Criteria for Spontaneous Processes Derived from the Global Point of

Article pubs.acs.org/jchemeduc

Criteria for Spontaneous Processes Derived from the Global Point of View Eric A. Gislason*,† and Norman C. Craig‡ †

Department of Chemistry, University of Illinois at Chicago, Chicago, Illinois 60680, United States Department of Chemistry and Biochemistry, Oberlin College, Oberlin, Ohio 44074, United States

‡

ABSTRACT: Starting with the fundamental and general criterion for a spontaneous process in thermodynamics, ΔStot ≥ 0, we review its relationships to other criteria, such as ΔA and ΔG, that have limitations. The details of these limitations, which can be easily overlooked, are carefully explicated. We also bring in the important example of electrical energy to show how criteria for spontaneity are properly applied to electrochemical cells. The analysis in this article gains clarity from use of the “global” formulation of thermodynamics and from carrying out finite changes of thermodynamic properties rather than manipulating differential changes. Several examples are given. KEYWORDS: First-Year Undergraduate, Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Problem Solving/Decision Making, Calorimetry/Thermochemistry, Electrolytic/Galvanic, Equilibrium, Heat Capacity, Thermodynamics

A

properties are tabulated. Two such state properties are the Helmholtz and Gibbs energies. In this article we develop spontaneity criteria for various processes involving particular experimental arrangements; several of these criteria involve the Helmholtz and Gibbs energies.4−9 Constraints on the use of changes in these energy functions, which are entropy changes in disguise, affect their range of applicability. In addition, criteria for assessing spontaneity in coupled chemical reactions and in electrochemical processes are developed. Examples of the various processes are given in the article. We use the terminology “local” formulation and “global” formulation of thermodynamics to distinguish between the standard, local formulation that focuses on the reactive system and the global formulation that places all energy reservoirs on an equal footing.10 While emphasizing the global formulation, we provide links to the more familiar local formulation expressed in terms of heat q and work w. It should be emphasized, however, that q and w are not needed to develop the various criteria for spontaneity. The derivations presented in this article involve finite changes of thermodynamic properties, rather than the manipulation of differential changes, because we believe this type of derivation gives more insight. In addition, the distinction between integrals of exact and inexact differentials, which is essential in the local formulation, is absent in the global formulation. All changes in the global formulation are for state functions in various subsystems. Care is taken to ensure that the exact experimental conditions are specified so that the spontaneity criteria are not misused.

full understanding of the concept of a spontaneous process is fundamental to the application of thermodynamics. All spontaneous processes involve the irreversible change of an isolated system as it moves toward equilibrium.1−3 This article shows how the “global” formulation of thermodynamics aids in an understanding of the analysis of spontaneous change. Consider a chemical process where a reactive system undergoes a change from state i to state f. The second law gives the criterion for this process to be able to occur spontaneously: ΔStot = ΔS + ΔSsurr ≥ 0

(1)

Here ΔS = Sf − Si is the entropy change of the reactive system, ΔSsurr is the entropy change of the surroundings, and ΔStot is the total change in entropy of the universe of the experiment during the process. It should be emphasized that ΔStot > 0 does not guarantee that the process i → f will occur, only that it is not forbidden by thermodynamics. Put another way, ΔStot > 0 is a necessary but not sufficient condition for the process to occur. If ΔStot > 0, the process can occur spontaneously; if ΔStot < 0, the intended process cannot occur. As ΔStot becomes less and less positive and reaches zero, the initial state approaches equilibrium with the final state. The equilibrium condition (i.e., the equal sign) is included in eq 1 as a reminder of this important limit. Although the master relationship in eq 1 applies to any process, it has proven useful to develop criteria for spontaneity that involve only changes in state properties of the reactive system. These criteria have the advantage of using a single thermodynamic state property of the reactive system, and these © 2013 American Chemical Society and Division of Chemical Education, Inc.

