings-based” definitions of heat and work based on energy changes in the surroundings.3,7,9−11 These differ from the conventional “system-based” definitions based on energy transfers at the system boundary. The surroundings-based definitions are natural accompaniments of the global formulation. This article compares the global and local formulations and the definitions of heat and work that accompany them. The article begins with definitions of some important thermodynamic terms, followed by general descriptions of the two formulations. Both formulations are then applied to four examples of experiments involving closed systems. The concluding Discussion points out some pedagogical advantages and disadvantages of both formulations.

principle can be described by the sequence of states through which the system passes. Note that this description of a process is confined to what is happening within the system and at its boundary, without regard to what is happening in the surroundings outside the boundary. The definition of a reversible process used in this article is described in detail elsewhere.12 During a reversible process, the system passes through a continuous sequence of equilibrium states, each of which if present in an isolated system would exhibit no perceptible change on the time scale of observation. (This kind of process involving equilibrium states is sometimes called quasistatic.) The sequence of equilibrium states can be approached, as closely as desired, by a real process (i.e., one that can actually take place) carried out sufficiently slowly, and the same sequence in reverse chronological order can be approached by another real process carried out sufficiently slowly. Thus, the reversible path followed by the system in the limit of infinite slowness must be the same in both directions there must be no hysteresis.13 A reversible process, then, is a process in an idealized reversible limit. The present definition of a reversible process differs from another common one that requires the reverse sequence of equilibrium states to be accompanied by a reversal of conditions in the immediate surroundings, with recovery of the initial states of both system and surroundings. This article, in order to make it clear that a reversible process refers only to reversibility within the system, will refer to an internally reversible process14 and a limit of internal reversibility. A process may be infinitesimally close to an internally reversible limit at

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DEFINITIONS This article considers only closed systemsthat is, ones in which no matter crosses the system boundaryas these are the only kinds of systems treated by the global formulation. The macroscopic properties of a system at any given instant determine the thermodynamic state of the system at that instant. For the purposes of this article, a thermodynamic process is a change in the state of the system over time that in © XXXX American Chemical Society and Division of Chemical Education, Inc.

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the same time that finite irreversible changes are taking place in the immediate surroundings.

A rearranged version of eq 3 allows the system’s energy change to be determined from measurements in its surroundings:

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ΔEsys = −∑ ΔEi

OVERVIEW OF THE LOCAL FORMULATION The local approach, which is the conventional one, focuses on transfers of energy across the boundary of the system during a thermodynamic process. Energy transfers in the forms of heat q and work w are taken as positive for energy transferred into the system and negative for energy transferred out of the system. Gislason and Craig9 call q and w “system-based” heat and work. A more appropriate term might be “boundary-based”. The internal energy Usys, enthalpy Hsys, and entropy Ssys are state functions of the system. (The subscript “sys” denotes the system.) The internal energy change during the process is given by the first-law expression ΔUsys = q + w

The system’s energy change is related to its internal energy change ΔUsys by16 ΔEsys = ΔE k,sys + ΔEp,sys + ΔUsys

(1)

q Tsys

ΔEi = ΔEiQ + ΔEiW

(2)

ΔEsys = −∑ ΔEiQ − i

OVERVIEW OF THE GLOBAL FORMULATION The original version of the global formulation,1,4−6 in its simplest form, assumes that a thermodynamic event can be described by changes in three parts of the physical universe: a reactive system, its thermal surroundings (such as a thermal reservoir), and its mechanical surroundings (such as a weight or piston whose gravitational potential energy changes during the event). More recently, Gislason and Craig8 have extended and generalized the original version of the global formulation as follows. A thermodynamic event involves various subsystems or parts, any one of which can be called the system. The totality of the parts is the “universe of the event”, which for the sake of brevity in this article will be called the supersystem. No matter or energy is transferred into or out of the supersystem. In what follows, the supersystem is indicated by subscript “ss”, the system is indicated by subscript “sys”, and index i is used to label the parts of the system’s surroundings within the supersystem. The energy changes of the various parts of the supersystem during a thermodynamic event, measured in a reference frame fixed in the laboratory, are ΔEsys and ΔEi (i = 1, 2, ...). The firstlaw expression of the local formulation (eq 1) is replaced by a statement of the conservation of energy within the supersystem: i

∑ ΔEiW i

(7)

