Calculating Entropy Changes at Different Extents of Reaction Tim Brcsnan Institute of Education, University of London, 20 Bedford Way, London WC1 OHA, England In an article in this Journal' Smith rightly bemoaned the ahsence in chemistry texts of demonstrations of entrnpy maxima at equilibrium. In the United Kingdom, the introduction of a new chemistry course for 16-18-year-old students has orovided the o~oortunirvto rectifv this omission. The ~ e v i s e dNuffield ~ h i m i s t course2 r~ us& a simple statistical approach to entrow as a unifving idea in its treatment of ihermodynamics.-Entropy changes are used to explain the direction of chemical reactions and why they "stop" before completion. I t was for students with this background that I developed the method of calculatine entronv chanees a t different extents of reaction illustracd be& I t is essentially a simplified version of Smith's aooroach usine mathematical models designed to be accessidle to students taking the Nuffield course and will run on the &bit machines (e.~.,the BBC microcomputer) that predominate in schools i'n this country. I t uses a spreadsheet to calculate the entropy change at constant pressure and temperature of both a chemical system and its surroundings a t various extents of reaction and then exports the calculated values to an associated graphdrawing program. The Chemical Svstem
In deciding on the reaction and thermodynamic values to be used, the following points were considered: 1. Although the approach can be applied to any reaction, that of the Haber process was chosen because of its economic importance and the fact that i t is used to illustrate other important thermodynamic points in the Nuffield course. 2. The enthalpy and equilibrium constant values used
were those listed in the Nuffield Book of Data,3 which is used extensively in the Nuffield chemistry course. Currently, these are only accurate to about three significant figures, but using this source has the advantage that the students can look UD the raw data for themselves and can use similar data inin the'book to construct models for other reactions-r deed for this reaction a t other temperatures. 3. Atemperature of 400 K was chosen so that a discernible maximum could be seen on the graph. Since the eauilibrium constant varies greatly with temperature, there areonly very small amounts of either products or reactants present when equilibrium is reached a t slightly higher or lower temperatures. In both these cases the maximum would lie too close to the ends of the graph to be noticed. At 400 K the equilibrium constant is 40.7, and at equilihrium about two-thirds of the nitrogen has reacted. 4. ~ n discussion y of thermodynamic quantities requires the specificntion of a reference zero. This. of course. can be chosen arbitrarily so the most natural zero.was chosen-that of a mixed but unreacted stoichiometric mixture of hydrogen and nitrogen. In this way the entropy change as the reaction proceeds can be easily seen. 5. The entropy changes of the system and the surroundings are calculated separately so that the student can see that it is the sum of these that provides the driving force for the reaction rather than either one on its own. 'Smith, N. 0.J. Chem. Educ. 1985, 62,5043. Stokes. B., Ed. Revised NuffieldChemistry; Longman: London, 1984 el seq. Ellis, H., Ed. Revised Nuffield Advanced Science: Book of Data: Longman: London. 1984.
Table 1. Spreadsheet Showlng Entropy and Free Energy Changes during the Haber Process Reactlona A
48
B
C
D
Journal of Chemical Education
E
F
G
H
I
.I
K
I
Explanation ot the Worksheet The sheet shown in Table 1was produced as follows: 1. Column A contains the extent of reaction, E. Since the initial mixture is of 1mol of nitrogen and 3 of hydrogen: this is also the fraction of a mole of nitrogen that has been converted into ammonia. 2. Column B calculates the entropy change in the surroundings a t each step. The molar value is taken as ASe(surr) = -AHe/T and so a t each step it is obtained by multiplying the molar value bv the extent of reaction. (The formulas used to perform t i i s and subsequent calcul.&ions are illustrated i n ~ a ble 2.) 3. Columns C, D, and E contain the entropies of the nitropen, hydropen, and ammonia without correcting for their nnnst&da& nartial nressures. 4. he nexi three lolumns calculate the partial pressures of each component. If the total pressure is held constant a t 1 atm, then the partial pressure of each component is numerically equal to its mole fraction:
Table 2.
Formulas Used In the Pmductlon ol the Worksheets Fwmula in fir* box
Column
Nitrogen: (1- EM4 - 2E) Hydrogen: 3(1 - .9/(4 2€),i.e., three times as large as that of nitrogen Ammonia: 2(/(4 - 2:)
-
where is the' fraction of a mole of nitrogen that has reacted-the extent of reaction. 5. In column I the uncorrected entropy values are added together and corrected for the nonstandard partial pressures calculated above using the equation The entropies of the initial stoichiometric mixture of nitrogen and hydrogen (also corrected for a total pressure of 1 atm) are then subtracted to give the entropy change of the system. (The formulas in the boxes in this column can be shortened from that given in Table 2, hut the expanded version is given so that the origin of all the numbers can be seen.) 6. Column J contains the total entroov chanee a t each step. This is found by adding together thevalues-contained in columns B and I. As can he seen. it has a maximum corresponding to an extent of reaction' of between 0.65 and 0.70. To define this calculated maximum more precisely, a second worksheet was constructed. I t is identical to the first except that it was restricted to the area around the maximum and used a step size of 0.01. It is shown in Table 3. 7. In column K, the Gibbs free energy change is calculated (relative to the same zero) from the total energy change. This was calculated using the relationship AG = -AS(tot)lT
-
J K L
' H ~ ' w ( H ~ ) ) 16.669 415.5 A 4 + 14 ~4:(-400) (H42)/((Gd3).F4)
TaMe 3. A
1 2
+ c4 L D+~E4
-
ZOOS
-
i.e., by dividing the total entropy change a t each step by -400. 7. As an additional check on the accuracy of the model, the final column calculates the "equilibrium constant" a t each extent of reaction. I t can be seen that, within the limits of the accuracy of the data used. the real eauilibrium constant is the same as that predicted from the p&tial pressures at the calculated total entropy -~ mini. maximum/free energy mum. The results of the calculations are more easily digested when presented graphically, as in the figure. So that the ~
Spreadsheet Showlng the Entropy and Free Energy Changes near the Posltlon of Equlllbrlum*
B
C
D
E
F
G
H
I
Ssur
SN2
SH2
SNH3
(eu)
leu)
(eu)
pN2 (atm)
pH2 (atm)
pNH3 (atm)
Ssp
leu)
leu)
J
St01 (eu)
Volume 67 Number I
K
G Jlmol
January 1990
L
"K"
49
0
* Total ('10) * System
C
g
g- -200 -100
m
i l
400 0.0
0.2
0.4
0.6
0.8
1 .O
Extent of reaction Emr~pychanges w i n g the Haber process reaction
maximum can be better seen, the total entropy change is shown as 10 times its calculated value. Summary The model allows entropy and free energy changes during a reaction to be calculated. From this i t can be seen that the
position of equilibrium corresponds to a maximum in the total entropy change rather than those of either the system or the surroundings. The presence of an equivalent minimum in the Gibbs free energy, and the reason for this is also easily demonstrated. Despite the approximations and rounding errors inherent in lookina a t small differences between two larger numbers, the calcuiated maximum is extremely close to the correct value of 6 This is confirmed hy the very close proximity of the entropy maximum with the value o f t corresponding to the equilibrium constant. I agree wholeheartedly with Smith' that this kind of modeling is of value in making concrete an otherwise abstract concept. In particular I believe that the graphical representation of the results helps students get a qualitative "feel" for what is going on and, therefore, I hope, more sense of what the math is about. Acknowledgment The figure and tables used in this article were originally prepared for the Nuffield-Chelsea Curriculum Trust and are reproduced with their kind permission.
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