Thermodynamic Diagrams

In the Classroom

Thermodynamic Diagrams S. Chaston Faculty of Applied Science, University of Canberra, ACT, 2601, Australia

Thermodynamic data such as equilibrium constants, standard cell potentials, the molar Gibbs free energies of formation, and standard entropies of substances can be a very useful basis for an organized presentation of knowledge in diverse areas of applied chemistry. Thermodynamic data can become particularly useful when incorporated into thermodynamic diagrams that are designed to be easy to recall, to serve as a basis for reconstructing previous knowledge, and to determine whether reactions can occur exergonically or only with the help of an external energy source. Few of the students in our chemistry-based courses would need to acquire the depth of knowledge or rigor of professional thermodynamicists (1, 2). But they should nevertheless learn how to make good use of thermodynamic data in their professional occupations that span the chemical, biological, environmental, and medical laboratory fields. This article discusses examples of three thermodynamic diagrams that have been developed for this purpose in our courses. They are the thermodynamic energy account (TEA), the total entropy scale, and the thermodynamic scale diagrams. Diagrams such as these can produce a better understanding of chemical change not only for processes studied by chemists, but in biochemistry and biology, and in industrial and environmental processes. They have the effect of putting the imagination to work towards the understanding of abstract thermodynamic functions. They serve as a confident starting point for the development of a more advanced knowledge of the laws of thermodynamics, and they allow the reader to keep track of specialist thermodynamic discourses in the literature (3–5). There have been a number of articles in this Journal describing such diagrams for teaching and learning the thermodynamics of chemical change (e.g., Barrow’s universe of the system diagrams [3], double scales for equilibria [6 ], the conjugate acid–base chart [7 ], potential ladders for electrochemistry [8], predominance diagrams [9, 10], and entropy diagrams [11]). The diagrams discussed here may be less complex and easier to set up and work with than the ones in some of the above references. The Thermodynamic Energy Account The thermodynamic energy account (TEA) diagram in Figure 1 shows graphically how the various thermodynamic

P = Pext Chemical or phase change

T = 25°C

Ew = –Pext ∆V Gw = ∆rG – qirr q = T ∆rS + qirr

∆r A – qirr ∆r H

∆rU

kJ/mol

Figure 1. The thermodynamic energy account (TEA) diagram that is set up for chemical or phase changes under conditions of constant external pressure and temperature.

216

state functions and path functions relate to each other under conditions of constant external pressure and temperature. The circle represents the system, and the arrows point to the surroundings. The three functions—expansion work (Ew), Gibbs work (Gw), and thermal energy (q)—are each observable and their values can be measured directly. The value of Ew is obtained from measurements of the volume change and the external pressure. Gibbs work, Gw, is the non-expansion work produced by an exergonic reaction that drives an endergonic process in a coupled subsystem. Thermal energy, q, is measured by calorimetry. The TEA diagram shows how these values are related to the thermodynamic state functions in the equations Ew = – Pext ∆V Gw = ∆rG – q irr q = T ∆rS + qirr Ew + Gw = ∆ rA – qirr Gw + q = ∆ rH Ew + Gw + q = ∆rU Under the sign convention in the diagram, values of Ew, Gw, q, and qirr are positive when they contribute to an increase in the internal energy of the system and are negative when they decrease it (12). The TEA diagram was designed to use published values of standard thermodynamic functions such as ∆rH°, ∆rG°, and ∆rS°. These equations therefore apply for reactions that occur at a constant temperature and at a constant system pressure that remains in mechanical equilibrium with the applied external pressure Pext . When one battery of greater emf is used to recharge another battery of lower emf, the battery of lower emf is the coupled endergonic subsystem that receives and retains energy Gw from the battery of greater emf. Other examples of exergonic reactions driving coupled endergonic processes are the synthesis of ATP as a product of aerobic respiration (12), or the driving of an electrical engine by an electrochemical cell, or driving a heat engine by a combustion reaction (13). In these circumstances the Gibbs energy ∆rG for the exergonic system splits into two path functions, Gibbs work Gw and irreversible heat qirr. In Example 1, Gw is the work retained that produces the muscle contractions of an exercising athlete and q irr is observed in her body warmth. The surplus energy from ∆rG that is not retained as work by a coupled endergonic subsystem becomes thermal energy, q irr. This energy was described by Craig (13) as “potentially useful energy that has been degraded into thermal energy”. Craig analyzed some constant-pressure irreversible physicochemical processes in terms of an increase in total entropy (∆Stot) of the system and its surroundings. The link between q irr as described here and Craig’s ∆Stot for these processes—provided that the system pressure remains in mechanical equilibrium with a constant external pressure—is found in the equation qirr = – T ∆Stot