The surroundings-based definitions of heat and work, which Gislason and Craig8 describe as an obvious extension of the global formulation, identify −∑iΔEQi as heat and −∑iΔEW i as work. This article uses the symbols q′ and w′, respectively, for these surroundings-based quantities, and reserves the symbols q and w for the conventional system-based heat and work. Note that q′ is the negative of the total thermal energy change, and w′ is the negative of the total mechanical energy change, in the surroundings. They do not necessarily represent energy transfers across the system boundary as do q and w. With these definitions and symbols, eq 7 for a stationary system becomes

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∑ ΔEi = 0

(6)

Equation 4 becomes

The inequality in eq 2 is for an irreversible process, and the equality is for an internally reversible process.15 Equations 1 and 2 are valid only for processes in closed systems. Once ΔUsys and ΔSsys have been described in terms of heat and work, the conventional thermodynamic formulation leads on to the concepts of Gibbs energy, chemical potentials, equilibrium constants, and the other important tools of classical thermodynamics.

ΔEss = ΔEsys +

(5)

where ΔEk,sys and ΔEp,sys are changes of the system’s macroscopic kinetic and potential energies, respectively. In the examples in this article, the system is stationary so that ΔEk,sys and ΔEp,sys are zero; thus, ΔEsys and ΔUsys are equal in these examples. Gislason and Craig assume there is no chemical reaction or phase change in the surroundings. They divide the energy change of part i in the surroundings into mechanical and thermal contributions.8 If a portion of ΔEi can be converted into the lifting or lowering of a weight in the earth’s gravitational field, this is a “mechanical” energy change denoted ΔEW i . Any remaining portion of ΔEi is a “thermal” energy change denoted ΔEQi :

The enthalpy is defined by Hsys ≡ Usys + psysVsys, where p is pressure and V is volume. From the second law of thermodynamics, the entropy change in a system of constant temperature Tsys can be expressed in the form ΔSsys ≥

(4)

i

ΔEsys = ΔUsys = q′ + w′

(8)

Craig and Gislason consider the role of the atmosphere when the system is open to the atmosphere at pressure patm. They envisage a column of the atmosphere resting on the surface of the system that is lifted or lowered as the system volume Vsys changes.17,18 The column is treated as one of the parts of the supersystem. Its gravitational potential energy change is ⎛m g ⎞ ΔEatm(sys) = matmg Δhatm = ⎜ atm ⎟(A sΔhatm) = patm ΔVsys ⎝ As ⎠ (9)

where matm is the column’s mass, Δhatm is its elevation change, As is its cross-section area, and g is the acceleration of free fall. The sum of the energy changes of a stationary system and a column of the atmosphere resting on it equals the system’s enthalpy change:8 ΔEsys + ΔEatm(sys) = ΔUsys + patm ΔVsys = ΔHsys

(10)

By the same reasoning, the sum of the energy changes of a thermal reservoir (res) open to the atmosphere and the column

(3) B

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The simplest application of the global formulation assumes the surroundings are entirely mechanical and consist of the external weight of mass m and the column of the atmosphere resting on the water:

of the atmosphere resting on it equals the enthalpy change of the reservoir: ΔEres + ΔEatm(res) = ΔUres + patm ΔVres = ΔHres

(11)

This enthalpy change can be evaluated from ΔHres = Cp ,resΔTres

ΔUsys = ΔEsys = −∑ ΔEi = −(mg Δh + patm ΔVsys) (12)

i

(15)

where Cp,res is the reservoir’s heat capacity at constant pressure. Under favorable conditions, the global formulation allows the system’s entropy change ΔSsys to be evaluated from measurements in its surroundings. The entropy change of a thermal reservoir is given by ΔSres = ΔHres/Tres. Any part of the supersystem whose energy change is purely mechanical has constant enthalpy and entropy. Thus, the entropy change of a supersystem with thermal energy changes in only the system and a thermal reservoir is given by ΔSss = ΔSsys + ΔHres/Tres

From eq 15, the enthalpy change defined by ΔHsys = ΔUsys + patmΔVsys is found to equal −mgΔh. The weight’s elevation change Δh is easily measured, so the global formulation is the appropriate one for this experiment. Realistically, the surroundings are more complicated. Consider James Joule’s determination19 of the “mechanical equivalent of heat”. His apparatus had two weights linked to a paddle wheel by strings and pulleys. To accurately calculate ΔHsys from the elevation change of the weights, Joule made corrections for (i) the kinetic energy gained by the weights before they hit the laboratory floor, (ii) friction in the connecting strings and pulley bearings, and (iii) the loss of potential energy of the stretched strings when the weights reached the floor and tension was relieved.

