Conventions Defining Thermodynamic Richard M. Noyes University of Oregon Eugene
Properties of Aqueous Ions and Other Chemical Species
Chemical thermodynamics is concerned with changes in properties associated with changes in state resulting from chemical reactions. Since only changes in the values of most thermodynamic properties are of interest, the absolute values of these properties are unimportant. In such a situation, it is customary to adopt a convention arbitrarily defining the value of a property for some specified reference system. Once such a convention has been adopted, the value of the property is also defined for any other system provided it is possible to measure the change in the property associated with the change in state between the two systems. A familiar non-chemical example involves the potential energy of an object in the earth's gravitational field. For different types of calculation, this potential energy could conveniently be assigned the value zero a t the center of the earth, at sea level, at some reference point in the laboratory, or a t an infinite distance from the earth. These conventions lead to very different absolute values for the potential energy of the object in any specific situation, but consistent application of any of these conventions will give the same value for the change in potential energy associated with any change in position of the object. A chemical reaction can conveniently be generalized by the equation
where the R's are reactants and the P's are products. The n's are rational numbers whose relative values are determined by the stoichiometry of the reaction. The equation must be balanced with respect to elements and charges by methods familiar to all chemists, and sufficient properties like temperature and pressure must he specified so that the state of each of the species in equation (1) is uniquely defined. If proper chemical conventions are adopted, the values of properties like enthalpy, entropy, and free energy can be tabulated for species in specified states so that quantities like AH, AS, and AF can be computed for any reaction like equation (1) provided tabulated data are available for all of the species in the equation. Such tabulations based on the same set of conventions have been prepared for many chemical species a t 25'C and 1 atmosphere by the Bureau of Standards1 and by Latimer.2 It is very useful to tabulate values based on a convention provided everybody understands the convention. Most persons using these tables for thermodynamic calculations do indeed know the conventions 2
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Journal of Chemical Education
and use them consistently for the problems of interest. Probably comparatively few of them realize t,hat the conventions with regard to aqueous ions are not internally consistent for certain types of reactions and that an uncritical use of tabulated data can lead to violations of some of the basic equations of thermodynamics. The purpose of this paper is to summarize the conventions presently used to tabulate thermodynamic properties of chemical. species, to show the reaction types for which these tabulations are and are not applicable, and to suggest alternative conventions that could extend the applicability of the tabu1ct'lons. Conventions for Observable Chemical Systems Tabulations of thermodynamic properties1. V i s t the quantities AH,', AFfo, and So. Conventions defining these quantities are: Convention 1. The quantity AH,' at any temperature is the enthalpy change per mole of species formed in its standard state from the elements in the standard states in which they exist at the same temperature. Of course A H f o will be zero for an element in this standard state. Convention 2. The quantity AF; is the free energy change for formation from the elements under the same conditions as in convention 1. Convention 3. The quantity Soat any temperature is the molar entropy in the standard state provided the entropy of a perfect crystal of that species is defined to approach zero as the absolute temperature approaches zero. The value of S o may be based on therma! measurements that include heat capacity data extended to very low temperatures; it may also be based on computations that employ the principles of statistical thermodynamics. The two methods of assigning entropies give the same results to within the accuracy with which they can be compared. The above conventions refer to standard states. For a liquid or crystalline solid (designated by 1 and c, respectively), the usual standard state is pure substance at 25OC and 1 atmosphere pressure. For a gas (designated by g), the standard state a t 2.5% has the same enthalpy, entropy, and free energy as one atmosphere of a hypothetical gas identical with
' "Selected Values of Chemical Thermodynamic Properties," Circular of the National Bureau of Standards No. 500, U.S. Government Printing Office, Washington, D. C., 1952. * LATIMER,W. M., "The Oxidation States of the Elements and Their Potentials in Aqueous Solutions," 2nd ed., Prentice-Hall, Inc., N e w York, 1952.
the real gas a t very low pressures but obeying the ideal equation of state during compression. Since most gases at one atmosphere fugacity have pressures very close to one atmosphere, and since thermodynamic properties of condensed phases are not very sensitive to small changes in pressure, a reaction in which reactants and products are all in their standard states can be regarded as a reaction a t constant pressure for the purpose of relating thermal effects to changes in thermodynamic properties like A H . The designation uq indicates a dilute aqueous solution. In the standard state a t 25'C and 1 a h mosphere pressure, the solute bas the same enthalpy as in an infinitely dilute solution. The free energy and entropy of the solute in this standard state are those computcd for a hypothetical ideal solution at unit activity if the activity of solute in a very dilute solution is equal to its molality. This standard state is not necessarily identical with any state that is attainable in reality. Table 1 contains values a t 25'C of thermodynamic propcrties based on the above conventions for hydrogen, oxygen, sodium, and chlorine and for many of the compounds that can be formed from these elements. The values arc taken from the Bureau of Standards listing.' Table 1 also contains entries for C,' which will be discussed later. These conventions are a little more complicated than the problem of potential energy in a gravitational field discussed previously. If the elements in their standard states a t 25" were assigned zero enthalpy and free energy, the values of H o and F o for other species would be the same as the values of AHfo and A F f o in Table 1, and people using such tables often treat the cntries mentally as though they were H o and Fo. However, the cntries in the table would then violate the equation F"
=
H" - TS"
(2)
Because of this difficulty, it is better to call the entries "enthalpies and free energies of formation" rather than "enthalpies" and "free energies." The entropies in Table 1 are not hased on zero values Table 1 .
