Research: Science and Education
Spontaneity and the Equilibrium Constant: Advantages of the Planck Function Robert M. Rosenberg* and Irving M. Klotz Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3113; *
[email protected] In a recent note (1), Frances H. Chapple delineated the seeming incompatibility in the different dependencies on temperature of the degree of spontaneity of a chemical reaction, as measured by ∆G °,
∂∆G ° ∂T
Since the pressure is constant, we can add VdP to the numerator on the right side of eq 3 without changing the value, so that
dU + PdV + VdP T
dS ≥
dU + d PV T
= {∆S ° P
and as measured by the equilibrium constant K for the same reaction, ∂ ln K = ∆H ° ∂T P R T 2 A related inconsistency is that the temperature of a maximum or minimum of ln K does not correspond to an extremum in ∆G ° for the same reaction. These paradoxes vanish when we use the Planck function Y = S – H/T as our criterion of spontaneity instead of the Gibbs function G. Furthermore, Y provides a more direct connection to the combined first and second laws of thermodynamics, since It seems appropriate, therefore, to elaborate on these advantages of the Planck function for characterizing the temperature dependence of the spontaneity of chemical reactions.
from which it follows that
dS – dH ≥ 0 T
dS – dH + H dT ≥ 0 T T2
(1)
That this function is indeed a criterion of spontaneity and equilibrium at constant temperature and pressure can be shown starting with the combined first and second laws expressed as
dS ≥
dU – DW T
If the change of state is carried out at a constant pressure equal to the pressure of the surroundings, DW = { Pexternal dV = { PdV ; and dS ≥
(7)
The greater-than sign refers to an irreversible process, whereas the equal sign refers to a reversible process. When the system is at constant pressure and temperature, an irreversible process can occur only if the change of state is spontaneous, and a reversible process can occur only as a result of small fluctuations about a state of equilibrium. Thus Y is a criterion of spontaneity and equilibrium at constant temperature and pressure.
Temperature and Pressure Dependence of the Planck Function From eq 1,
Y =S – H T it follows that
DQ T (2)
dS ≥
(6)
If we convert eq 1 to the differential of Y, we obtain the left side of eq 6. Consequently, dY ≥ 0
Y =S – H ={G T T
(5)
Since the temperature is constant, we may add (H/T 2)dT to the left side of eq 5 without changing the value of the expression. Then
Planck Function
A Criterion of Spontaneity and Equilibrium A century ago Planck suggested an alternative to the Gibbs function as a criterion of spontaneity and equilibrium at constant temperature and pressure (2, 3).1 The Planck function, symbolized by Y, is defined as
(4)
dS ≥ dH T
∆Y = ∆Ssystem + ∆Ssurroundings
1448
dS ≥
dU + PdV T
(3)
dY = dS + H dT – dH 2 T T dU + PdV + VdP dY = dS + H dT – 2 T T DQ + DW + PdV + VdP dY = dS + H dT – 2 T T
Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
(8)
Research: Science and Education
For a reversible process, DQ = TdS, DW = {PdV, and
dY = H 2dT – V dP T T
(9)
Therefore,
∂Y ∂T
= H 2 and P T
∂Y ∂P
={V T T
(10)
Dependence of KP on Temperature From eq A3.10, Appendix 3, we can see that
∂∆Y m° ∂T
=R P
∂ ln K P ∂T
(11) P
Figure 1. Temperature dependence of ln Ka of acetic acid in water. A maximum near 298 K is clearly evident.
We can deduce from eq 10 that
∂∆Y m° ∂T
= P
∆H m° T
(12)
2
Clearly, then,
∂ ln K P ∂T
= P
∆H m° RT
2
∂∆Y m° =1 R ∂T
(13) P
Thus the variation of ln KP with temperature is directly proportional to the variation of the measure of spontaneity with temperature (4 ). Nonparallel Temperature Dependencies of ln KP and DG ° The surprising feature that two correlated but different thermodynamic parameters for a chemical reaction have different temperature dependencies is strikingly illustrated by a concrete example, the ionization constant of acetic acid: HC2H3O2(aq) + H2O = H3O+(aq) + C2H3O2{(aq) Classical very precise measurements of the ionization constant of acetic acid as a function of temperature were made many decades ago (5). The dependence of ln Ka on temperature is illustrated in Figure 1. As the temperature is increased from 0 °C, the degree of ionization increases gradually, reaches a maximum just below 25 °C, and then decreases with increasing temperature. In comparisons of reactions at the same temperature, the reaction with the more positive value of ∆G ° is less “spontaneous”, less capable of progressing from reactants to products. On the other hand, as is evident from Figure 2, in comparisons of a given constant-temperature reaction at different temperatures, the temperature at which ∆Gm° is more positive may be the one at which the transformation is the more spontaneous. For example, for acetic acid, the value of ∆G ° at 0 °C is 25.005 kJ mol{1, whereas at 25 °C it is 27.152 kJ mol{1. Nevertheless, the degree of ionization is greater at 25 °C, with Ka = 1.754 × 10{5, compared to K a = 1.659 × 10{5 at 0 °C. In contrast to the Gibbs function, the Planck function varies with temperature in the same direction as the extent of reaction at equilibrium, as measured by K or ln K. Thus, for the ionization of acetic acid, a graph of ∆Ym° against T
Figure 2. Temperature dependence of ∆Gm° for the ionization of acetic acid in water. The values rise monotonically and show no extremum in the experimental range of temperatures studied.