(13)

As the supersystem is thermally isolated from the rest of the universe, the second law can be expressed in the form ΔSss ≥ 0 where the inequality applies if an irreversible change occurs anywhere in the supersystem, and the equality applies if there is no such irreversible change. In the limit of reversibility for the supersystem as a whole, if such a limit exists, ΔSss is zero and eq 13 becomes

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ΔSsys = −ΔHres/Tres

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EXAMPLE 2: GAS EXPANSION WITH EXTERNAL FRICTION Gislason and Craig9 used the device shown in Figure 2 to illustrate differences between system-based and surroundings-

(14)

EXAMPLE 1: JOULE PADDLE WHEEL EXPERIMENT To compare the local and global formulations, four instructive experiments are examined. In the classic Joule paddle wheel experiment (Figure 1), an external weight sinks, a cord attached to the weight turns a

Figure 2. Cylinder-and-piston device to illustrate expansion work of a gas. The piston is shown at its initial elevation h = h1.

based work. The experiment also illustrates a connection between internal reversibility and friction in the surroundings. A sample of gas is confined below a piston of mass m in a vertical cylinder of cross-section area As. Above the piston is a vacuum. The cylinder is in thermal contact with a large volume of water serving as a thermal reservoir (res) of practically constant temperature Tres. The system is the gas. Initially, the piston is held in place at elevation h1 by catches, and the gas is in an equilibrium state at the temperature Tres of the thermal reservoir. When the catches are removed, the piston moves upward and may oscillate. In intermediate states of the process the gas temperature is not necessarily constant or uniform. The piston eventually comes to rest at a final elevation h2, greater than h1, with the gas again in an equilibrium state at temperature Tres. In this final state, the upward force exerted on the piston by the expanded gas balances the downward gravitational force. In the local formulation of the process, the system-based expansion work is given by20−22

Figure 1. Schematic of the paddle wheel experiment.

paddle wheel immersed in water, and frictional drag forces at the rotating paddle wheel cause the temperature of the vessel and its contents to increase. The system can be taken as the vessel, the water, the paddle wheel, and the portion of the vertical shaft immersed in the water and is assumed to be thermally insulated from its surroundings. The purpose of the experiment is to evaluate the enthalpy increase ΔHsys and the system’s heat capacity Cp,sys = ΔHsys/ΔTsys. In the local formulation of this experiment, ΔHsys is given by ΔUsys + patmΔVsys with ΔUsys equal to the sum of shaft work wshaft and expansion work −patmΔVsys. Thus, ΔHsys is equal to wshaft. In principle, wshaft can be calculated as the internal torque in the shaft where it enters the system, integrated over the angle of rotation. As a practical matter, the internal torque is unknown, so that ΔHsys cannot be evaluated by the local formulation unless Cp,sys is already known. C

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∫ pb dVsys

w′ = w −

(16)

∫h

h2

Ffric dh

1

where pb is the instantaneous pressure of the gas at the piston face, that is, at the moving portion of the system boundary. (The subscript “b” stands for boundary; Gislason and Craig9 use the symbol Ps for this pressure.) When the piston moves with finite velocity, the value of pb is unknown, although it can be approximated with an expression derived from kineticmolecular theory.23,24 If the gas is ideal, ΔUsys is zero for this process and the entropy change is given by the well-known formula ΔSsys = nR ln(V2/V1). For a nonideal gas, however, the evaluation of these quantities by the local formulation is not practical because, as Gislason and Craig8 point out, the system-based quantities q and w cannot in general be measured. In the global formulation, the mechanical energy change in the surroundings is due almost entirely to the piston’s elevation change, so the surroundings-based work is9 W w′ = −ΔEpiston = −mg (h2 − h1)

Because Ffric and dh have opposite signs or are zero, the integral ∫ Ffric dh is negative or zero. It is convenient to rewrite eq 24 using the absolute value of this integral: w′ = w +

Q W ΔEres = ΔEres − ΔEres = ΔUres + patm ΔVres = ΔHres

(18)

The heat capacities of the piston and cylinder are assumed to be negligible, so the surroundings-based heat is (19)

