Practical Calculation of the Equilibrium Constant and the Enthalpy of Reaction at Different Temperatures

Practical Calculation of the Equilibrium Constant and the Enthalpy of Reaction at Different Temperatures Knud Andersen University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen 0, Denmark

During a course on the theory of chemical reactions i t is important to demonstrate how to make use of a table of thermodynamic data for calculating equilibrium constants or enthalpies of reaction. If the given temperature equals 298.15 K, an abundance of thermodynamic data is available for this purpose, for example, from the NBS Table of Chemical Thermodynamic Properties ( I ) . If the given temperature T differs from 298.15 K, however, the situation gets somewhat more complicated. This article is devoted to a discussion of procedures for obtaining values of Afl(T) andKO(T) over a range of temperatures higher than 298.15 K.

Use and Advantages of Empirical Expressions Sources of the necessary thermodynamic data are, for ex-

ample, JANAF Thermoehemical Tables (21, CODATAKey Values for Thermodynamics (31, or The Chemical Thermo- dynamics of Organic Compounds (4). These tables all present values of

a t different temperatures. Temperatures chosen for the tables are usually multiples of 50 K or 100 K. However, in the excerpts presented in the book by Wyatt (5) or in many ele- mentary textbooks (e.g., Alberty and Silbey (6)) the temperature intervals can be as large as 500 Kor 1000 K

In most cases, the temperature T a t which we want to know the value of an enthalpy of reaction or an equilibrium constant differs from any of the fixed temperatures chosen for the table. The tabulated thermodynamic functions can be combined in various ways to produce approxi- mations to A&' or KO a t the temperature T. Our problem has been todrwsc a procedure for-mak~ngth~ b ~ s t possible use of the mformatwn presented in the table.

The most obvious solution to this problem appears to be thc aoolicationofsome kind oflincar interwlation directlv to thethermodynamic function we want tb evaluate. procedures of this type will be discussed in the last section of this article. An alternative approach would be to develop empirical expressions for the standard thermodynamic functions.

For example, eq 13 is an empirical expressions for the standard enthalpy of reaction and contains four coefficients. Eauation 18 has five coefficients and expresses the equilibrium constant us a functmn of temperature. If the coeffic~cnts are calculated from valucs sclccted from a thermodynamic table, they are considered to be an extract of the information in the table. Values of equilibrium constants calculated by the method described here turn out to be at least 5 times more accurate than those found by linear interpolation.

From the ooint of view of the instructor re~lacine inter- - polation with the use of empirical expressions provides another benefit. For the chemical reaction under considera- tion, the function in F ( T ) is actually known (e.g., in the form of eq 18). I t could be differentiated according to van't Hoff's equation to obtain an expression for the standard

enthalpy of reaction. Aformula for 4C;(T) could be derived by Kirchhoff's equation, and @(T) could also be found easily

When we turn our attention to a pure substance, eq 8 gives us the standard molar enthalpy as a function of temperature. An analogous expression for the standard molar entropy could be derived similarly. All other standard functions for the substance could then be found from these two equations by simple mathematical operations. Within this framework the close relationship between the various thermodynamic functions can be demonstrated, and the relationship can be explained from the fact that the Gibbs function is a thermodynamic potential.

The third advantage associated with the application of em~irical ex~ressions has to do with data reduction. Table 1 cimtains f;vc cocfficicnts for each suhstancc. From this set ofcoefficients we can calculate the standard thermodvnamic functions at any temperature between 298 K and 1200 K. Very little of the precision present in the original tables is lost in this process. Inother words, we have man- aged to reduce 13 lines of 6 values in the thermodynamic table to five coefficients. Due to the practical limitation of any course material (textbooks, notes, etc.) given to the student, a data reduction of this magnitude would allow the instructor to enlarge the total number of compounds to be used in examples and problems.