substance
H
F 01
Thermodynamic Properties of some Observable Chemical Species at 25'C" AHID
State koal/mole 0 D
B
52.089 0.000 59.159 0.000
Arlo kosL/mole
48.575 0.000 54.994 0.000
So CPO oal/mole dea eal/rnole dep 27.3927 31.211 38.4689 49.003
4.9680 -6.892 5.2304 7.017
'Them entries are based on conventions 1-3 snd are applicable to reaotions of type A.
for the elements a t 25" because the third law of thermodynamics permits more fundamental expressions that relate entropies of different chemical species to the same scale without the need of a separate convention for each element. Even though the convention for entropy is different from those for enthalpy and free energy, the data for every species in Table 1 will he found to satisfy the equation AFlS
=
AHJ" - T A S I o
(3)
where ASf' has the same type of meaning as A H J o and AFf0. If a table of such entries is available. standard changes in thermodynamic properties can be computed for a reaction like equation (1) by means of the relations
If the change of state indicated by the equation is carried out, the changes in thermodynamic propcrties will agree with those computcd from tabulated data by means of these equations. The use of the equations can he illustratcd by the reaction 2Na(c)
+ 2B0(1)
-
Hdg)
+ 2NaOH(ay)
(7)
Since sodium hydroxide is a strong electrolyte, the last entry could be replaced by 2Na+(aq) 20H-(aq). Application of data from Table 1 gives
+
AH' AF"
= =
-224.472 -200.368
+ 0 + 136.6348 - 0-87.837 + 0 + 113.3804 - 0 =
kcal/mole
(8)
=
-86.988 kcal/mole (9)
AS"
=
23.8
+ 31.211 - 33.432 - 24.4
=
-2.8 cal/mole deg
The self-consistency of the conventions is illustrated by the fact that these data satisfy the equation AS'
=
AH" - AF' T
(11)
Any other balanced equation hased on the entries in Table 1 could he used to show that these entries are self-consistent. The above conventions are not sufficient to permit application to all conceivable equations for reactions. Thus conventions 1 and 2 effectively assign values to thermodynamic properties for each element and assume that the nuclear properties of that element are unaffected by the reaction. Similarly, since the elements in their standard states are electrically neutral, these conventions are only sufficient for application to equations such that each side is electricdly neutral. Even though a solution of a strong electrolyte like NaOH(aq) may be dissociated into ions,. the total concentrations of positive and negative charge in the solution must be equal. Any equation for which these conventions are sufficient must satisfy this requirement of electrical neutrality even if the equation as written includes ions that do not enter into the reaction. Volume 40, Number I, January 1963
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It is convenient to summarize these requirements and to clmsify types of reactions to which different conventions may be safely applied. Reactions of type A involve no change in nuclear structure or in isotopic composition; they involve no charged species other than aqueous ions; and both sides of any equation are electrically neutral. Equation (7) is an example of a reaction of type A. Although very many chemical reactions belong to type A, other situations are also of interest. Additional or alternative conventions must he adopted if they are to be treated. Some of the possibilities are considered in the following sections.
chemical species containing that element, and the simplest procedure is to neglect the entropy of isotopic mixing. However, it is well to remember that sueh neglect is involved in convention 3 and in Table 1. Conventions for Reactions of Gaseous lons
Reactions of type D involve gaseous ions and may also involve gaseous electrons. They involve no nuclear or isotope fractionation reactions, and any aqueous ions must satisfy the electroneutrality requirement for reactions of type A. Examples are
+ -
N4c)
Nuclear Reactions
Nab)
Reactions of type B involve some change in nuclear structure. An example is :H
+ 7,Li
-
2 $He
Isotope Fractionation Reactions
Reactions of type C involve change in isotopic composition of the atoms in one of the chemical forms of some element. Convention 3 defines the entropy of a species so as to neglect the mixing of isotopic species of the same element. Such a convention is satisfactory for application to any reaction snch that all species containing that element have the same isotopic distribut,ion. If that distribution changes during reaction, additional entropy effects must be considered. If X,is the mole fraction of an isotopic species, and if only one atom of an element is present per mole of a chemical substance, a treatment that accounted for isotope effects would increase the entropy of that substance in Table 1 by an amount =
- R Z X I In X A k
H+(g)
+ e-(g)
Na+(v)
+ H(r)
(14)
(15)
I n order to treat such reactions, an additional convention is necessary.
(12)
Such a reaction cannot be treated by means of conventions 1-3. Physicists talk about the energy changes associated with such reactions, but the necessary conventions must be based on the energies of fundamental particles or of some specified nucleus sueh as oxygen-16. Often these reactions are carried out under conditions such that it is impossible to talk about temperature or entropy, hut reactions of thermal neutrons in a uranium pile and of ions in a hot plasma would lend themselves to treatments like those of chemical thermodynamics. Any snch treatment would require different conventions and would be outside the scope of this paper.