Journal of Chemical Education • Vol. 76 No. 2 February 1999 • JChemEd.chem.wisc.edu

In the Classroom

This equation provides a link between an observable path function q irr and the abstract index of reaction, ∆Stot. For reactions that are irreversible, exergonic, and equilibriumseeking, qirr must be nonzero and negative. When a system is not coupled to an endergonic subsystem and reacts irreversibly, q irr captures all the useful non-expansion work as thermal energy, so that Gw = 0 and qirr = ∆rG. When the reaction is carried out reversibly on a coupled endergonic subsystem, q irr equals zero and Gw equals the maximum nonexpansion work, ∆rG, that the reaction can produce. Noyes used the symbol qirr previously in a different sense from its meaning here (5). The TEA diagram in Example 1 uses values for the relevant standard thermodynamic functions for aerobic respiration, in which reactions occur under conditions of constant temperature and constant system pressure. The results below are more easily worked out and understood when they are pencilled onto a TEA diagram.

Example 1 The oxidation of glucose to CO2 and H2O, eq 1, by the aerobic respiration reaction that provides energy for biochemical processes that maintain life and exercise. C6H 12O6 + 6O2 → 6H 2O + 6CO 2

(1)

The energy Gw for the respiration reaction is stored in cells by the conversion of 38 moles of ADP per mole of glucose into 38 ATP (12). This energy is harnessed in the citric acid cycle for muscle contraction and other body needs, and amounts to about 40% of the standard Gibbs energy of this reaction. On this basis Gw = 0.40 × ∆rG°, and qirr = 0.6 × ∆rG °. Standard thermodynamic data for this reaction at 298 K are ∆rS ° = 182.4 J K᎑1 mol᎑1, ∆ rH° = ᎑2808 kJ mol᎑1 (12). In this example these values are used as approximate values for the blood temperature 310 K. The values of the remaining thermodynamic functions in kJ/mol can be calculated by using equations that can be reconstructed with the help of the TEA diagram: T ∆rS° = 56.5 ∆rG° = ᎑2864 Ew = 0 Gw = ᎑1146 q irr = ᎑1718 q = ᎑1662 ∆rA° = ᎑2864 ∆rU° = ᎑2808 ∆Stot = 5542 J/K mol

310 × 182.4/1000 ∆rG° = ∆rH° – T ∆rS° no increase in number of moles of gas Gw = 40% of ∆rG ° Gw = ∆rG ° – q irr ; qirr = 60% of ∆rG ° q = ∆r H ° – Gw ; q = T ∆rS° + q irr Ew + Gw = ∆rA° – qirr ∆rU° = Ew + Gw + q ∆Stot = ᎑q irr/T

Irreversible Expansions The equations from the TEA diagram can be made more general to include processes where the system pressure is not at mechanical equilibrium with the external pressure, such as the irreversible expansion of an ideal gas against a constant external pressure in Example 2. For this mechanically irreversible process, where wirr = ᎑Pext ∆V, the equations on the TEA diagram become Ew = w irr Gw = ∆ rG – q irr q = T ∆ rS + qirr – w irr

Ew + Gw = ∆ rA – q irr + wirr Gw + q = ∆ rH – wirr Ew + Gw + q = ∆rU qirr – wirr = – T ∆Stot

Example 2 A mole of an ideal gas expands irreversibly against a constant pressure Pext at constant temperature when the initial pressure is P*. The expansion stops when final pressure is Pf . The volume change is ∆ V. Ew = wirr = ᎑Pext ∆V irreversible expansion work against constant pressure Pext Gw = 0 no Gibbs work is done ∆U = 0 ideal gas expansion q = ᎑ wirr ∆U = Ew + Gw + q = 0, Gw = 0, q = ᎑Ew ∆G = RT ln(Pf /P*) state function ∆G depends on Pf and P* only ∆ S = ᎑R ln(Pf /P*) state function ∆S depends on Pf and P* only q irr = ∆G Gw = ∆G – qirr = 0 ∆H = 0 ∆H – wirr = q + Gw = ᎑ wirr ∆ A = ∆G ∆A – qirr + wirr = Ew + Gw = wirr ∆ Stot = ᎑R ln(Pf /P*) + Pext ∆V/T ᎑T∆ Stot = q irr – wirr

When this gas is allowed to expand into a vacuum chamber, Pext = 0, and final pressure is Pf : Ew = wirr = 0, Gw = 0, q = 0, ∆H = 0, ∆U = 0, q irr = ∆G = ᎑T∆S = RT ln (P f /P*), and ∆Stot = ∆S = ᎑R ln (Pf /P*) (2, 3). Total Entropy Scales