Input Data Initially we introduce the notation e for the reference

temperature. For the thermodynamic tables mentioned in this article the value 0 = 298.15 K has been chosen. For all substances J participating in our chemical reaction we need to know the following values.

@f@), the standard enthalpy of formation at 298.15 K Sj(tl), the standard molar entropy at 298.15 K q, J, the standard molar hest capacity at constant pressure at a number of different temperatures (See the section on evaluation of the coefficients.)

Our final expression for the standard enthalpy of reaction (eq 13) will he obtained by integrating 4C; as a function of temperature. Similarly, our equilibrium constant expression (eq 18) will be found by integrating A,P(T). During integration possible errors in the expression for the integrand tend to cancel, and the resulting expression gen- erally leads to more accurate values than results obtained from differentiation or internolation. This is our main ar- hmment for choosing the preceding thwe items as input data ruther than other coml~inations of tabulated values.

Heat Capacity Expression

As a basis for the equations that we will derive, we must decide on an empirical expression for C",,J as a function of the temperature T. Examples from the chemical literature are given below.

474 Journal of Chemical Education

Table 1. Coefficients for 10 Elements and 32 Compounds Cap,J(T) =d(o,~,+ do, ~ ) ~ ' + d ( z , $ ~ (5 )

YO. 4

4.34022

4.61378

3.761 36

13.54404

14.63781

13.67315

14.45640

7.28342

13.61659

14.19083

10.60956

13.36489

13.68010

12.95266

4.581 37

8.04966

10.06653

16.41054

4.14002

4.651 76

10.14605

16.25918

3.731 92

4.87596

4.56544

3.69070

5.90381

6.90248

9.13579

15.52405

4.64737

4.74789

7.82923

4.45601

8.01464

17.82462

4.891 91

7.871 09

7.84423

11.18239

4.70522

The number of adjustable parameters in each of the five expressions has been chosen as 3. Equation 3 was introduced by Kelley (7) and is suggested by many textbooks (e.g., Lewis and Randall (8)). Alberty (6) follows the recommendation of Clarke and Glew (9) and uses eq 2, which is a polynomial in T. The possibility of expressing @,, J as a polynomial in In T has been investigated by the author (10). In eq 4 the symbol 0 stands for the reference temperature.

The application of eq 5 to represent heat capacity data was proposed by Bernstein (11). The idea of using a polynomial in the reciprocal temperature originates from the fact that T appears in the denominator in the e-*JkT terms in the partition function from which the thermodynamic functions can be derived. Apart from that it is difficult to fmd a theoretical justification in favor of any particular empirical expression.

All the following equations will be derived from eq 5. I t is convenient to divide eq 5 by the gas constant R.

New coefficient c o , ~ - i s dimensionless; and the units of c,l, J, and C(Z, Jl are K (kelvin) and IC, respectively

Evaluation of the Coefficients In this section our method of calculating

the coefficients cii, J,'S (i = 0, 1,2) in eq 6 for a given substance J will he explained. As the first step we choose the following 17 temperatures in K: 100, 200, 250, 298.15, 350, 400,450,500,600,700,800,900,1000,1100, 1200, 1600, and 2000. In JANAF Thermo- chemical Tables (2) we find the correspond- ing values of @,, J. Using the value of the gas constant given in the JANAF tables, we pre- pare a table of 17 values of the dimensionless heat capacity function C ; , J (~ IR .

A Spline Function in a Limited Temperature Range

Our next s t e ~ is to set UD a Drocedure for calculating a cubic splinefunction CJ(T ' I ~ a s s i n a through the 17 known ~oin ts . t T ' . @,.,IR). when CJ(T') is available, vdues of q, J(T)IR a t any temperature in the interval from 100 K to 2000 K can be esti-

-271 .3l8 mated very precisely. Only for tempera-

4367 tures just above 100 K or just below 2000 4.74470 -278.886 8479 K do these values become less accurate.