.,,,S .i
Na+(r)
(13)
Of course, Z X x = 1.
Convention 4. The energy of a gaseous electron is zero except for its kinetic energy. According to this convention, the molar enthalpy of an electron gas is ( 5 / 2 ) R T ,and the difference in energy of a positive ion from that of an atom is equal to the minimum energy necessary to cause ioni7,a t'lon. Standard tabulations of thermodynamic quantities', assign enthalpies of formation to gaseous ions in this way but do not state convention 4 specifically. They do not make any attempt to assign entropies or free energies to these ions. However, the entropy of a monatomic species depends only on the mass and the distribution of electronic energy levels. Since the mass of an ion is insignificantly different from that of the parent atom, entropies of gaseous ions differ from those of atoms only by easily calculable terms involving electronic partition functions. Free energies of formation of gaseous ions can then be defined to give consistency with equation (11) for any reaction. The thermodynamic properties of gaseous electrons can also be calculated by standard equations of statistical thermodynamics with application of convention 4. Table 2 lists thermodynamic properties of electrons and of some gaseous ions derivable from the elements of Table 2.
6ubatanoe
Thermodynamic Properties of Some Gaseous lons at 25°C. Stat,?
1.481
P-
0":"-
AHP keal/mole
0 9 "
367.088
-
374.609 >v
,"
AX," kcal /mole
-
0.008 363.985 370.879
So
cn"
eal/mole dee cal/mole deg 4.994 26.0153 37.0109
4.9680
4.9680
4.9680
k
As an example, natural chlorine contains 75.4% of W1 'nd 24.6% of S'Cl. Hence the entropies of Cl(g) and of HCI(g) in Table 1 would be increased by 1.109 cal/mole deg if this fact were considered. If any chemical species contains more than one atom per mole, the entropy corrections depend upon whether the atoms are indistinguishable in position as in Nz or are distinguishable as in NNO. Although isotopic fractionation chiefly affects entropy values, enthalpy values of some compounds can also be influenced by isotopic fractionation of the lighter elements. During most chemical reactions, the isotopic composition of an element is virtually the same in all 4
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Journal of Chemical Education
These entries are based on oonventions 1-1 and are applicable t o rcaotiona of tym D.
T ~ h l e1. The entries are based on all four of the conventions that have been introduced. They satisfy equation (31, and application to a balanced equation will be self-consistent with regard to equation (11). Existing Conventions for Reactions of Aqueous lons
Reactions of type E usually involve no charged species other than aqueous ions. If they do involve gaseous charged species, such gaseous species must satisfy the
type of electroneutrality restriction imposed on reactions of type A. No neutrality restrictions are placed on aqueous ions provided the equations are balanced with respect to charge. These equations involve no nuclear or isotope fractionation reactions. Of course reactions of type A are included in type E. Additional examples not belonging to type A are
Although any real solution of ions must contain equal concentrations of positive and negative charge, there is write equation (16) to include specifically the negative ions that are also present throughout the entire reaction. Just as we cannot prepare a solution of sodium ion and measure its properties independent of any negative ions, we cannot get a direct measurement of the contribution of any single type of ion to the thermodynamic properties of an electrolyte solution. However, we can adopt a "bookkeeping" convention that arbitrarily assigns the properties of some one ion. Such a couvention can be combined with experimental data to define the properties of all other ions. The following convention is generally applied in order to assign thermodynamic propert,iesto individual ions. Aqueous hydrogen ion in its Convention 50. standard state at any temperature has zero values for the quantities AH,", A F T , and So. Table 3 presents the values of thermodynamic properties for several aqueous ions derived from the elements of Table 1 and calculated with the use of conventions 1, 2,3,and 5a. A comparison of Tables 1 and 3 will show that the properties for strong electrolytes in Table 1 are t.he sums of the properties for the individual ions in Table 3. As was indicated, convention 5 a is a bookkeeping convention determining how the total value of some property will be divided among constituent ions. Thermodynamic properties listed in Table 3 are never considered to have the same sort of experimental significance as do those in Table 1. It is less commonly realized that simultaneous adoption of arbitrary conventions for the three quantities AH,', AF,", and So leads to tabulated quantities that violate equation (3). I n other words, convention 5a overdefines the thermodynamic properties of ions so that some applications not only lack absolute experiTable 3. Thermodynamic Properties of Some Aqueous Ions at 25'C Based on Presently Accepted Conventionsa
Substance
State
H OH-
ep
Ns+
ao
+
c1c101c104-
ay ay aq ap
S" AH," AF," cal/mole kcnl/mole kcal/rnole deg 0.000 -54.957 -40.023 -23.50 -31.41 -57.279
0.000 37.595 -31.350 - 0.62 - 2.57 -62.589
Cn' eal/mole deg
0.000 0.000 2.510 -32.0 13.17 -30.0 39.0 -18. 43.5 14.4 5.4
-
These entries are based on conventions 1-3, 5a, and 88 a d
mental significance but also lack self-consistency in terms of the basic identities of thermodynamics. Some caution is necessary in the application of the entries from Table 3. Ionic Reactions Permitting Valid Application