Example 3 Water evaporates isothermally and irreversibly into a container. Its vapor pressure throughout the change remains in mechanical equilibrium with the external pressure that is exerted by the walls of the container. Figure 2 is a gas–liquid vaporization scale for water that is based on values of total entropy change, ∆Stot J/K mol, from eq 2. H2O(l) → H2O(g)

(2)

Craig introduced entropy diagrams to reinforce entropy analyses (11). His diagrams used a vertical axis for entropy values instead of the horizontal version presented here. Each of the three positions on the scale represents eq 2 as a gas–liquid couple for water at its standard state Pext = P°, its equilibrium state P ext = P*, and at an expanded/dilute state Pext < P*. In the standard state the liquid is pure water and its vapor pressure is equal to P° = 1 atm. In the equilibrium state at 25 °C the –2 Standard gas liquid H2O(g) H2O(l) ∆rS tot

–28.8 ∆rS °tot

Equilibrium gas liquid H2O(g) H2O(l)

2 Expanded Dilute gas liquid H2O(g) H2O(l)

0.0 29.5 ∆rS °tot – R ln(P*/P °) = 0 R ln Xw – R ln(P/P*)

Figure 2. Total entropy change diagram for the vaporization of water. Saturation vapor pressure at 25 °C, P* = 0.03126 atm.

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In the Classroom

liquid is pure and its vapor pressure is equal to its saturation vapor pressure P* = 0.03126 atm. In the general expanded/ dilute state, the liquid may contain a solute and its vapor pressure is any value of P = Pext . An arrow above a couple indicates the direction that the reaction takes when the system shifts towards equilibrium. The arrow also indicates the direction of movement of the couple along the scale as the reaction proceeds. A decrease in vapor pressure shifts the scale position of the couple to the right, and an increase moves it to the left. Dilution of the liquid phase by addition of a solute moves the couple to the left. The standard state position is based on the value of ∆rS°tot that was obtained by calculation from eq 3, where the standard enthalpy ∆rH ° and entropy ∆rS ° functions are for evaporation of water at 298 K. ∆ rS °tot = ∆rS ° – ∆rH°/T

(3)

These standard values were obtained by calculation from the standard enthalpy of formation, ∆H °f , and standard entropy, S°, data for water in its liquid and gas phases from values published on the Internet (14). The other positions were obtained from the expression for total entropy change (11) ∆Stot = ᎑R ln(Pext /P*) where P* is the equilibrium saturation vapor pressure of the pure liquid. At each point on the scale, the vapor pressure of the liquid equals the applied external pressure, Pext, that is exerted by the walls of the container. As the solvent continues to evaporate irreversibly, the walls of the container progressively exert a pressure that remains equal to the pressure of the vapor. For the TEA diagram this example gives Gw = 0, ∆G = qirr = ᎑T∆Stot . Addition of an nonvolatile solute to the liquid phase reduces the value of Xw (the mole fraction of the solvent after dilution by a solute) from unity to a fractional value. Dilution of the liquid by solute shifts the gas–liquid couple to the left by an amount calculated from the expression R lnXw. The equation for calculating ∆Stot would then be determined by the combined effects of dilution of solvent and the external pressure in the equation ∆S tot = R lnXw – R ln(Pext /P*) Raoult’s law, P = X w P*, is derived from this equation when ∆S tot = 0 and Pext equals the equilibrium vapor pressure, P, of a liquid that contains an nonvolatile solute (11). Figure 2 shows visually the difference between thermodynamic values for standard states and equilibrium states. These tend to be confused in students’ minds because of the important equation,

equilibrium position on the scale. This is evident from eq 3, where the enthalpy factor ∆ rH°/T decreases in value as the temperature rises. The standard and the equilibrium positions share the zero position on the scale when the temperature is at the standard boiling point, which is where the equilibrium vapor pressure P* equals standard pressure P°. Thermodynamic Scales Thermodynamic scales are a further development from “double scales” (6 ). They can be used in any area of chemistry, where thermodynamic constants are available, to predict the direction and extent of reactions that involve the exchange of a particular particle, such as the transfer of atoms, protons, ions, electrons or ligands in either the solution phase or gas phase (6 ). The scale in Figure 3 is based on an oxygen atom transfer scale. It has features, described below, that are typical of all thermodynamic scales. Each position on this O atom transfer scale consists of a pair of substances called a “couple” that differ only in the number of O atoms, eg. NO2ⴢ|NOⴢ. The substance NO2ⴢ on the left of the couple, for example, has one more oxygen atom than the substance on the right. Unlike the total entropy scale where each couple represents a full reaction, this couple represents a dissociation half-equation, NO2ⴢ → NOⴢ + 1 /2 O2