Therefore, the limits of the integrations described in the following paragraph will - - - -

be chosen so that the necessary evalu- (2) ationsofCdT') can be confined to the interval 298.15 K

< T < 1 0 0 0 ~ . G, J(T) = f a 4 + f i l , ~ ) ~ + f i z , JIT' (3) The heat capacity spline function CJ(T1) may be considered to be a many-parameter representation of the true

Cap, AT) = e(o,a+e(l, JI In heat capacity hctionC;, J(T)/R. Because we have decided

(4) to restrict the number of parameters to 3, we are going to

Volume 71 Number 6 June 1994 475

approximate CJ(T1) with the polynomial in T' on the right-hand side of eq 6.

GT1) - c(o, J) + CII, 4F ' + ccz,.nF2

A Minimization to Determine the Coefficients

This approximation will be based on the continuous leastsquares method. In other words, the values of

90, JI c ~ ~ . a c~z.a are determined as those that minimize the following integral.

The lower limit of the integral over T1 is chosen to be C' where T is 1000 R This choice is a compromise: A larger integration interval would imply that the results could be extended to higher temperatures, but with reduced overall accuracy.

The requirement that IJ be minimized with respect to variation of the cli,~;s leads to the following set of equations.

8.'

= J IP~c,(T~)~F~ -1 i = 0 , 1 , 2

(7)

All calculations described in this section were carried out with an accuracy of 16 decimal digits using Mathcad, ver- sion 3.0. Included in Mathcad are standard numerical methods for setting up the spline function, evaluating the three integrals on the right-hand side of eq 7, and solving the linear equations in the unknowns q0, a, eel, J) and c12, ,,,, Exolessions for the Other Thermodvnamic Functions

Integratingeq 6, we obtain the following expressions for the standard molar enthalpy as a function of temperature.

We will now consider an arbitrary chemical reaction r among substances J for which the input data are known.

The standard enthalpy of reaction for r as a function of temperature can be written in the following way

4HO(Q=x VJ@J(~)

J

= C VJ@J@) + Z v J ( ~ n - w e ) ) J J

Substitutingeq 8 into the last term of this equation, we get

where

For reaction r we introduce the following constants.

Thus, we arrive at our final expression for AJP.

To derive an expression for the logarithm to the standard equilibrium constant we use van't Hoff's equation.

Inserting eq 13, we get

where

Then we replace b, in the last line of eq 15 with the right- hand side of eq 12, collect the constant terms, and define the constant.

The following expression is then a t our disposal for calculation values of standard equilibrium constants.


Determination of Coefficients and Use of a Programmable Calculator

Equations 13 and 18 will be ready for use as soon as the constants cc0,,,, c(~,., ccz, ,,, b,, and a, have been calculated from eqs 11,12, and 17. For these calculations we need the values of SJ, h ~ , ~(0.4, c(l,n, and q z , ~ , for all substances participating in the reaction r as defined by eq 9. Both s~ and h~ are calculated directly from the input data, as described by eqs 16 and 10, respectively. The set of coefficients cci,~,'s (i = 0, 1,2) is obtained by solving eq 7. The resulting constants for 42 selected substances have been listed in Table 1. All the input data originate from JANAF Thermochemi- cal Tables (2).

When the necessary constants are known, this method for calculating reaction enthalpies and equilibrium constants is very well suited for a programmable pocket calculator. Subroutines for input, for calculation of A J P and KD, and for finding the root T, of the equation

P ( T J = a given eonstant

require 150 bytes of the memory of a HP 32.911 calculator. An additional 80 bytes are used by the variables. On re- quest, I will mail a list of program codes.

Example Reaction Consider the water gas reaction.

Problem

Evaluate AJP andKD at T = 873.15 K, and find the temperature T, at which P(TJ = 0.8000.

Solution

Use Table 1 to set up the following list of coefficients.