Since the equction for a reaction of type E involves no deviation from electroneutrality except for aqueous ions, the algebraic sum of the charges on such ions is the same on both sides of the equation. For such a recetion, the conventions are self-consistent, and any application to valid data will satisfy equation (11). I t will also be susceptible to experimental verification. Such consistency can be demonstrated with any reaction of type E involving the data of Table 3. A more extreme example is provided by the reaction
Tabulated data1 for the species in this equation give A S o = 452.5 cal/mole deg and ( A H o - AFo)/T = 450.6 cal/mole deg. The agreement is well within the indicated uncertainties of the data. Attempted Application to Half-Reactions
When a reaction of type E involves oxidation and reduction of species, it is often convenient for purposes of discussion to break it down into a sum of "halfreactions" involving free electrons. Reactions of type F involve both electrons and aqueous ions but no other charged species. They involve no nuclear or isotope fractionation reactions. Examples of such reactions are Na(c)
-+
Nat(aq)
CIOa-(ap) f 6Ht(ap)
6ec-
+ eCI-(up)
(19)
+ 3Hs0(1)
(20)
Since half-reactions are used for purposes of balancing equations or for separating reactions at different electrodes in a cell, the state of the electron is usually not specified. Any convention for discussing thermodynamic changes in such reactions will want to introduce the least complication from the presence of electrons. The most obvious one is: Convention 6a. The electron in a half-reaction has zero values for AH;, AF,", and So. An electron in such a state shall henceforth be designated er(0). It is possible in principle to define the state edO) . . so that convention 4 is simu1taneousl.v satisfied. This convention is not self-consistent with conventions 1-5a. The inconsistency can be illustrated by applying the data of Table 3 to reaction (19). The results are AH" = -57.279 - 0 = -57.279 kcal/mole AF" = -62.589 - 0
=
-62.589 keal/rnole
AS" = 14.4 - 12.2 = 2.2 cal/male deg (AH" - AFm)/T= 17.80 cal/mole deg
(21) (22) (23) (24)
Rossini3is one of several authors who have pointed out this lack of consistency. It will he involved in any application of these conventions to a half reaction. The discrepancy could be removed by an alternative convention. a R o s s r ~ F. , D., "Chemical Thermodynamics," John Wiley and Sons, Inc., New York, 1950, pp. 374.f.
Volume 40, Number I, Jonuary 1963
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Convention 6b. The electron in a half-reaction has zero values for AH," and for AF,", but it has a value of So equal to half the molar entropy of hydrogen gas, which at 25°C is 15.606 cal/mole deg. The reason for such an assignment will become apparent from the discussion below. Although such an entropy assignment seems arbitrary, i t is little more so than assigning the value zero: If no other reaction types caused trouble, this modification of convention 6a would be sufficient to attain self-consistency in the tabulation of thermodynamic data for chemical reactions. However, the conventions lead to difficulty with another type of reaction also. Attempted Application lo Ion Transport Reactions
Reactions of type G involve net transport of charge between gaseous and aqueous phases. They involve no nuclear or isotope fractionation reactions. An example is Na+(g)
-
Na+(aq)
AH* = -57.279 - 146.015 = -203.294 keal/mole (26) AF' = -62.589 - 139.12 = -201.71 keal/mole
(27)
AS" = 14.4 - 35.34 = -20.9 eal/mole deg
(28)
=
-5.30 cal/mole deg
(29)
The discrepancy is again 15.606 cal/mole deg within experimental error. This lack of consistency in the conventions was discovered somewhat accidentally during work on the absolute changes in thermodynamic properties associated with processes like reaction (25). Some discussions with Professor Henry S. Frank of the University of Pittsburgh helped greatly to define the nature of the inconsistency. The reasons become apparent from an examination of the conventions presented above. The data presented in Table 2 and based on convention 4 permit calculation of quantities like AH,' for the species Na+(g) e-(g) as in equation (14). Such quantities satisfy equation (3), and the entries for e-(g) satisfy equation (2). The data presented in Table 3 and based on conventions 5a and 6a permit calculation of similar quantities for the species Na+(aq) e-(0) as in equation (19). Equations (21) to (24) show that use of these conventions will violate equation (3). If convention 66 is used instead of convention Ba, the thermodynamic changes for formation of Na+(aq) e- satisfy equation (3), hut the satisfaction is achieved by assigning properties to the electron that violate equation (2). Since electrons do not appear on either side of equation (25), the entries from Tables 2 and 3 cannot he combined consistently unless the conventions about electrons in both tables satisfy equation (2) or differ from it in consistent fashion. Strong statistical mechanical arguments can be raised against a modification of
+
+
+
6
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Journol of Chemical Education
The Electrochemical Conventions
If a modified convention is to he adopted for aqueous ions, it should recognize the electrochemical conventions of long standing in discussing the potentials of single electrodes. The potential of a chemical cell is the potential of the ri-ht electrode relative to the left, and the reaction of the cell is the chemical process that takes place when current passes so that reduction takes place ~t the right electrode. For this situation
(25)
Of course it is impossible to study the thermodynamics of this process by direct observation. However, the thermodynamic changes associated with reactions like this are of great interest to the understanding of the properties of aqueous ions. The lack of self-consistency can he illustrated easily for this reaction by applying data from Tables 2 and 3.