Figure 3. Temperature dependence of ∆Ym° for the ionization of acetic acid in water. The dome-shaped curve parallels the curve for ln K a and the maximum occurs at the same temperature, approximately 298 K.
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
1449
Research: Science and Education
(Fig. 3) presents the same form as the plot for ln Ka in Figure 1 and exhibits an extremum at the same temperature, approximately 298 K. We can see the distinction between the temperature dependence of ∆Gm° and the temperature dependence of ln K analytically from the empirical equations that Harned and Owen developed for the ionization of acetic acid (6 ). They found that the following empirical equations for ∆Gm° and log K fit their data: log K = { (A*/T ) + D * – C *T ∆Gm° =2.303R[A* – D *T + C *T 2]
(14)
From these equations we can deduce that the temperature of the extremum in log K is
Appendix 1. The Joule–Thomson Coefficient Since Y is a state function, the two crossed partial derivatives are equal; that is, ∂ ∂Y ∂P ∂T
P T
= 1 ∂H 2 T ∂P
T
= { 1 T ∂V 2 ∂T T
= V – T ∂V ∂T
µJT = { 1 ∂H C P ∂P
Conclusion It is apparent from the preceding discussion that the Planck function provides a direct basis for comparisons of the degree of spontaneity of a given transformation at different temperatures. The trend in ∆Ym° reliably expresses the variation of the equilibrium constant with temperature. ∆Ym° also offers conceptual advantages for didactic purposes. It follows from eq 1 that, for an isothermal transformation,
(17)
Thus the Planck function provides a more direct and explicit expression of the combined first and second laws of thermodynamics. The Planck function merits more attention in classical thermodynamics. Note 1. A reviewer asked whether G. N. Lewis knew of the Planck function. We refer the reader to a footnote on p 158 of Lewis, G. N.; Randall, M. Thermodynamics; McGraw-Hill: New York, 1923. Reference 3 was an early exponent of the importance of the Planck function.
Literature Cited 1. Chapple, F. H. J. Chem. Educ. 1998, 75, 342. 2. Planck, M. Treatise on Thermodynamics, 3rd ed.; Dover: New York, 1926. 3. Strong, L. S.; Halliwell, H. F. J. Chem. Educ. 1970, 47, 347. 4. See also Schellman, J. K. Biophys. J. 1997, 73, 2960. 5. Harned, H. S.; Ehlers, R. W. J. Am. Chem. Soc. 1933, 55, 652. 6. Harned, H. S.; Owen, B. O. The Physical Chemistry of Electrolytic Solutions, 3rd ed.; Reinhold: New York, 1958.
– V (A1.1) P
(A1.2) P
T
µJT = { 1 V – T ∂V CP ∂T
D* (16) 2C * and the numerical values of the two temperatures are respectively 297.5 and 173.9 K.
1450
∂H ∂P
T P
Using eq A1.2, we can readily derive an alternative expression for the Joule–Thomson coefficient:
(15)
G T0 =
∆Y = ∆S – ∆H T ∆Y = ∆S system + ∆S surroundings
∂ ∂Y ∂T ∂P
=
Consequently,
1/ 2
K T 0 = A* C* whereas the temperature of the extremum in ∆Gm° is
T
(A1.3) P
Thus, from the temperature and pressure derivatives of Y, we are able to obtain expressions for (∂ H/∂ P)T and for µ JT .