Published: April 12, 2013 584

dx.doi.org/10.1021/ed300570u | J. Chem. Educ. 2013, 90, 584−590

Journal of Chemical Education

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w′ = −ΔEelec

DERIVATION OF FIRST LAW RELATIONS AND SPONTANEITY CRITERIA

These results allow us to recast the first law expression in eq 2 in the local formulation as

First Law Relations

ΔU − q − wPV − w′ = 0

Consider a simple experiment where the reactive system sits in a cylinder under a movable piston that at equilibrium exerts a pressure Pext = mg/A on the system. Here m is the mass of the piston, g is the acceleration of gravity, and A is the surface area of the piston. Above the piston is a vacuum. The piston’s height above the cylinder floor is denoted h, and the volume of the system is Ah. For simplicity, we assume that both the cylinder and the piston have no relevant molecular-level degrees of freedom so their temperatures remain constant. In addition, the system may or may not be in thermal contact with a calorimeter whose temperature is T. Consider a process where the system changes from state i to state f. We anticipate a later elaboration of the system and assume that the system in carrying out its process can produce (or absorb) electrical energy through coupling to an electrical system. Then, conservation of energy, the f irst law in the global formulation, can be written10 ΔU + ΔHcal + ΔEpist + ΔEelec = 0

Spontaneity Criteria

The second law is given in eq 1. There is no entropy change associated with either ΔEelec or ΔEpist. The reason for this characteristic is that entropy is a measure of dispersal of energy at the microscopic level, which is described by the variables temperature T and pressure P. Neither P nor T are involved in ΔEelec or ΔEpist. Instead, these energy changes are fully described with a few macroscopic variables, such as piston height and the voltage of the capacitor that holds ΔEelec. For the experiment considered here, any thermal energy received or given up by the calorimeter at temperature T causes a state function change in the calorimeter given by14

(2)

(3)

We assume that the piston is at rest at the beginning and end of the process but moves during the process. Here, ΔEelec is positive if the process creates electrical energy that is transferred to the electrical surroundings and is negative if electrical energy from the surroundings is absorbed by the system in the process. The quantity −ΔEelec is equivalent to a work effect term (see below). In addition, U and V are the internal energy and volume of the system, and Hcal is the enthalpy of the calorimeter. ΔH is used for the calorimeter with the understanding that it is under a constant pressure of 1 atm applied by the atmosphere.10 ΔEpist is initially written in eq 3 in terms of piston-characteristic variables (m, g, and h) and then transformed into variables connected to the reactive system, P and V. In the global formulation, the student sees readily how to accommodate additional energy exchanges, such as with an electrical system. We now consider the f irst law in the alternative, local formulation. There are two protocols to define work w and heat q in general use.11−13 The surr-based definitions determine work and heat from changes in the various subsystems of the surroundings,11 whereas the sys-based definitions determine these quantities from changes in the system.12 In the experiments under consideration in this article, the surr-based values of w and q can always be determined but that is not the case for the sys-based definitions if there is friction between the piston and the cylinder as the piston moves. In addition, surrbased definitions are the natural definitions to use in conjunction with the global formulation of thermodynamics,10 which includes eq 2. For these reasons, we use surr-based definitions in this article. The expressions for the heat and work effects during the process are then given by11−13

q = −ΔHcal

(4)

w = wPV + w′

(5)

wPV = −ΔEpist = −mg Δh = −PextΔV

(6)

(8)

The work term wPV relates to changes in height of the piston in the gravitational field of the earth. In many processes an equivalent term expresses energy changes in the atmosphere for a system where the atmosphere plays the role of the piston.10 The work term w′ is often referred to as the “useful” work, since wPV arises from the obligatory expansion or contraction of the system, typically against the piston or the atmosphere, and cannot be used for other purposes. We use −ΔEelec as a common example of w′ throughout the article, but other types of work such as surface work or magnetization would also qualify as “useful” work.

where ΔEpist = mg Δh = (mg /A)(AΔh) = PextΔV

(7)

ΔSsurr = ΔScal = ΔHcal /T

(9)

This result can be substituted into eq 1 and is essential for evaluating entropy changes. For a reversible process ΔS + ΔSsurr = 0, so ΔS = −ΔHcal/T.