(AH" - aF0)/T
convention 4, which provides the basis for the electron entries in Table 2. Although convention 6a satisfies equation (21, its use causes all entries for aqueous ions to violate equation (3); use of convention 6b permits these entries to satisfy equation (3) but does so by violating equation (2) for electrons in half-reactions. More fundamental changes in convention will be necessary if tabulated data are to be applied to transport processes like equation (25).
AF' = - 8 s ~ '
(30)
where E o is the potential of the cell when all reactants and products are in their standard states, 5 is the Faraday constant, and AFo is the standard free energy change (in joules per mole) for the cell reaction associated with passage of z moles of electrons. Since the absolute potential of a single electrode cannot be measured as accurately as the difference in potential between two electrodes, it has been customary to adopt a convention about potentials: Conoention 7. The potential of a standard hydrogen electrode is zero at all temperatures of interest. A standard hydrogen electrode consists of platinum black bathed with hydrogen gas a t one atmosphere fugacity and immersed in a solution containing hydrogen ion in its standard state. The oxidation reaction a t such an electrode can be written '/s
Hdr)
-
H+(aq)
+ e-(PC
(31)
where e-(Pt) indicates an electron in the highest energy level normallv occu~iedin ~latinummetal. Comhination of equation (30) with convention 7 is equivalent to assigning zero values to AH0, AFo, and ASo for reaction (31). Suggested Revision of Conventions
The requirements are now apparent for conventions that will permit the assignment of thermodynamic properties to individual aqueous ionic species such that equation (3) is satisfied, that will assign properties to the electron that satisfy equation (2), and that will satisfy the electrochemical convention 7. Conventions that satisfy these requirements most simply and that retain the most of former conventions seem to be: Convention 5b. Aqueous hydrogen ion in its standard state at any temperature has zero values for the quantities AH," and AF," ; the value of So at this temperature is equal to half the molar entropy of hydrogen gas at one atmosphere fugacity and the same temperature. Conoention 6c. An electron in the state e-(Pt) has zero values for the quantities AH;, AFT, and So.
If these conventions are adopted, entropies tabulated a t 25" for all positive ions of charge +z will he 15.6062 cal/mole deg more positive than formerly, and entropies of all negative ions of charge -2 will be 15.6062 caljmole deg more negative than formerly. Entropies calculated according to these conventions are included in Table 4. Table 4. Entropy ond Heat Capacity Values of Some Based on Proposed Conventions* Aqueous Ions at 25'C
Substance
H+
OHC1-
c10,c10,Nn+
State
so
eal/mole deg
C, "
cal/mole deg
aq aq aq aq
an
These entries replace those for the same properties in Table 3. Values for AH," and AFj" from that table should continue to be used. These entries are besed on conventions 1-3, 5b, and 8 b and are applicable with thermodynamic consistency to reactions of types A, E, F, and G , provided that for reactions of type F the electron is in state e - ( 0 ) if convention 6a is used and is in state e-(Pt) if convention 6e is used. Applications to reactions of types F and G do not give "correct" values for changes in thermodynamic properties.
confirm the predictions of calculations from these tabulations. Reactions of types F and G are not subject to direct observation, although we can discuss them as hypothetical processes. They combine various chemical species in ways so that the arbitrary parts of the different conventions do not cancel out. Even though the conventions have been selected so that tabulated data can be applied with thermodynamic consistency to reactions of these types, calculated changes in thermodynamic properties will differ from the "correct" changes that we are convinced would be observed if the reactions could be carried out. The problem can be illustrated with equation (25) as an exmple of a reaction of type G. This reaction can be written as the sum of three reactions
a
Since entropies of positive and negative ions are changed in opposite directions by this change of conventions, reactions of types A and E would give the same changes in thermodynamic properties whether entropies from Table 3 or from Table 4 were used to make the calculations. Application to reactions of type F would be self-consistent, provided the state of the electron were assumed to be e-(Pt), and ASo in equation (23) would become 17.8 cal/mole deg as necessary for consistency. Application to reactions of type G would also be self-consistent, and ASo in equation (28) would become -5.3 cal/mole deg. For application to reactions of type F with electron state e-(O), convention 6a would have to be substituted for convention 6c. These proposed changes in convention are not new. Klotz4 has made an almost identical proposal. Van Rysselberghe5 has also proposed such a change in the entropy assigned to aqueous hydrogen ion.
Naf(g)
Na(c)
X m m , I. M., ''Chemical Thermodynamics," Prentice-Hall, New York, 1950, p. 352. P., Anales Real Sac. Espan. Fis. y 6 V m RYSSELBERGHE, Ruim., 5 6 8 , 515 (1960).