Using eqs 13 and 18, we can now calculate the values

A,P(873.15 K) = 135.72 kJ/mol

Alternatively, if only data valid at 298.15 K had been available, the use of the approximate, but simple formula,

would lead to F(873.15 K) = 0.1356, which is 42% too small.

Based on eq 18 a procedure for calculating the function PIT) has been set up. Using the SOLVE facility of the calculator to search for a root ofKD(T) = 0.8000, we amve at the result, T = T, = 934.37 K.

Comments on the Results Comparing Calculated Values with Data from Thermochemical Tables

For a discussion of the accuracv of reaction enthalnies and equilibrium constants calcul&l from eqs 13 an2 18 we have chosen the formation reaction for a m m o ~ a . as a representative example.

0 = NH&) - '/zNz(g) 3/2H2(g)

Our standard of comparison is the AfHO and A f l columns for NHdg) in JANAF Thermochemical Tables (2). Aforma- tion constant obtained from a tabulated A@ value using

q = ,-W~ITJIRT

may carry a round off error up to 0.02%; in most cases this error will be smaller than the round off error associated with application of the log @ column.

Errors Due to the Limited Number ofparameters

The error bars in Figure 1 show the differences between the values of 4 H O calculated from eq 13 and the table val-

and similarly,

Temperature in kelvin

Figure 1. Error in J mol-' on 421 for NH3(g) as a function of T(in K). An error Is defined as the value calculated from eq 13 minus the value

400 Sm Bm t~ 1x0

Temperature in kelvin

Figure 2. Relative error in percentage on KP for NH,(g) as a function of T (in K). An error is defined as the value calculated from eq 18 minus the value of In as calculated directly from 4Go in JANAF Thermochemical Tables (2).


ues. Figure 2 is the analogous error plot when eq 18 is applied to calculate In K;!. We emphasize that these devia- tions occur as a direct consequence of our choice of working with only three adjustable parameters in the heat capacity expression. (See eq 6 and the section on evaluation of the coefficients.) The error on In @ is approximately equal to the relative error on the formation constant K;! itself

Curves drawn through the endpoints of the error bars oscillate around zero. The numerical value of the errors on w below 1050 K are smaller than 0.03 kJ/mol, and the relative errors onK;! below 1200 K are smaller than 0.08%.

Maximum Errrors within Tempemture Intervals

We now turn our attention to the formation reactions for all 32 compounds included in Table 1. The column for eq 5 in Table 2 shows that for 24 of these compounds the relative errors on@ in the interval are smaller than 0.1%, and errors larger than 0.25% do not occur. We may conclude that eq 18 produces remarkably accurate results.

We have now verified one of the postulates in the introduction: Within certain temperature limits (see below), the five numbers per substance in Table 1 contain the same information as the A&?, A$?, and log K;! columns in JANAF Thermochemical Tables (2).

We must not forget that all calculated values originate from parameters fitted to heat capacity data in the interval between 298 K and 1000 K. (See the section about evaluation of the coefficients.) In the case of NH3(g) the maximum error on A,P is equal to 0.03 kJ/mol within the temperature interval from 298 K to 1050 K , and the maximum relative error on K;! is 0.08% within the temperature interval from 298 K to 1200 K.

At temperatures above' these upper interval limits we fmd constantly increasing errors. At T = 1500 K, for example, the relative error on @ for NH8(g) reaches the value 0.64%. This result and similar results for other compounds imply that eq 18 also may be used for extrapolations, whenever high accuracy is of minor importance.

It should be added that cco,n, c(I,J,, and c(z,~, inour initial numerical experiments were calculated using the better known method of discrete least squares on the same input data. These first results produced error curves for in K;! with oscillations not centered around zero and with larger maximum errors. We therefore consider the continuous leasbsquares method to be essential whenever the resulting coefficients should be used for calculation of an inte- grated function.