(5)

Numbers on the scale are based on “᎑ 1 /2 log fO2 ” values, where

fO2 represents the fugacity or activity of O 2. The couples on the scale in Figure 3 are set at their standard positions at values that are characterized by either their pK or ∆ rG° values. Each oxygen atom transfer reaction involves two couples. The reaction represented by eq 6, for example, involves the couples O3|O2 and NO2ⴢ|NOⴢ in Figure 3. O3 + NOⴢ → O2 + NO2ⴢ

(6)

When the reactants are the “outsiders” on the scale, such as O3 |…| NOⴢ, the reaction is “major” because most of the reactants will have decomposed into the products O 2 and NO2ⴢ before the reaction approaches equilibrium. Major reactions are exergonic and equilibrium seeking, and have negative ∆ rG values. During the course of a major reaction the positions of the two couples move closer together on the scale, and share eventually the same position on the scale at equilibrium. The two arrows in the Figure 2 for eq 6 also indicate a major reaction because they point inwards towards each other. The tails of the arrows are placed above the reactants and the arrowheads above the products. As the reaction proceeds each couple moves along the scale in the direction indicated by its arrow.

∆ r G° = ᎑RT lnK The equilibrium constant K in this equation represents the adjustments that have to be made to the activities (molar volumes, concentrations and partial pressures) of the reacting species in order to bring them from their standard states to an equilibrium. This becomes evident from inspection of the equilibrium position on the entropy scale in Figure 2, where K = P*/P° and ∆S tot = ∆ r S°tot – R lnK = 0 When the temperature increases, the liquid approaches its boiling point, and the standard position moves closer to the 218

6 9

9 •

{O+NO } NO

–40.6 pK

•

O3 O2

–28.6 pK

6 1/

2 O2

*

0.0

•

•

RO2 RO NO2 NO•

?

•

6.3 pK

1/

• 1/ 2

2 RO2

R•

?

Figure 3. Thermodynamic scale diagram for oxygen atom tranfer reactions in a polluted urban atmosphere. The scale is based on values of “– 1/2 log PO 2“, where PO 2 is the equilibrium partial pressure of O2 .

Journal of Chemical Education • Vol. 76 No. 2 February 1999 • JChemEd.chem.wisc.edu

In the Classroom

A “minor” reaction is a major reaction in reverse. There are two types of a minor reaction. The first type requires the two couples to move further apart on the scale as the reaction proceeds. This type is an endergonic, equilibrium opposing reaction that can only happen with the help of an external energy source. The second type of minor reaction occurs when reactants are present only at the beginning of a reaction and products are absent. This is an equilibrium-seeking type of reaction in which only a minor amount of the reactants decomposes into products before reaching equilibrium. Minor reactions of this second type can be quite significant. Acid dissociation reactions, for example, are minor reactions that determine the pH of solutions of weak acids.

Example 4 The production of ozone in photochemical smog. The oxygen atom transfer scale in Figure 3 is used here as an aid to the understanding of the particularly complex chemical system that produces photochemical smog (15, 16 ). It is based on a recent and very interesting article by Baird on atmospheric reactions. The scale helps to identify those reactions that are major or minor, and to distinguish minor reactions from slow major reactions. Each of the couples on the scale is on its standard position where all substances are in their standard states. The representative hydrocarbon couples, 1/2RO2ⴢ|1/2 Rⴢ and RO2ⴢ|ROⴢ, were put in approximate positions marked (?) that are consistent with their reactions in photochemical smog. The symbol Rⴢ represents small radicals such as methyl, ethyl, and propyl, that occur from the photolysis of aldehydes—which are in turn derived from a free radical reaction sequence beginning with alkene molecules (15). The arrows in Figure 3 represent two of the major reactions, eq 6 and eq 9, in photochemical smog. Additional arrows can be inserted to indicate the other major reactions such as eqs 10 and 11. The reaction in eq 7 is minor, being the reverse of the major reaction in eq 6. O2 + NO2ⴢ → NOⴢ + O 3

(7)

Equation 7 therefore does not make a significant contribution to the chemistry of the troposphere in the absence of sunlight. This reaction can become a major reaction, however, with an input of energy from the absorption of photons into NO2ⴢ. Photons from sunlight in the 300 to 400 nm range cause the photolysis of NO2ⴢ, as in eq 8. NO2ⴢ + hν → {NO ⴢ + O}