+ HYaq) '/nHdg)
N&(c]
Nac(aq)
HYad
'/>Hdg)
e-(0)
(32) (33) (34)
Equation (32) can he treated with data from Table 2 and regarded as a reaction of type D if conventions 4 and 6g (instead of 6c) are applied to electronic states. Equation (33) is half of equation (16) and is a typical reaction of type E giving the same value for calculations with data from Table 3 (convention 5a) or from Table 4 (convention 5b). The conventions presented above assign zero values to changes in thermodynamic properties associated with equation (34). If the changes in these properties are not truly zero, the changes computed for equation (25) will not be "correct" even though they are self-consistent with regard to thermodynamic relations. If it is desired to know "correct" changes in thermodynamic properties associated with reactions of type G , we must abandon convention 56 and determine the true changes associated with reaction (34). Because of the importance of this specific reaction, the subscript H will be used to designate properties associated with it. An examination of experimental data on ionic hydrationBhas recently led to the following estimates: AXx'
Estimation of "Correct" Thermodynamic Changes
Conventions 1 4 , 5b and 6c form a set that also satisfies convention 7. If data for individual species are tabulated on the basis of these conventions, thermodynamic consistency (as defined by satisfaction of equation (11)) can be obtained by application of these data to reactions of types A, D, E, F, and G. Reactions of types A, D, and E can he observed directly, and calculations from tabulated data are subject to experimental test. In addition to conventions 1-3, reactions of type D use only convention 4, and reactions of type E use only convention 56. By the time balanced equations of these types have been written, the arbitrary assignments of thermodynamic properties have all cancelled out just as they do in discussions of potential energy changes in a gravitational field. Our confidence in thermodynamics assures us that any experimental test of such a reaction would
+ - + + e-(0)
=
99.17 kcal/rnole
(35)
AFx0 = 104.81 kcal/mole
(36)
ASx" = -18.9 cal/male deg
(37)
If these values are used with conventions 1 4 and 6a, thermodynamic properties of aqueous ions can be tabulated in such a way that they can be applied to give "correct" values for changes in thermodynamic properties during reaction of types F and G (provided reactions of type F assume the state e-(0) for the electron). These tabulated values will be as valid as the others for application to reactions of types A, D, and E. Table 5 presents such a tabulation. Note that conventions 5 and 7 have been discarded in the preparation of this table. Although it might be desirable in principle to discard conventions 5 and 7 and to list properties of aqueous ions according to a convention that would give "correct" values for reactions of all these types, the practical difficulty arises that the numerical values in equations (35-37) are not known with the same precision as are 8
NOYES, R. M., J . Am. Chem. Soe., 84, 513 (1962). Volume 40, Number 1 , January 1963
/
7
thermodynamic properties of many electrolytes. Quantities like AF,' are very subject t o future modification and refinement, and any such modification would change all the values in Table 5 . Until the changes in thermodynamic properties associated with reaction (34) are known with an accuracy a t least comparable to the best data for formation of aqueous electrolytes from the elements, it is better to retain some form of convention 5 (and preferably convention 5b because of the Table 5.
"Correct" Thermodynamic Properties of Some Aqueous Ions at 25°C"
Substance
State
Let AHP(T) for a substance be the enthalpy change associated with formation of the substance in its standard state at temperature T from elements in their standard states also at temperature T. Also, let ACroI be the change in standard state heat capacity associated with forming the substance from the elements. Then
S" eal/mole deg
" These entries are based on conventions 1-3 and 6a combined with equations (35) to (37). They specifically avoid use of conventions 5a, 5b, and 7. They are applicable to reactions of types A, E, F, and G and will give 'correct" values for $1 including types F and G.
consistency problems described above). Such retention requires the recognition that the conventions do not give "correct" values when applied to reactions of types F and G. Application to "Absolute" Electrode Potentials
Convention 7 effectively assigns changes in thermodynamic properties to reaction (31). This reaction differs from reartion (34) only in the state of the electron. As has been indicated above, the property rhanges assoriated with reaction (34) have now been estimated with moderate confidence. If we could also evaluate the changes in thermodynamic properties assoriated with the potentially observable reaction, e-(Pt)
-
(38)
c(g)
the results could be combined to estimate property changes for reaction (31). Such data could then be used to compute "absolute" potentials for individual electrodes. The problem of "absolute" electrode potentials is similar to the one of "correct" ~ r o ~ e r tchanees v for reactions of types F and G. If the standard free energy change of reaction (31) were known with sufficient confidence so that equation (30) could be used to assign the standard hydrogen electrode a potential known to better than v, convention 7 could be abandoned in favor of a tabulation of "absolute" oxidation potentials. Until such time (which looks very far in the future), it is better to continue the present convention 7 while recognizing that application of tabulated data to half-reactions (type F) will not give "correct" values for changes in thermodynamic properties. A
.