The system internal energy change can therefore be evaluated from measurable quantities: (20)

∫h

The conclusions of Gislason and Craig regarding the relation between w and w′ in the device of Figure 2 are based on the following derivation.7,9 During the expansion process, the net upward component of the force acting on the piston is given by Ftot = pb A s − mg + Ffric

h2

pb A s dh − mg (h2 − h1) +

1

h2

Ffric dh = 0

1

h2

1

pb A s dh =

∫V

V2

1

pb dVsys = −w

(25)

Ffric dh max

⎞ ⎛h h = nRTres⎜ 1 − ln 1 − 1⎟ h2 ⎠ ⎝ h2

(26)

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EXAMPLE 3: GAS-PHASE REACTION The global formulation is readily applied to a gas-phase reaction taking place in a cylinder-and-piston device immersed in a thermal reservoir. Gislason and Craig8 use the combustion of hydrogen as an example of this kind of experiment. The experimental device is the same as for example 2, but without the catches shown in Figure 2. The system is the contents of the cylinder below the piston. Initially, the piston is stationary and the cylinder contains the reactants, at least one of which is a gas. The reactants are at the temperature Tres of the thermal reservoir and at a pressure p1 = mg/As determined by the mass m of the piston resting on the gas.

(22)

The first integral on the left side of eq 22 is

∫h

Ffric dh

In the limit of infinite c, with the value of the frictional work at its upper limit, the expansion process is infinitely slow and internally reversible. In principle, this limit of internal reversibility could be approached by increasing the viscosity of the lubricant used for the piston seal. The important point is that for the process to approach a limit of internal reversibility in the device of Figure 2, the expansion must be slowed by irreversible friction in the surroundings. Thus, there is always irreversibility in the supersystem as a whole, and ΔSss can never be zero. For this reason, eq 14 cannot be used to evaluate ΔSsys.

(21)

∫h

h2

1

where pbAs is the upward force exerted by the gas, mg is the downward gravitational force, and Ffric is a frictional force acting in the direction opposite to the piston’s velocity. The friction occurs at the lubricated seal between the edge of the piston and the inner surface of the cylinder, and is a form of external friction. According to the work-energy principle of classical mechanics, the integral ∫ Ftot dh is equal to the piston’s net kinetic energy change, which in this experiment is zero. Replacing the integrand Ftot with the expression of eq 21 gives the relation

∫h

h2

|∫ Ffric dh| is the frictional work performed by the piston on the rest of the surroundings. I used a computer program to model a two-fold expansion of an ideal gas in the device of Figure 2. The calculations of the piston’s elevation as a function of time t were based on Newton’s second law, Ftot = md2h/dt2. The calculations assumed the frictional force depends on the piston velocity according to Ffric = −cdh/dt, where the constant c is a positive viscous damping coefficient. Details are given in the Supporting Information. The results of varying the value of c in the model calculations, while keeping other parameter values constant, may be summarized as follows. The piston behaves as a damped oscillator. If the value assumed for c is close to or equal to zero, the piston is underdamped and oscillates about its final elevation h2 before coming to rest at that elevation. The value c = 0 corresponds to a frictionless piston. In this case, the oscillations are damped by the finite rate of thermal conduction and by pressure inhomogeneities in the gas. If the value assumed for c is sufficiently large, the piston is overdamped and its elevation approaches h2 without oscillation. The greater is the value assumed for c, the slower is the expansion. As c is increased without limit, the frictional work |∫ Ffric dh| increases and its value asymptotically approaches a finite upper limit. In the Supporting Information, the following expression is derived for this upper limit for an ideal gas:

The thermal reservoir is at atmospheric pressure patm and has small temperature and volume changes during the process. Its mechanical energy change is ΔEW res = −patmΔVres, and its thermal energy change is equal to its enthalpy change:

ΔUsys = q′ + w′ = −(ΔHres + mg Δh)

∫h

1

(17)

Q q′ = −ΔEres = −ΔHres

(24)

(23)

Making substitutions from eqs 17 and 23 in eq 22 and rearrangement yields the relation obtained by Gislason and Craig:25 D