Comparison with Other Results From each of the other heat capacity expressions (i.e.,

eqs 14) we have derived a set of formulas for AJP and In KD analogous to eqs 13 and 18. Compared to eqs 13 and 18 these other formulas are less well suited for calculating

Table 2. Relative Errors on Formation Constants Kf'

Maximum errorsfound at 298 K to 1200 K for 32 Compounds

thermodynamic functions over an extended temperature range. We shall briefly describe the work that led us to this conclusion.

Among various criteria upon which the comparison could be based we have given first priority to optimizing the accuracy of the formation constant values. Tables of the temperature-dependent errors for the five different expressions for in @ were then calculated for the 32 compounds in Table 1. For each expression and compound the largest error in the temperature interval from 298 K to 1200 K was oinoointed. This lareest error could either be located . . - at a maximum or a minimum pomt on the error curve or at the end~olnt of the interval As shown in Table 2 we have grouped the largest errors according to thrir numerical values. For each heat capacitv 6:x~n~s;iion Table 2 disolavs . . . . . the total number ot'compounds with maximum errors within the intervals wrltten in the first column.

We have also compared the extrapolation properties of the five different in IQ expressions. We rather arbitrarily selected 1500 K as a test temperature outside the interval with error curves oscillations. A set of tables of 5 x 32 errors on in @(1500.K) has been worked out, and the distri- bution among the compounds according to the numerical value of the relative errors is presented in Table 3.

The numbers in Tables 2 and 3 support each other and allow us to set up a classification of the five heat capacity expressions according to the ability of their integrals to re- produce the correct data on KP. For the 32 compounds as a whole we arrive at the following rule: The higher the heat capacity equation number (eqs 1-51, the smaller the ex- pected errors. The mutual difference between eqs 4 and 5 is insigmkant.

Tables 2 and 3 show that a polynomial in T1 (eq 5) for the heat ca~acitv leads to more accurate exoressions for . "

other thermod.ynamic functions than does a polynomial in T Itself ~ e o 21. A oossible reason can be based on the obser- . . . vation that in analogy to eq 8 is obtained by integrating A,€;(T), and lnK;!(T) is derived from W(n by yet another integration (eq 14). The error curve for A,€; is therefore an important clue to the explanation we seek.

The error curves for A&P (Fig. 1) as well as in K;! (Fig. 2) could in f ad have been found by integrating the Wp error curve. Plots of the Wp error curve associated with eq 2 versus T and the @; error curve associated with eq 5 versus !P1 both display positive and negative errors evenly distributed between 298 K and 1000 K. Apositive G error at some temperature will be accumulated during the integration process so that it may be only partially can- celled by the negative GP errors following at higher temperatures.

The integration described by eq 8 uses T as a variable. In Figure 3 the G errors associated with eq 5 are plotted versus T instead of II". The oscillations then become com-

Table 3. Relative Errors on Formation Constants Kf'

Evaluated at 1500 K lor 32 Compounds

Number of compounds for Calculation of n? based on eq which the largest relative error

is less than 1 2 3 4 5

Number of compounds for which calculation of @based on eq the relative error is less than

1 2 3 4 5


Figure 3. Ermr in J lC1mol" on 4CDpfor NHdg) as a function of T(in K). An ermr is defined as the value of Z V J G J with the CDp, j s calculated from eq 6 and Table 1 minus the value of Z V J ~ ~ , J with the G, J'S found in JANAF Thermochemical Tables (2).

pressed towards the low temperature end of the integration interval. The positive errors are rapidly succeeded by negative errors. Consequently, the unavoidable error accu- mulation during the integration decreases. In other words, when we use empirical expressions based on polynomials in T' (eq 5) rather than T (eq 21, the errors on A&? and in IQ will be more equally balanced around zero and therefore numerically smaller. ' Interpolation in the JANAF Tables

In the first part of this section we describe the conven- tional procedure for obtaining equilibrium constants by interpolation in JANAF Thermochemical Tables (2). Then we propose an alternative interpolation method for finding Ko(T). Finally, we discuss the precision of the results.