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ADIABATIC PROCESSES For an adiabatic process, the system is thermally isolated from the surroundings so ΔHcal = −q = 0, and from eq 9 ΔScal = 0. Then, the global conservation-of-energy expression in eq 2 becomes ΔU + ΔEpist + ΔEelec = 0

(10a)

and, for comparison, the local equivalent in eq 8 becomes ΔU = wPV + w′

(10b)

The criterion for spontaneity from eq 1 becomes merely ΔS ≥ 0. We next consider two special cases of adiabatic processes. Table 1 summarizes the changes in various thermodynamic quantities for both cases. Case 1: Adiabatic Process at Constant Volume

In this case, ΔEpist = 0, so conservation of energy from eq 10a becomes ΔU = −ΔEelec = w′ and the criterion for spontaneity remains ΔS ≥ 0. Case 2: Adiabatic Process at Constant Presssure

For this case, the initial and final values of the system pressure P, namely Pi and Pf, equal Pext. Conservation of energy is given by eq 10a, and it can be rewritten in terms of the enthalpy of the system, H = U + PV, as ΔH = ΔU + (Pf Vf − PV i i ) = ΔU + Pext ΔV = ΔU + ΔEpist = −ΔEelec = w′ 585

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Table 1. State Function Changes and Spontaneity Criteria for Four Processes Variable ΔHcal ΔScal ΔEpist wPV ΔEelec w′ w = wPV + w′ q ΔU = w + q ΔH = ΔU + Δ(PV) System-oriented criterion for spontaneity

Adiabatic

Adiabatic

Const-T

Const-V

Const-P

Const-V

Const-P

0 0 0 0 ΔEelec −ΔEelec −ΔEelec 0 -ΔEelec VΔP − ΔEelec ΔS ≥ 0

0 0 PextΔV −PextΔV ΔEelec −ΔEelec −PextΔV − ΔEelec 0 −PextΔV − ΔEelec −ΔEelec ΔS ≥ 0

ΔHcal ΔHcal/T 0 0 ΔEelec −ΔEelec −ΔEelec −ΔHcal −ΔEelec − ΔHcal VΔP − ΔEelec − ΔHcal ΔA + ΔEelec ≤ 0

ΔHcal ΔHcal/T PextΔV −PextΔV ΔEelec −ΔEelec −PextΔV − ΔEelec −ΔHcal −PextΔV − ΔEelec - ΔHcal −ΔEelec − ΔHcal ΔA + ΔEelec + PextΔV ≤ 0 and ΔG + ΔEelec ≤ 0

We see that for a “constant pressure” process, the enthalpy function absorbs an energy change PextΔV that occurs in the surroundings (see eq 3). This subtlety, which is illuminated by the global formulation,10 must always be remembered in applications of the enthalpy function. We emphasize that “constant pressure” applies to Pext as well as to Pi and Pf, but not to the reactive system pressure P during the process. The criterion for spontaneity for this adiabatic process remains ΔS ≥ 0.

ΔU = q + w = q − mg Δh + w′

CONSTANT TEMPERATURE PROCESSES In a constant temperature process, the system is in thermal contact with a calorimeter whose temperature is T. Thus, the initial and final temperatures of the system, Ti and Tf, equal T. For such a process, combining eqs 1 and 9 gives

ΔA = ΔU − T ΔS

(12)

ΔA + ΔEpist + ΔEelec ≤ 0

and in local terms as ΔA − w ≤ 0

(14)

Case 3: Constant Temperature, Constant Volume Process

For this case, ΔEpist = −wPV = 0. Conservation of energy from eqs 16a and 16b can be written