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Consider a thermodynamic process initiated when the switch is closed and terminated when the switch is reopened after a period of time Δt. During the process, an electrical circuit exists that includes an ammeter and an external resistor of electric resistance Rext. The total charge that enters the cell at the positive terminal is Q = −|I|Δt, where I is the electric current measured with the ammeter. The charge is related to the advancement ξ of the cell reaction by Q = −zFξ, where z is the electron number of the cell reaction and F is the Faraday constant. The cell potential Ecell can be measured with the highimpedance voltmeter shown in Figure 3. Ecell is taken as positive if the right-hand terminal is positive as in the figure. When the switch is open, Ecell is the equilibrium cell potential Ecell,eq whose value depends on the cell’s temperature, pressure, and composition according to the Nernst equation. During the process, when there is a finite electric current, Ecell is less than Ecell,eq on account of the cell’s internal resistance. The electrical work during the process is given by wel = QEcell = −zFξEcell and is negative: the cell does electrical work on the surroundings. Williamson and Morikawa27 show that in the limit as Rext approaches infinity, the current approaches zero and the rate of the cell reaction becomes infinitesimal. The process therefore has a limit of internal reversibility. (To confirm that the process can be reversed internally, a different kind of surroundings is needed, such as a slidewire potentiometer.) In the reversible limit, Ecell has its zero-current value Ecell,eq and the reversible electrical work is given by wel,rev = −zFξEcell,eq. The thermodynamic quantities of interest are the enthalpy change ΔHsys, the entropy change ΔSsys, and the Gibbs energy change ΔGsys. When these quantities are divided by the advancement ξ, they become (for small ξ) the molar reaction quantities ΔrH, ΔrS, and ΔrG. The thermodynamic equilibrium constant K of the cell reaction can be evaluated from ΔrG° = −RT lnK where ΔrG° is the value of ΔrG extrapolated to standard-state conditions. In the local formulation of the process, w is the sum of expansion and electrical work: w = −patmΔVsys + wel. The internal energy change is ΔUsys = q + w (eq 1) and the enthalpy change is

When the reaction is initiated, for instance by ignition, it proceeds spontaneously and irreversibly. During the process, the piston moves in response to the changing force exerted on it by the gas and by friction, and both the temperature and pressure within the system are nonconstant and nonhomogeneous. When the process is complete, the piston is again stationary and the cylinder contains an equilibrium mixture of reactants and products at the same temperature and pressure as in the initial state. The thermodynamic quantity of greatest interest is the enthalpy change ΔHsys. This is also the reaction enthalpy ΔH(rxn), defined as the enthalpy change of a reaction process in which the initial and final temperatures are equal and the initial and final pressures are also equal. In the local formulation of this process, ΔHsys is equal to q + w + p1ΔVsys. As in example 2, q and w cannot in general be measured in this experiment and the evaluation of ΔHsys in the local formulation is purely conceptual. In the global formulation, using eq 8, the system enthalpy change is given by ΔHsys = ΔUsys + p1 ΔVsys = q′ + w′ + p1 ΔVsys

(27)

From eq 17, the surroundings-based work is ⎛ mg ⎞ w′ = −mg (h2 − h1) = −⎜ ⎟(A sΔh) = −p1 ΔVsys ⎝ As ⎠

(28)

Substitution of this expression for w′ in eq 27 shows that ΔHsys is equal to q′,8 which is equal to −ΔHres (eq 19). Thus, ΔHsys is equal to −ΔHres and can be evaluated using eq 12. In practice, a gas-phase reaction is unlikely to be carried out in a cylinder-and-piston apparatus. Instead, the gas is usually contained in a constant-volume vessel. For example, a combustion reaction is carried out in a bomb calorimeter. To evaluate ΔH(rxn) from the observed temperature change of a bomb calorimeter, many corrections are needed that do not fit into a purely global formulation.26

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EXAMPLE 4: GALVANIC CELL The experimental setup of Figure 3 is similar to one described by Williamson and Morikawa.27 The system is a galvanic cell without liquid junction, such as a lead storage battery. The cell is open to the atmosphere and is in thermal contact with a thermal reservoir (res1) of essentially constant temperature Tres1.