The standard molar Giauque function (also called the Gibbs energy function) is defined by

As mentioned in the introduction @(T) is tabulated in JANAF Thermochemical Tables (2) at temperature intervals equal to 50 K or 100 K. The equilibrium constant for the reaction r (as defined by eq 9) can be found from the following expression.

where

Approximate values of A,W(T) at an arbitrary temperature T may be found by linear interpolation. This method is recommended by a number oftextbooks, for example, by McGlashan (12). Assumingaj to be - linear in T

linear in In T linear in T'

Table 4. Relative Errors on Formation Constants K? Linear Interpolation in JANAF Thermochemical Tables

Maximum errors at interval 298 K to 1200 K for 32 Compounds

Number of compounds for which largest relative error is less than

we have carried out a number of test interpolations using values from the JANAF Tables. The maximum errors on A@' turned out to be almost independent of our choice of variable. From eq 19 we find

4 G 0 0 = A R ( O ) - T4@(n (20)

We now know that A,W(T) within an interval can be approximated with a linear function in T-'. According to eq 20, can then be equally well approximated with a linear function in T itself For all compounds J included in JANAF Thermochemical Tables (2) their values of A P J are listed in a separate column. Thus, values for

for a ~ v e n chemical reaction can be found at the lixed tem- oeratures used bs the table. Rv simole linear interwlation in T we then o b k n ArGo(T), and &om ArGa(T) we finally calculate Ko(T). This alternative method is somewhat more convenient than interpolation of 4W and subsequent use of eq 19.

The method described above was applied to find interpolated values of the formation constant IQ for the 32 compounds listed in Table 1 at many different temperatures. At the same temperatures very precise values of

were obtained from a spline function in the variable T passing through the tabulated values of WJ(T). The relative errors on the interpolated lQ values can then be calculated. The results are presented in Table 4. A comparison to the numbers in Table 2 shows that linear interpolation applied to the JANAF tables may result in errors 5 to 10 times as large as the errors associated with the use of a five-parameter expression like eq 18.

Literature Cited 1. Wagman, D. D. The NBS Tables of Chemical Thermodynamic Roperties: J. Phys.

Cham Rpf Doto 1982, 11, oupp1. no. 2. 2. Chase, M. W JI ct d. JANAF Thennochemical Isbles, 3rd ad.; J Phys Chem. R p f

Dola 1986, 14, ouppl. no. 1. 3. C a i o f ~ K e y BluesforTherma(yMmies: c m , J. D : wagman. D. D.:Medvedev,Y A ,

Eds; Hemisphere: New York 1989. 4. St"l1, D. R : westnun, E. F: Sinks, G. C. The Chemiml Tharmodynamicsof01gonic

Compounds, Wiley: New York, 1961. 5. Watt. P A. H.A ThormaivnomiBvooss GOTOLoeK: The Roval Soeietv of Chem- . . .. - .

i&w, Monographs for Teachers No. 35: Iondon, 1982. 6. Alberty, R. A ; Silky, R. J. Physimi Chemistry Wiley: New Ymk 1992. 7. Kelley, K. K U. S Bur Mines Bulls. 1949,476: 1960,584. 8. Lewis, G . N.: Randall, M. Thermodynamics: Revised bv Rtzer, K. 8.; Brewer, L.:

Mc0r.w-HilL New York, 1961: p 66. 9. Clarke, E. C. W.: Clew, D. N. %ions Fomdoy Sa. 1966.62,53%547 10. Anderse", K unpublished results. 11. Bematein, H. J. J. Chem Phys. 1956.24.911-912. 12. McOlashan. M. L. Chemieol Thermodynamics; Academic: Lmdm, 1979: p 164.


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Practical Calculation of the Equilibrium Constant and the Enthalpy of Reaction at Different Temperatures