(15a)

ΔU = −ΔHcal − ΔEelec = q + w′

The equivalent local formulation is ΔU − T ΔS − wPV − w′ ≤ 0

(15b)

(19)

The criterion for spontaneity at constant-T and constant-V in eq 15a can be rewritten in terms of the Helmholtz energy as

Although eqs 15a and 15b involve energies, their derivation shows they are entropy expressions in disguise. The inequality sign signals this fact, because ΔStot can grow, whereas total energy is always conserved as in eq 2. We now apply eqs 2 and 15 to some particular experimental arrangements.

ΔA + ΔEelec ≤ 0

(20a)

and in local terms as ΔA − w′ ≤ 0

General Constant Temperature Process

(20b)

These results are summarized in Table 1. We emphasize that ΔA alone is not a criterion for spontaneity at constant volume when ΔEelec = −w′ is not zero. This result also shows that the maximum useful work that can be done by the system (−w′ = ΔEelec) for the case of a process at constant temperature and volume is −ΔA, the limit when eqs 20a and 20b equal zero.

For this process, we allow both the system’s volume and pressure to change. The expression for conservation of energy from eq 2 can be written ΔU = −ΔHcal − mg Δh − ΔEelec

(18b)

We emphasize that ΔA alone is not a criterion for spontaneity when ΔEpist + ΔEelec = −w is not zero. We also see that the maximum amount of work that can be done by the system (−w) in any process carried out at constant temperature is −ΔA, the limit when eq 18b equals zero.

After multiplying through by −T, which changes the sense of the inequality, this equation can be recast as an energy expression: ΔU − T ΔS + ΔEpist + ΔEelec ≤ 0

(18a)

(13)

an expression known as the Clausius inequality. The general criterion for spontaneity is obtained by combining eqs 2 and 12: ΔStot = ΔS − [ΔU + ΔEpist + ΔEelec]/T ≥ 0

(17)

so the criterion for spontaneity in eq 15a can be rewritten as

and substituting eq 4 gives the well-known result ΔS ≥ q/T

(16b)

Here, we have written −ΔEpist = wPV = −mgΔh to emphasize that the PV-work done by the system is always equal to the energy exerted to lift the piston through a height Δh. (We assume that the piston is at rest before and after the process takes place.) This condition is so even in the extreme case where the initial and final pressures of the system, Pi and Pf, are different from each other and also different from Pext, which is defined by the third equality in eq 3. It is useful at this point to define the Helmholtz energy A = U − TS in terms of reactive system variables. At constant-T

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ΔStot = ΔS + ΔSsurr = ΔS + ΔHcal /T ≥ 0

Const-T

(16a)

and in the local equivalent 586



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Case 4: Constant Temperature, Constant Pressure (Pext) Process

the Helmholtz energy nor the Gibbs energy is conserved, even when −w (ΔEpist + ΔEelec) and −w′ (ΔEelec) are zero, except in limiting reversible processes. It is useful to summarize the results given above for the spontaneity criteria for the various cases discussed above. Table 2 gives this summary for various processes where the system

For this process, all of the terms in eqs 2, 15a, and 15b are nonzero. Conservation of energy from eq 2 can be written in two useful ways. One is ΔU = −ΔHcal − ΔEpist − ΔEelec

(21a)

Table 2. Criteria for Spontaneous Processes

or its local equivalent

ΔU = q + w

Criterion

(21b)

ΔStot ≥ 0 ΔS ≥ 0 ΔA + ΔEpist + ΔEelec = ΔA − w ≤ 0

The other expression for conservation of energy is ΔH = −ΔHcal − ΔEelec

(22a)

or its local equivalent

ΔH = q + w′

ΔG + ΔEelec = ΔG − w′ ≤ 0

(22b)