ΔHsys = ΔUsys + patm ΔVsys = q + wel

(29)

In the limit of internal reversibility, the system temperature has the constant value Tres1. The equality of eq 2 becomes ΔSsys =

qrev Tres1

=

ΔHsys − wel,rev Tres1

(30)

Rearrangement yields (ΔHsys − Tres1ΔSsys) = wel,rev, or ΔGsys = wel,rev = −zFξEcell,eq

(31)

The molar reaction Gibbs energy is then given by Δr G = ΔGsys /ξ = −zFEcell,eq

(32)

The molar reaction entropy and enthalpy can be evaluated from the temperature dependence of Ecell,eq: Figure 3. Galvanic cell immersed in a thermal reservoir, where V is a high-impedance voltmeter, A is an ammeter, and Rext is the electric resistance of the indicated segment of the external resistor. The dashed rectangle indicates the boundary of the supersystem in the global formulation.

Δr S = −

dEcell,eq dΔ r G = zF dT dT

Δr H = Δr G + T Δr S E

(33) (34)

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The global formulation involves the supersystem whose boundary is indicated as a dashed rectangle in Figure 3. The supersystem includes a second thermal reservoir (res2) of temperature Tres2. The energy transferred from the cell by electrical work ends up as energy increases of reservoir res2 and the column of the atmosphere above it. Equation 3, with substitutions from eqs 10 and 11, becomes ΔEss = ΔHsys + ΔHres1 + ΔHres2 = 0

with a phase-change calorimeter whose temperature is exactly constant at constant pressure. The enthalpy change of an ice calorimeter, for example, can be determined from its volume change. A phase-change calorimeter, however, has the limitation that its temperature must be that of an equilibrium phase transition. The systems of examples 2, 3, and 4 are immersed in thermal reservoirs that serve to return the system temperature in the final state of the process to its initial value. For this purpose, an experimental setup is more likely in practice to replace the thermal reservoir with a stirred thermostat bath. The bath needs a stirrer motor and provisions for regulating its temperature, so it cannot easily be included in the constantenergy supersystem of a global formulation. Other difficulties with global formulations are the corrections needed in the Joule paddle wheel experiment (example 1), and the corrections for bomb calorimetry mentioned in connection with example 3. Regarding heat and work, an operational advantage of the surroundings-based quantities q′ and w′ is that they can be evaluated from changes in state functions in the surroundings, whereas the values of the system-based quantities q and w may be unknown for an irreversible process.8 Whenever mechanical or electrical energy is dissipated in the system’s surroundings within a supersystem, w′ is greater than w; otherwise they are equal. Thus, w′ and w are equal if all changes in the supersystem as a whole are reversible. They can also be equal if the process is irreversible both internally and in the surroundings, as illustrated by example 2 when the piston is frictionless and the external irreversibility is due to temperature gradients. As Gislason and Craig9 point out, q′ and q are equal if and only if w′ and w are equal, as can be deduced by comparing eqs 1 and 8. There are conceptual issues when surroundings-based heat and work differ from their system-based counterparts. For instance, Gislason and Craig7 describe an experiment in which the system is a steel ball with an outer layer assumed to be a perfect thermal insulator. When the ball is dropped into a container of water, the potential energy lost by the ball is converted by frictional work into increased energy of the water, whose temperature increases slightly. Because the system is thermally isolated, q is zero, whereas the temperature increase in the surroundings makes q′ negative. The finite surroundingsbased heat in this experiment is quite unlike the conventional concept of heat as an energy transfer across the system boundary by thermal conduction or radiation. Using the term “heat” for q′ and “work” for w′ can only confuse students. It would be more accurate to call −q′ the “thermal effect” and −w′ the “mechanical effect” in the surroundings. Gislason and Craig9 claim that a serious conceptual problem with system-based work is that it does not always satisfy the theorem of maximum work. (The theorem states that the maximum possible value of −w for an isothermal process is −ΔAsys, where A is Helmholtz energy.) This claim is refuted in the Supporting Information for this article, where their example of a supposed violation of the theorem in the expansion process of example 2 is shown to be based on conditions that are physically impossible. There is a more general issue: aside from electrical work, experimental values of heat and work are seldom needed. Electrical work is easily measured and is commonly used to evaluate heat capacities of substances and reaction quantities of galvanic cells. The importance of heat and nonelectrical work,

(35)

Thus, ΔHsys can be evaluated from the enthalpy changes of the thermal reservoirs: ΔHsys = −ΔHres1 − ΔHres2

(36)

The evaluation of ΔSsys and ΔGsys from enthalpy changes in the surroundings requires an abbreviated supersystem (ss′) that includes only the cell, switch, voltmeter, and reservoir res1. In the limit as Rext approaches infinity and the electric current approaches zero, all changes in supersystem ss′ (but not in the full supersystem ss) approach a reversible limit. In this limit, provided supersystem ss′ is thermally isolated, ΔSss′ is zero and ΔSsys is given by −ΔHres1/Tres1. Note that the dissipation of electrical energy in this experiment, resulting in the irreversible heating of reservoir res2, plays the same role as external friction in example 2. In both experiments, the approach to a limit of internal reversibility requires irreversibility in the surroundings.