Appendix 2. Standard States for the Planck Function Like U and H, Y can only be determined relative to some reference state or standard state. The standard states that have been agreed upon are given in Table A2-1. The most stable form is the form with the highest value of Y. Table A 2-1. Standard States for the Planck Function State
Standard State Conditions
Solid
Pure solid in most stable form at 1 bar (100 kPa) and the specified temperature
Liquid
Pure liquid in most stable form at 1 bar (100 kPa) and the specified temperature
Pure gas at unit fugacity;a for ideal gas, f= 1 when pressure is 1 bar (100 kPa); at specified temperature a The term fugacity has not been defined here. Nevertheless, it is used in this table because it is the most general definition for the standard state of a gas. For now, the standard state of a gas may be considered to be that of an ideal gas, 1 bar pressure. Gas
Appendix 3. Relationship between DYm° and the Equilibrium Constant for Gaseous Reactions In the preceding section we established the properties of the Planck function as a criterion for equilibrium and spontaneity of transformations at constant temperature and pressure. Thus, from the sign of ∆Y, it is possible to predict whether a chemical transformation can proceed spontaneously. Further considerations show that it is possible to calculate the equilibrium constant for a reaction from the value of ∆Y under certain standard conditions. After agreement on a standard state for Y, we can derive the expression relating the standard change in Y to the equilibrium constant for a reaction. In a multicomponent system, as in a gaseous reaction, Y is a function of the mole numbers of the reactants and products as well as of T and P. Then the differential of Y can be expressed as
dY = ∂Y ∂T
P,n i
dT + ∂Y ∂T
T,n i
dP + Σ ∂Y ∂n i
Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
T,P, n j
dn i
Research: Science and Education which, at constant temperature and pressure, can be reduced to
dY = Σ ∂Y ∂n i
(A3.1) T,P, n j
Since Y = { G/T,
∂Y ∂n i
T,P, n j
= { 1 ∂G T ∂n i
Then
dY = {Σ
={ T,P, n j
µi T
0 = ∆Y m° – R ln
µi dn i T
(A3.2)
Let us represent a chemical transformation by the equation (in which lower-case letters are the stoichiometric coefficients): aA(g,p A ) + bB(g,p B) = c C(g,p C ) + d D(g,p D )
(A3.3)
Each reactant and product is assumed to be an ideal gas at a given partial pressure ( pA , p B, etc.). For more general circumstances, the same argument would lead to an equilibrium expression in terms of fugacity or activity. The reaction of eq A3.3 is associated with a change in the Planck function, ∆Ym. From eq A3.2 we can express ∆Y for the reaction as
dY = {
µA µ µ µ dnA – B dn B – C dn C – D dn D T T T T
(A3.4)
From the stoichiometry of the reaction, we know that
dn B dn C dn D dnA = c = = dξ { a ={ b d
(A3.5)
where ξ is the degree of advancement of the reaction and has values from 0 to 1. Then we can express eq A3.4 as
dY = a
µA µ µ µ dξ + b B dξ – c C dξ – d D dξ T T T T
(A3.6)
p µ i = µ i° + R T ln i P°
(A3.7)
we can substitute from eq A3.7 for each chemical potential in eq A3.4 and rearrange to obtain c
a
pC P° pB P°
d
b
or
pC P° pA P°
c
a
pD P° pB P°
d
(A3.9)
b equilibrium
∆Ym° = R ln K P
(A3.10)
where K P is the equilibrium constant in terms of partial pressures. Since R is a constant, K P is also a constant, independent of initial composition and total pressure, because the change in the Planck function under standard conditions is constant at a given temperature. Therefore, we can write
KP =
pC P° pA P°
c
a
pD P° pB P°
d
b
= constant
(A3.8)
(A3.11)
equilibrium
The numerical value of KP can be evaluated if ∆Ym° for the reaction is known, or vice versa. The choice depends on which quantity is more accessible for a given reaction. It should be emphasized that although KP refers to equilibrium pressures, it is calculated from data for ∆Ym° that refer to the reaction occurring with reactants and products in their standard states. If we use the symbol Q * for the quotient of pressures (nonequilibrium as well as equilibrium) on the right side of eq A3.8, then we can write ∆Ym = ∆Ym° – R ln Q * ∆Ym = R ln K – R ln Q *
Since the chemical potential of an ideal gas is given by the expression
pC P° ∂Y = ∆Y m = ∆Y m° – R ln ∂ξ T,P pA P°
where ∆Ym is the integral of (∂Y/dξ)T,P from ξ = 0 to 1 at constant composition; that is, when one mole of reaction occurs in an infinitely large system. At equilibrium at constant temperature and pressure, dY and hence (∂Y/dξ)T,P and ∆Ym are equal to zero, so we may write
(A3.12)
∆Ym = { R ln (Q */K ) Thus, if the initial quotient of pressures is greater than KP, ∆Ym is negative and the reaction of eq A3.3 will be spontaneous to the left, whereas if the initial quotient of pressures is less than KP, ∆Ym is positive and the reaction will be spontaneous to the right. The form of the equilibrium constant in eq A3.11 is different from that usually presented in introductory courses. It has the advantages that (i) it is explicit that K is a dimensionless quantity; (ii) it is explicit that the numerical value of K depends on the choice of standard state, but not on the units used to describe the standard state pressure; the equilibrium constant has the same value whether P° is expressed as 750.062 torr, 0.98692 atm, 0.1 MPa, or 1 bar.
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
1451