As discussed below eq 11, ΔH has absorbed the work term ΔEpist = −wPV. The reader is undoubtedly familiar with the result that for a constant pressure process ΔH = qP, provided no other work other than PV-work is done during the process.10 Equation 22b gives the more general result, when w′ = −ΔEelec is not zero (see also eq 11). In terms of the Helmholtz energy, the criterion for spontaneity at constant-T and constant-P in eq 15a can be written ΔA + ΔEpist + ΔEelec ≤ 0

a T may vary in the system during the process. bP may vary in the system during the process.

goes from initial (i) to final (f) state. It is not assumed in a “constant-T” process that the temperature of the system is constant during the process; in fact, processes where the system temperature is not even well-defined during the process are allowed. Similarly, it is not assumed in a “constant-P” process that the pressure of the system is constant during the process; in fact, processes where the system pressure is not even welldefined during the process are allowed. Constant-T and constant-P simply refer to the initial and final states, i and f, for the process and to the conditions in the surroundings.

(23a)

or in local terms ΔA − w ≤ 0

(23b)

Comparison of eqs 23a and 23b shows that w typically includes wPV (−ΔEpist = −PextΔV) in addition to an electrical energy change w′ (−ΔEelec). The results in eqs 18, 20, and 23 all demonstrate that the maximum amount of work that can be done by the system (−w = ΔEpist + ΔEelec) in any process carried out at constant temperature is −ΔA, the limit when eq 23b equals zero. Because the present process is also being carried out at constant external pressure, Pext = Pi = Pf, the criterion for spontaneity can be written in a second form by defining the Gibbs energy G = U + PV − TS in terms of reactive system variables. Thus, ΔG = ΔU + P ΔV − T ΔS = ΔH − T ΔS

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We now analyze several examples of processes illustrative of the cases described in the previous section. Not only do these examples provide a deeper understanding of the results in this article, but they provide the student of chemistry examples of the calculation of changes in important state function changes such as ΔU, ΔS, ΔA, and ΔG. We encourage students to check these calculations by consulting thermodynamic tables in their textbook or in The Handbook of Chemistry and Physics.15 As an example of a constant temperature process where both pressure and volume change, consider the process

(24)

(25a)

or the global equivalent ΔG + ΔEelec ≤ 0

EXAMPLES

General Processes

Equation 11 shows that for a constant-P process ΔH = ΔU − wPV, so when eqs 15b and 24 are combined, we obtain the criterion for a constant-T and constant-P process as ΔG − w′ ≤ 0

Description of the Process Any process Adiabatic process (−ΔHcal = q = 0) (a) System in thermal contact with calorimeter at temperature T (b) Ti = Tf = T (constant-T process)a (a) System in thermal contact with calorimeter at temperature T (b)Ti = Tf = T (constant-T process)a (c) System in mechanical contact with surroundings that exert a pressure Pext on the system at equilibrium (d) Pi = Pf = Pext (constant-P process)b

1 mol Ar (2 atm, 298 K) → 1 mol Ar (1 atm, 298 K) (25b)

where the Ar(g) sits in a cylinder under a movable piston in thermal contact with a calorimeter at 298 K. The piston mass is such that when moving it exerts a constant pressure down on the Ar of Pext = 1/2 atm. The piston is initially clamped in place so that the Ar pressure is 2 atm. When the piston is released it rises rapidly and is then caught and clamped again so that the final pressure of the Ar is 1 atm. We assume that the Ar behaves as an ideal gas, so U = U(T) and, consequently, ΔU = 0. For this process, ΔEelec = 0. The energy change of the piston, ΔEpist, is PextΔV. Because Pext = 1/2 atm and ΔV = RT[1/(1 atm) − 1/(2 atm)], ΔEpist = (1 mol)RT/4. The entropy change of the gas16 is ΔS = (1 mol)R ln(P1/P2) = (1 mol)R ln 2. Thus, we see from eq 17 that ΔA = −(1 mol)RT ln 2 and ΔA + ΔEpist = −(1