"
-
Effects of Changing Temperature
The above discussion has been confined to applications a t 25'C. The heat capacity a t constant pressure is defined by the equation
8 / Journol of Chemical Education
Since heat capacity is a property having an absolute m3gnitude, no additional convention is needed in order to use these equations for preparing a table like Table 1 for the properties AHlo, AFIo, and So a t any temperature other than 25". It is only necessary that values of Cvoand ACSo, be known as functions of temperature in the interval of interest. Reactions of type A can be treated appropriately. Since the heat capacity of e-(g) is (5/2)R, the data can also he used without additional conventions t o prepare a table like Table 2 for any other temperature and to treat reactions of type D. If ionic reactions of type E are to he treated, an additional "bookkeeping" convention is necessary. The one presently adopted1is:
Convention 8a. The partial molal heat capacity of aqueous hydrogen ion in its standard state is zero. Data based on this convention are presented in Table 3. Convention 8a represents an unfortunate choice because it cannot be combined with some of the other conventions without violating basic equations of thermodynamics. The arguments are similar to those developed for the inconsistencies in applying convention 5a: Convention 8a can be applied to reactions of type E without inconsistency. The data tabulated in Table 3 can be used satisfactorily to predict temperature effects on changes in thermodynamic properties associated with reactions of this type. If the electron in state e-(Pt) is assigned a heat capacity of zero, conventions 7 and 8a are inconsistent. Convention 7 requires AH" for reaction (31) to remain zero as the temperature is changed, and convention 8a does not permit this. Attempts to discuss electrode potentials and properties of ions a t temperatures other than 25' will result in various contradictions and inconsistencies. The discrepancy could be removed without violating convention 8a if the heat capacity of e-(Pt) were assigned a value equal to half that of hydrogen gas, which a t 25°C would be 3.446 cal/mole deg. Such a solution is not very satisfactory. If the electron in a half-reaction (type F) is assigned a heat capacity of zero, combination of convention 8a with AHlo values based either on convention 5a or 5b will lead to predictions for such reactions that violate the basic equation
be found. Until ACD0, has been estimated, persons using data like those in Table 4 must realize that they give thermodynamically consistent results a t different temperatures hut that they do not give "correct" values for reactions of types F and G.
Thermodynamic consistency can also be saved for reactions of type F by assigning the electron a molar heat capacity equal to half that of hydrogen gas. The inconsistency is at least as bad for reactions of type G, and there does not seem to be any way to correct the situation while retaining convention 8a. Consider the reaction H+(gl
-
H+(ad
Effects of Changing Pressure
Pressure is the property usually used in addition to temperature to deiine the state of a chemical system. Of course
(45)
When data from Tables 1 and 2 are employed in equation (40),
(T)? =
4.9680 - 3.446 = 1.522 cal/mule deg for H+(g)
If molar volumes and coefficients of thermal expansion are known for various chemical species, thermodynamic properties can be calculated for temperatures and fugacities other than one atmosphere. KO conventions are necessary for application to reactions of types A and D or for determining values for the species listed in Tables 1 and 2. Application to aqueous ions does require a "bookkeeping" convention in order to assign the separate contributions of positive and negative ions to the volumes and coefficients of expansion of electrolytes. If the partial molal volume of aqueous hydrogen ion were set equal to half the molar volume of hydrogen gas, all negative ions would have absurdly large negative partial molal volumes. The most satisfactory convention seems to be :
(46)
Either convention 5a or 5b requires that
(%)*
=
0 for H+(aq)
If these results are applied to equation (4), then for reaction (45) we obtain 3AHo
=
(T)p
-1.522 callmole deg
(481
However, the data from Tables 2 and 3 for the same reaction give AC,'
=
-1.9680 callmole deg
(49)
Equation (44) is clearly violated. The discrepancies can be avoided by rejecting convention 8a and adopting the following one:
Conoention 9. The partial molal volume of aqueous hydrogen ion and its temperature and pressure derivatives are all zero. This convention permits the potential of a hydrogen electrode to change with pressure, but no difficulty is introduced by this fact. Convention 7 requires the potential of that electrode to be assigned the value zero when the fugacity of the hydrogen gas is one atmosphere, but change of the potential with pressure can be handled by standard equations.
Conoention 8b. The partial molal heat capacity of aqueous hydrogen ion in its standard state at any temperature is half the molar heat capacity of hydrogen gas in its standard state at the same temperature. The heat capacity of e-(Pt) is zero, and of wurse that of e-(0) is also. Ionic heat capacities based on convention 8b are presented in Table 4. This convention permits convention 7 to be retained and also permits consistent application of tabulated data to reactions of types F and G. Of course convention 86 will not give %orrect2' values of AC,' for reactions of types F and G. Although an estimate of AC,', for rea.ction (34) could he made by the same type of procedure used to estimate the changes in other properties for this reaction, no such calculations have been made. These calculations will be undertaken in the near future if the necessary heat capacity data can Table 6.
Convention comhinations~
- -
Summary
Many of the basic conclusions of this paper are presented in Table 6. They can he summarized as follows: Chemists are not usually concerned with reactions that change atomic nuclei (type B) or that discriminate significantly among isotopic species of the same element (type C). The conventions usually adopted specifically exclude the possibility of such reactions.