(8)

ⴢ

Photodissociated NO2, shown as {NO +O}, turns eq 7 into the major reaction eq 9. O2 + {NO +O} → NO + O3 ⴢ

ⴢ

(9) NOⴢ

The ozone-producing reaction eq 9 produces and O3 in equal amounts. But these two products of eq 9 destroy each other in the major reaction (eq 6) that brings the system back to its original state. For these reasons, in clean air and strong sunlight the concentrations of NOⴢ and O3 reach a peak at roughly equal values (16 ), but then decline in concentration in the absence of sunlight. Motor vehicle exhaust emissions produce hydrocarbon radicals such as Rⴢ, HCOⴢ, and CH3Oⴢ, which lead to other major reactions as in eqs 10 and 11.

1

/2Rⴢ + 1 /2O2 → 1/2 RO2ⴢ

RO2ⴢ

+ NO → ⴢ

NO2ⴢ

+ RO

(10) ⴢ

(11)

Hydrocarbon radicals in reactions such as eq 11 consume the NOⴢ and make it unavailable for the destruction of O3 in eq 6 (15). The ozone produced by eq 9 therefore accumulates during hot sunny afternoons in hydrocarbon-polluted atmospheres. The pK or ∆rG° values used on thermodynamic scales are usually available directly from tables of thermodynamic data. The calculation of these values from primary data is treated as a separate exercise. The standard position of the NO2ⴢ|NOⴢ couple in Figure 3, like all the other couples, is based on the value of standard Gibbs function, ∆rG°, of eq 5 where (᎑ 1 /2 log fO2 ) = ∆rG°/2.303RT = pK. Thermodynamic data for calculating these ∆G° values were obtained from the Internet (14). Some additional reactions mentioned by Baird (15) that consume the available NO ⴢ can be worked out by adding to the scale the following three couples: HO2ⴢ|HOⴢ at 2.0, 1/ (HO ⴢ + H CO)| 1/ CH Oⴢ at 10.7, and 1 / (HO ⴢ + CO)| 2 2 3 2 2 2 2 1/ HCOⴢ at 12.4. These are the standard positions for these 2 couples that were worked out from the published thermodynamic data (14 ). The standard position of the couple NOⴢ + O2|{O + NOⴢ} is at 40.6 and is off the scale to the right. Typical tropospheric concentration values are in the following ranges (17): [O3] = 2–20 pphm; [NO2ⴢ] = 5–20 pphm; [NOⴢ] = 2–15 pphm; [O] = 5 × 10᎑7 pphm. These values would put the couples {NOⴢ + O}|NOⴢ at ~ ᎑26; O3|O2 at ~ ᎑21; 1/2 O2|* at 0.34; NO2ⴢ|NOⴢ at ~6; and NOⴢ + O 2| {O + NOⴢ} at ~26. The actual positions of these couples on the scale under tropospheric conditions therefore remain in the same order from left to right as their standard positions in Figure 3. Reactions predicted as major or minor from Figure 3 remain the same as for these tropospheric conditions. The five major oxygen atom transfer reactions considered above are the most important for producing relatively high ozone concentrations under smog conditions of sunlight and hydrocarbon pollution (15, 16 ). However, there are 15 major reactions and an equal number of minor ones that can be worked out from the six couples in Figure 3. The addition of a seventh couple would add another six major reactions. Polluted air is a complex chemical system with up to 200 reactions (15). Many of these reactions are oxygen atom transfer reactions, and the thermodynamic scale is a compact way of organizing a systematic presentation of them. Summary The thermodynamic energy account (TEA) diagram is easy to recall and reconstruct on paper. It allows students to begin with what they already know from their observations and measurements in the laboratory. It provides them with a logical process for working out all the equations that determine the relationships between the values of these laboratory measurements and the values for the more complex thermodynamic functions, ∆U, ∆H, ∆A, ∆G, ∆S, and ∆Stot . The total entropy and thermodynamic scale diagrams give a visual appreciation of the differences between types of experimental conditions as shown by the standard, equilibrium, and expanded/dilute positions of couples. Not shown in this article is the practical standard position for pKa and pKw values on

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acid/base thermodynamic scales that show the effect of ionic strength on the accuracy of calculations of pH (17). With thermodynamic scales you can see how ionic interactions affect thermodynamic values, and how shifts in concentration or pressure of reactants affects the direction and extent of chemical changes. Acknowledgments I am indebted to the authors whose works are cited in this article, for maintaining my interest and in deepening my knowledge of thermodynamics over the years. I am grateful to the referees for comments on the original paper.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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