for a spontaneous process. These results are summarized in Table 1. We emphasize that ΔG alone is not a criterion for spontaneity when w′ = −ΔEelec is not zero. This result shows that the maximum usef ul work that can be done by the system (−w′ = ΔEelec) for a process at constant temperature and pressure is −ΔG. It must be remembered that although ΔG is expressed in terms of system variables, it incorporates a change in the surroundings, namely ΔEpist = −wPV. Consequently, the pressure exerted by the surroundings on the system, Pext, must remain constant during the process for eq 25 to apply. When this subtle constraint is overlooked, incorrect conclusions can be reached. Because ΔA and ΔG are entropy changes in disguise, neither 587



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mol)RT(ln 2 − 1/4) = −0.443(1 mol)RT < 0. Equation 18a, which we emphasize contains more than ΔA as a criterion, confirms that this process can occur spontaneously. If the gas is not ideal, ΔU will have a small positive or negative value, but ΔA + ΔEpist + ΔEelec will be dominated by the ΔS term, so the process can still occur spontaneously. This result is guaranteed by the exact derivative (∂A/∂V)T = −P, which is less than zero for a gas. As an example of case 3 at constant T and V, we consider the reaction

Pf (atm) = (1 mol)RTf /V = (1 mol)Tf /[(3 mol)(298.15 K)] ΔU2 = (70.8 J/K)(Tf − 298.15 K) ΔS2 = (79.1J/K) ln(Tf /298.15 K) − R ln(Pf /1 atm)

Solving ΔU = ΔU1 + ΔU2 = 0 yields Tf = 346.2 K and Pf = 0.387 atm. From these results, we compute ΔS2 = 19.7 J/K and so ΔS = ΔS1 + ΔS2 = −276.3 J/K. The criterion for spontaneity for an adiabatic, constant volume process (see Table 1) is ΔS > 0. Clearly, this proposed process is impossible, even though it would be exothermic. The criteria involving ΔA and ΔG do not apply in this process that is not carried out at constant T. Considering the process in reverse, we conclude that N2F4(gas) can spontaneously decompose into its elementary substances. The entropy increase in the reactive system for the reverse reaction outcompetes the entropy decrease due to cooling the gas, so ΔS > 0.

CS2 (l, 25 °C, 1 atm) + O2 (g, 25 °C, 1 atm) → CO2 (g, 25 °C, Pf ) + 2S(s, 25 °C, Pf )

carried out in a constant-volume bomb calorimeter sitting in a calorimeter at 25 °C with the reactants in their standard states. A table of thermodynamic properties15 gives values of Gf°, from which we can determine ΔrG°, and then ΔrA°. Because ΔG = ΔA + Δ(PV), we have for the ideal-gas approximation Δr A° = Δr G° − RT Δngas

Coupled Reactions

The criteria discussed in this article lend themselves nicely to a discussion of coupled chemical reactions, whereby a reaction favored thermodynamically causes a second reaction that by itself is not favored to occur. An interesting example is the geologically relevant process of dissolving limestone, CaCO3(c), in water. The direct process is

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where Δngas is the change in the number of moles of gas in the reaction as written. For this reaction, Δngas = 0, so Pf = 1 atm, and, consequently, the products are also in their standard states. The final result is ΔA = ΔrA° = −459.0 kJ/(mol rxn). (The unit “mol rxn” is one round of the chemical equation as written in moles.)17 Because w′ = −ΔEelec = 0, eqs 20a and 20b confirms that this process can occur spontaneously. The criterion in eqs 25a and 25b applies to any chemical process carried out at constant-T in contact with the atmosphere, since here Pi = Pf = Pext = 1 atm. As an example of case 4, consider the proposed reaction with reactants and products in their standard states and with ΔEelec = 0:

CaCO3(c) + H 2O(l) → Ca 2 +(aq) + HCO3−(aq) + OH−(aq)