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DISCUSSION The examples above demonstrate some of the varying capabilities of local and global formulations of thermodynamic processes. A general conceptual advantage of the conventional local formulation is its focus on the system, which is where the process takes place. In this formulation, the surroundings can be whatever they happen to be in a given experiment. The global formulation, on the other hand, has rather stringent requirements: the supersystem of system and surroundings must be closed, thermally isolated, and of constant energy. The actual surroundings of an experiment may not meet these requirements. A general advantage of the global formulation is that measurements of state function changes in the surroundings allow irreversible processes to be analyzed as easily as reversible ones.8 This advantage is demonstrated by the irreversible processes of examples 1 and 3 in which q and w are unknown and ΔHsys must be evaluated from measurements in the surroundings. The galvanic cell of example 4 provides an instructive comparison of measurements in the two formulations. The equilibrium cell potential Ecell,eq can be measured electrically with great precision. In practice, molar reaction quantities of the cell reaction are evaluated with measurements of Ecell,eq using eqs 32−34 from the local formulation. The evaluation of these quantities by the global formulation, while theoretically possible, would be impractical because it relies on measurements of the temperature changes of the two thermal reservoirs. The temperature of a thermal reservoir is supposed to remain practically constant, so its change must be small and therefore difficult to measure with good precision. The difficulty concerning the temperature change of a thermal reservoir is also present in the global formulations of examples 2 and 3. It could be avoided by replacing the reservoir F

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Journal of Chemical Education

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(6) Craig, N. C. Entropy Analysis: An Introduction to Chemical Thermodynamics; VCH Publishers: New York, 1992. (7) Gislason, E. A.; Craig, N. C. General Definitions of Work and Heat in Thermodynamic Processes. J. Chem. Educ. 1987, 64, 660−668. (8) Gislason, E. A.; Craig, N. C. The “Global” Formulation of Thermodynamics and the First Law: 50 Years On. J. Chem. Educ. 2011, 88, 1525−1530. (9) Gislason, E. A.; Craig, N. C. Cementing the Foundations of Thermodynamics: Comparison of System-Based and SurroundingsBased Definitions of Work and Heat. J. Chem. Thermodyn. 2005, 37, 954−966. (10) Gislason, E. A.; Craig, N. C. The Proper Definition of Pressure− Volume Work: A Continuing Challenge. APS Forum on Education Spring 2005 Newsletter, 2005, 9−11; http://www.aps.org/units/fed/ newsletters (accessed Jan 2013). (11) Gislason, E. A.; Craig, N. C. Pressure−Volume Integral Expressions for Work in Irreversible Processes. J. Chem. Educ. 2007, 84, 499−503. (12) DeVoe, H. Thermodynamics and Chemistry, 2nd ed. [free e-book Online]; 2012; Sec. 3.2.1. http://www.chem.umd.edu/thermobook. (13) Adkins, C. J. Equilibrium Thermodynamics, 3rd ed.; Cambridge University Press: Cambridge, 1983; p 8. (14) de Heer, J. Phenomenological Thermodynamics with Applications to Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1986; p 42. (15) A reviewer asked why the relation ΔSsys = q/Tsys must be true for an internally reversible process. A familiar derivation is based on a Carnot cycle of an ideal gas. In calculating the work in each step of the cycle, the derivation assumes the gas at each instant is homogeneous, i.e., in an equilibrium state. The derivation does not require that the step, when reversed, returns the surroundings to their initial condition; internal reversibility is all that is required. (16) Gislason, E. A.; Craig, N. C. General Definitions of Work and Heat in Thermodynamic Processes. J. Chem. Educ. 1987, 64, 660−668. Eq. 16. (17) Craig, N. C. Entropy Analysis: An Introduction to Chemical Thermodynamics; VCH Publishers: New York, 1992; Chapter 2. (18) Gislason, E. A.; Craig, N. C. Why Do Two Objects at Different Temperatures Come to a Common Intermediate Temperature When Put in Contact? Entropy is Maximized. J. Chem. Educ. 2006, 83, 885− 889. (19) Joule, J. P. On the Mechanical Equivalent of Heat. Philos. Trans. R. Soc. London 1850, 140, 61−82. (20) Bauman, R. P. Maximum Work Revisited (letter to the editor). J. Chem. Educ. 1964, 41, 676−677. (21) DeVoe, H. Particle Model for Work, Heat, and the Energy of a Thermodynamic System. J. Chem. Educ. 2007, 84, 504−512. (22) DeVoe, H. Thermodynamics and Chemistry, 2nd ed. [free e-book Online]; 2012; Sec. 3.4.2. http://www.chem.umd.edu/thermobook. (23) Bauman, R. P.; Cockerham, H. L., III. Pressure of an Ideal Gas on a Moving Piston. Am. J. Phys. 1969, 37, 675−679. (24) Bertrand, G. L. Thermodynamic Calculation of Work for Some Irreversible Processes. J. Chem. Educ. 2005, 82, 874−877. (25) Gislason, E. A.; Craig, N. C. Cementing the Foundations of Thermodynamics: Comparison of System-Based and SurroundingsBased Definitions of Work and Heat. J. Chem. Thermodyn. 2005, 37, 954−966. Eq. 9. (26) DeVoe, H. Thermodynamics and Chemistry, 2nd ed. [free e-book Online]; 2012; Sec. 11.5.2. http://www.chem.umd.edu/thermobook. (27) Williamson, B. E.; Morikawa, T. A Chemically Relevant Model for Teaching the Second Law of Thermodynamics. J. Chem. Educ. 2002, 79, 339−342. (28) Mungan, C. E. Letter to the Editor: Why Distinguish Work from Heat? APS Forum on Education Fall 2005 Newsletter, 2005, 3−4; http://www.aps.org/units/fed/newsletters (accessed Jan 2013).