Applicability of Conventions to Reactions Involving Aqueous Ions
Tahb - -. nf ...
typical data
Reaction tvoe E
Reaction tvoe FD
Conventions 5a snd 8a
3
Valid
Inconsistent for 6%and 6"; consistent Inconsistent
Conventions 5b and 8b
4
Valid
Thermodynamic data far reaction (341 Thermodynamic data for reactions (31) and (38)
5
Valid
5
Valid
Consistent for 6a. or 6c if electron Consistent for 6e; not applic- Consistent able for 6a and 6h state specified; inconsistent for 6b Correct Correct for 6a; not applicable for Not applicable 6b or 6c Correct for 4; other conventions not Superceded by absolute Correct applicable electrode potentials
".
..
.".
Electrochemical convention 7
Reaction tvoe ., G
.
Inconsistent
fnr R h "-
a
All combinations asaume conventions 1-3. The last one also asaumes convention 4, as do all applications to reactions af type G.
* Application to reactions of type F requires specifying the state of the electron.
Volume 40, Number 1 , January 1963
/
9
If such reactions are excluded, it is permissible to make as many separate assignments of enthalpy and free energy as there are elements. These assignments are made by defining enthalpies of formation (convention 1) and free energies of formation (convention 2) that are zero by definition for each element in its standard state a t any temperature. Entropies (convention 3) are based on assigning the same (zero) value to all perfect crystals a t sufficiently low absolute temperature. These conventions are sufficient for treatment of reactions that include no charged species other than aqueous ions provided the equations are electrically neutral (type A). Extension to reactions of gaseous ions (type D) assigns no energy other than kinetic energy to a gaseous electron (convention 4). Extension to non-neutral equations for reactions of aqueous ions (type E) customarily assigns zero values to AH,", AF,", and S o for aqueous hydrogen ion in its standard state (convention 5a). This convention is unfortunate because tabulated data for aqueous hydrogen ion and for all other aqueous ions fail to satisfy the equation AF,"
=
AH,'
- TASl0
(3)
The inconsistency does not affect application of tabulated data to reactions in which the only charged species are aqueous ions (type E), but violations of basic thermodynamic equations result from attempted application to half-reactions (type F) and to transport of ions between gaseous and aqueous phases (type G). If the partial molal entropy of aqueous hydrogen ion is set equal to half the molar entropy of hydrogen gas (convention 5b), the entropies of all positive ions a t 25°C become more positive by 15.6062 cal/mole deg, and entropies of all negative ions become less positive by the same amount. This change of convention does not affect changes in thermodynamic properties computed for reactions of type E but permits thermodynamic consistency for properties computed for ion transport reactions (type G). It also permits consistency for half-reactions (type F) if the electron is assigned a state e-(0) having zero values of AH,', MIF,", and So and also consistent with convention 4 (convention 6a). Even though these conventions permit self-consistent application, they do not give 'Lcorrect" values for changes in thermodynamic properties for reactions of types F and G. "Correct" values could be obtained if it were possible to determine changes in thermodynamic properties for the reaction Reasonable estimates of these quantities are possible, but the probable accuracy does not justify abandonment of convention 5b (although the argument of thermodynamic consistency does justify substituting convention 56 for the present convention 5a).
10
/
Journal of Chemical Education
If zero values of AH,', AFlo, and So are assigned to an electron in the highest occupied energy level in platinum metal (convention 6e), the simultaneous application of convention 56 (but not of convention 5a) is consistent with the electrochemical convention that the potential of a standard hydrogen electrode is zero at all temperatures of interest (convention 7). The evaluation of the "absolute" potential of this electrode would require knowledge of changes in thermodynamic properties associated with reaction (34) and also with the reaction e-(Pt)
-
e-(g)
(381
The partial molal heat capacity of aqueous hydrogen ion is usually assigned the value zero (convention 8a). Such an assignment is satisfactory for reactions of type E, but it is not consistent with the electrochemical convention (convention 7). Application to reactions of types F and G leads to violation of the equation
These inconsistencies can be eliminated by assigning aqueous hydrogen ion a partial molal heat capacity equal to half the molar heat capacity of hydrogen gas in its standard state (convention 8b). This convention permits thermodynamically consistent but not "correct" calculations of temperature dependence of reactions of types F and G. Apparently no difficulties arise in the treatment of pressure changes if zero values are assigned to the partial molal volume of aqueous hydrogen ion and to the temperature and pressure derivatives of this quantity (convention 9). If this convention is adopted, the assigned potential of a hydrogen electrode becomes a calculable function of the fugacity of the hydrogen gas. Because of the inconsistencies with the electrochemical convention (convention 7) and during applications to reactions of types F and G, it is strongly recommended that the presently accepted conventions 5a and 8a be replaced by conventions 5b and 8b. Acknowledgments
This paper is an outgrowth of work on ions and ion pairs in solution that has been supported by the National Science Foundation under grants G7330 and G19646. Professor Henry S. Frank of the University of Pittsburgh provided helpful discussion a t the time I discovered the inconsistencies in application of existing conventions to ion transport reactions (type G). Professor Pierre Van Rysselberghe of Stanford University called a reference to my attention, and Professor T. F. Young of the University of Chicago encouraged me when I suggested the desirability of a discussion of these conventions. Professor John A. Schellman of the University of Oregon and Professor Irving M. Klotz of Northwestern University read the manuscript and made helpful suggestions.