With reactants and products in their standard states, ΔrG° = +68.6 kJ/(mol rxn),15 so CaCO3 is very insoluble in water. However, if an acid is added, as by CO2 saturation of groundwater, the H+ can react with the OH− in a coupled process H+(aq) + OH−(aq) → H 2O(l)

C2H5OH(l) + H 2O(l) → 2CH3OH(l)

for which ΔrG° = −79.9 kJ/(mol rxn).15 The net reaction for the spontaneous coupled process is

Here ΔrG° = +78.7 kJ/(mol rxn);15 so, not surprisingly, this process cannot occur spontaneously. A somewhat more challenging process to analyze is case 1 with the proposed reaction

H+(aq) + CaCO3(c) → Ca 2 +(aq) + HCO3−(aq)

for which ΔrG° = −11.3 kJ/(mol rxn). Thus, CaCO3 can dissolve in acidic solutions as shown above, and, if the H+ is in excess, CO2(g) can be released, carrying the equilibrium even farther to the right. The criterion for spontaneity in a coupled process at constant temperature and pressure with ΔEelec = −w′ = 0 is seen to be ΔrG° = ΔrG°1 + ΔrG°2 ≤ 0.

1 mol N2(g, 25 °C, 1 atm) + 2 mol F2(g, 25 °C, 1 atm) → 1 mol N2F4 (g, Tf , Pf )

carried out at constant volume V in a bulb with adiabatic walls. Since there are 3 mols of gaseous reactants, V = (3 mol)R 298.15K. We have ΔEelec = −w′ = 0 in this process as no electrical system is coupled, so ΔU = 0. This reaction is exothermic, which means that the temperature of the products, Tf, will be greater than 25 °C. Tf can be obtained from the ΔU = 0 requirement. We assume that the gases behave ideally, and that CP for 1 mol of N2F4, 79.1 J/K, is independent of pressure and temperature, as is CV = 70.8 J/K. The analysis of the process is broken into two steps. In step #1, both the reactants and products are in their standard states of 1 atm and 25 °C. For this process, a table of thermodynamic functions15 gives us ΔH1 = −8400 J, ΔU1 = ΔH1 − ΔngasRT = −3400 J, and ΔS1 = −296.0 J/K. In step #2, the N2F4 product is changed from its standard state to its final state described by Tf and Pf. For this process,16

Electrochemical Cells

The results derived in this article can be applied to reactions carried out in galvanic cells (chemistry dominant) or electrolysis cells (electrical system dominant). When an electrical system is linked to a chemical reaction, ΔEelec can be expressed in variables related to the reactive system as ΔEelec = VelF Δz

(27)

where Vel is the voltage of the electrical subsystem, F = 96485 C/(mol e−) is the Faraday constant, and Δz is (mol e−)/(mol rxn).17 Equation 25a shows the criterion for spontaneity for a process at constant-T and constant-P. Thus, Δr G + VelF Δz ≤ 0 588

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In a galvanic cell, the negative ΔrG term dominates over the positive electrical term; in an electrolysis process, ΔrG is positive and the negative ΔEelec dominates. At electrochemical equilibrium, that is, equilibrium for the overall electrochemical process, Δr G = −FEΔz

ΔEelec = (0.981 V/1.101 V)(212.6 kJ/mol rxn) = 189.4 kJ/(mol rxn)

In this case, ΔEelec = −w′ = 189.4 kJ/(mol rxn) is less than the maximum possible for this process, and correspondingly, the thermal energy released into the calorimeter through the heat effect is increased to −q = ΔHcal = 29.3 kJ/(mol rxn). The overall criterion for spontaneity at constant-T and constant-P in eq 28 gives ΔG + ΔEelec = −212.6 kJ/(mol rxn) + 189.4 kJ/ (mol rxn) = −23.2 kJ/(mol rxn), which is less than zero, as required. We reemphasize that ΔG alone is not the criterion for spontaneity.