both system-based and surroundings-based, is mainly conceptual. Gislason and Craig find that students new to thermodynamics comprehend the global formulation more easily than the local formulation.8 One reason, according to them, is that the global formulation is based on changes of easily understood state functions, whereas the energy transfers q and w of the local formulation are “rather vague” path-dependent quantities. In my opinion, the concept of an energy transfer is no more difficult than that of an energy change. The two concepts are closely related: the energy of a subsystem can change only if there is an energy transfer. Furthermore, because q and w are different kinds of energy transfer, the local formulation helps a student keep track of the effects that change the system’s energy.28 System-based heat q involves a temperature difference or gradient at the system boundary, and system-based work w involves changes in one or more work coordinates of the system. I agree with Gislason and Craig8 that the philosophy of the global formulation can deepen students’ understanding of the local formulation. The pedagogical advantages and disadvantages of each formulation depend on the intended purpose and on the kind of experiment to which it is applied. If the aim is conceptual understanding of the first and second laws, either formulation works and the choice is a matter of taste. The local formulation of a process is usually more straightforward and less complicated, as is evident in the examples above. For explaining how thermodynamic quantities can be evaluated from experimental measurements, a local formulation is better for some kinds of experiments and a global formulation for others.

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ASSOCIATED CONTENT

S Supporting Information *

A detailed description of model calculations for the gas expansion of example 2, and a derivation of the upper limit of the frictional work in this example. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS I thank Norman Craig and Eric Gislason for stimulating discussions on the topics of this article and the reviewers for helpful comments.

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REFERENCES

(1) Bent, H. A. The Second Law of Thermodynamics: Introduction for Beginners at Any Level. J. Chem. Educ. 1962, 39, 491−499. (2) Bent, H. A. The Second Law: An Introduction to Classical and Statistical Thermodynamics; Oxford University Press: New York, 1965. (3) Bent, H. A. A Note on the Notation and Terminology of Thermodynamics. J. Chem. Educ. 1972, 49, 44−46. (4) Craig, N. C. Our Freshmen Like the Second Law. J. Chem. Educ. 1970, 47, 342−346. (5) Craig, N. C. Entropy Analyses of Four Familiar Processes. J. Chem. Educ. 1988, 65, 760−764. G

dx.doi.org/10.1021/ed300797j | J. Chem. Educ. XXXX, XXX, XXX−XXX