(29)

where E is the emf measured with negligible current flow under essentially equilibrium conditions. Here, the tendency for change in the electrical system balances the tendency for change in the chemical system. Equation 29 becomes Δr G° = −FE°Δz

(30)

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for standard state conditions. E° and Δz are positive for the chemical reaction as written in the chemically spontaneous direction, guaranteeing that ΔrG° < 0, and negative for the reverse of the chemical reaction in electrolysis, guaranteeing that ΔrG° > 0.18 A good example of an electrochemical reaction is the galvanic cell

SUMMARY In this article, we have derived various criteria for physicochemical processes to determine if they can occur spontaneously. Students see how all spontaneous processes trace to ΔStot ≥ 0 and how subsidiary criteria are entropy changes in disguise. These criteria are summarized in Tables 1 and 2. It is very easy for students to misapply the criteria for spontaneous processes, but careful attention to the conditions and relationships in the summary in Table 2 will ensure that this does not happen. The relationship between these criteria and the maximum work of particular types that can be done in a process is also shown. None of the derivations involved the manipulation of differentials; instead, all of the results involve finite changes in thermodynamic state properties. We believe this makes the derivations and results more readily understandable. Several examples of the criteria described in the article are presented. Our work shows how the common practice of calling the Gibbs and Helmholtz functions “free energies” is mistaken. Rather, the “free” or available energies are the changes, ΔA and ΔG, under the appropriate constraints and circumstances described in this article.

Zn(c)|Zn 2 +(aq)||Cu 2 +(aq)|Cu(c)

A galvanic cell is one in which chemical energy is converted into electrical energy. The chemical reaction associated with this cell is Cu 2 +(aq) + Zn(c) ⇔ Cu(c) + Zn 2 +(aq)

For simplicity, we assume that the Cu2+(aq) and Zn2+(aq) ions are in their standard states (ideal solution, unit activity) and the entire cell is at 1 bar pressure. The cell sits in a calorimeter held at 25 °C. First, we consider the chemical reaction alone, which occurs with a perfect short circuit between the electrodes or by direct mixing of the reactants. A table of thermodynamic properties15 shows that ΔrH° = −218.7 kJ/(mol rxn), ΔrG° = −212.6 kJ/(mol rxn), and ΔrS° = −20.9 J/(K mol rxn). Clearly, this reaction can occur spontaneously. From the ΔrG° value, the standard state emf can be computed from eq 30 as E° = 1.101 V recognizing that Δz = 2 (mol e−)/(mol rxn). (Of course, one could also obtain E° from a table of standard electrode potentials.) If eqs 2 and 11 are combined, the first law expression for the electrochemical cell can be written ΔH + ΔHcal + ΔEelec = 0

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

(31)

We consider two experiments where the cell is run for 1 mol rxn, so that 1 mol of Cu(c) is plated out from the solution. In one case, the process is run at the reversible limit; in the other case, the process is run irreversibly with a significant current flow. In both cases, ΔH° = −218.7 kJ/(mol rxn) and ΔG° = −212.6 kJ/(mol rxn). In the first experiment, when the cell is run reversibly, Velec = E° = 1.101 V. In this case of electrochemical equilibrium, eq 30 shows that −ΔEelec = ΔG° = −212.6 kJ/(mol rxn). ΔHcal can be computed from eq 31 and q = −ΔHcal = −6.1 kJ/(mol rxn). For this reversible case, ΔEelec = −w′ = 212.6 kJ/(mol rxn) is the maximum possible and, therefore, the thermal energy released into the calorimeter through the heat effect, −q = ΔHcal = 6.1 kJ, is the minimum possible for this process. In the second experiment, we assume that the cell is run with a current of I = 30 mA and that the cell has a resistance of 4.0 Ω. In that case,

REFERENCES

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Velec = E° − IR = 0.981 V 589



Article

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Criteria for Spontaneous Processes Derived from the Global Point of

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