In the Classroom
Calculating Enthalpy of Reaction by a Matrix Method Mutasim I. Khalil Chemistry Department, King Saud University College of Science, Riyadh 11451, Saudi Arabia;
[email protected] The basis of many thermochemical calculations is the law of constant heat summation (i.e., the law of Hess), where the possibility of deriving an enthalpy of reaction from measurements on other reactions arises from the fact that thermochemical data can be treated algebraically. The conventional textbook method for calculating reaction enthalpies from given thermochemical data is often mentally tedious and time-consuming (1– 4 ). The thermochemical equations have to be manipulated and rearranged before they are added. Besides inspecting, at the beginning, the type of reactants and products of the thermochemical equations given, one has to multiply or divide the coefficients by some factor and change the direction of the thermochemical equation(s)—keeping in mind the rules governing such manipulations. I have recently introduced matrix algebra for the calculations of reactions enthalpies, ∆rH °, from given thermochemical data to my chemistry junior students at the College of Science of King Saud University. The method proved to be elegant, easily digested and assimilated by the students, and of direct applicability. In fact, the elegance of matrix algebra in the field of chemistry goes without saying, and its important applications for thermodynamics treatment (1, 5, 6 ) and chemistry kinetics (7 ) have recently been illustrated. Theoretical Background
The stoichiometric number matrix of a system of linear equations has a column for each reaction and a row for each reactant. For example, the stoichiometric number matrix of the set of equations [1] H2(g) + 1⁄2O2(g) = H2O(g) [2] H2O(g) = H2O() is
reaction 1
H2 g O2 g ν= H2O g H2O
(5)
reaction 2
1 1/2 +1 0
0 0 1 +1
(6)
Generally, the stoichiometric number matrix for a set of R reactions involving N species and the stoichiometric number matrix net for a particular net reaction are represented by (8)
ν 11 ν 12 ν 21 ν 22
ν 1R ν 2R
s1 s2
νN 1 νN 2
ν NR
sR
=
ν net1 ν net2
(7)
ν netN
Chemical reactions in general can be represented by N
0 = Σ νi B i i=1
(1)
where νi are stoichiometric numbers and Bi are the molecular formulas of the N substances involved in the reaction. These stoichiometric numbers are positive for products and negative for reactants. Accordingly, the thermochemical equation H2(g) + 1⁄2O2(g) = H2O(g) ;
∆rH ° = 242 kJ mol1 (2)
would be written 0 = H2(g) – 1⁄2O2(g) + H2O(g)
(3)
This is a linear equation, which can be abbreviated by writing only the rectangular array of numbers as
H2 g ν = O2 g H2O g
1 1/2 +1
where the row is Bi and the column is νi .
(4)
where the column matrix s is referred to as the pathway vector that shows the number of times the reactions would occur to give the net reaction. In solving for enthalpies of reaction by Hess’s method we are in fact looking for the number of times each thermochemical reaction has to occur in order to, when added, produce the net reaction. The stoichiometric number matrices and net can be written from the data given. The path is then easily calculated in simple cases by inspection. Multiplying the path by the stoichiometric number matrix yields the column vector for the net reaction net, and multiplying net by the vector of standard enthalpies of reaction yields ∆ rH ° for the net reaction. In complicated cases, a mathematical program (like Mathematica) or the systematic hand method described by Smith and Missen (9) can be used to calculate the path for a given net reaction. In Mathematica the pathway is calculated with Linear Solve: path = Linear Solve [numat, nunet] where path, numat and nunet represent the s, , and net matrices (8).
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In the Classroom
Method and Illustrations
SOLUTION. The matrix equation is
The following steps are observed when solving for the enthalpy change of a certain reaction. STEP 1. Find the solution s of the linear equations s = net
where is the given reaction matrix, net is the reaction for which the ∆ rH ° value is to be determined and s is the multiplier vector (path) that converts the given reactions to the specified one. STEP 2. Calculate the enthalpy of the given reaction by multiplying the path matrix s value found in step 1 by the column matrix of ∆ rH °. The following examples illustrate the ease of using the method.
Example 1 Using the matrix representation method and the following data (3) [1] CH4(g) + 2O2(g) = CO2(g) + 2H2O ()
∆ rH° = 890 kJ mol1
[2] 2CO(g) + O2(g) = 2 CO2(g)
∆ rH° = 566 kJ mol1
determine ∆ rH ° of reaction 3 below [3] 2CH4(g) + 3O2(g) = 2CO(g) + 4 H2O ()
CH4 g O2 g CO2 g H2O CO g
reaction 2
1 2 +1 +2 0
0 1 +2 0 2 numat
0 7 +4 +6 2 0
0 1/2 0 +1 0 1
s1 s2 s3
1 0 = 0 0 +1 2
path nunet
Hence, the enthalpy change of reaction 4 is +1/2 1/2 +2
2600 3210 286 = 1300 + 1605 – 572 = 267 kJ mol
1
Example 3 Given the following thermochemical equations (4 ): [1] H2(g) + 1/2O2(g) = H2O(g)
∆ rH ° = 242 kJ mol1
[2] H2(g) + 1/2O2(g) = H2O()
∆ rH ° = 286 kJ mol1
[3]
2 3 = 0 +4 +2
s1 s2 path
H2O (g) = H2O()
SOLUTION. The matrix equation is
H2 g O2 g H2O g H2O
nunet
reaction 1
reaction 2
1 1/2 +1 0
1 1/2 0 +1 numat
∴ path =
890 566 = 1780 + 566 = 1214 kJ mol
1
Example 2 From the following data (2) [1] 2C2H2(g) + 5O2(g) = 4CO2(g) + 2H2O() ∆rH° = 2600 kJ mol1 [2] 2C2H6(g) + 7O2(g) = 4CO2(g) + 6H2O() ∆rH° = 3210 kJ mol1 ∆rH° = 286 kJ mol1
Calculate the enthalpy of reaction 4 below using the matrix representation method. [4] C2H2(g) + 2H2(g) = C2H6(g)
186
2 5 +4 +2 0 0
reaction 3
+1/2 ∴ path = 1/2 +2
Hence, the enthalpy change of reaction 3 is
[3] H2(g) + 1⁄2O2(g) = H2O()
reaction 2
numat
∴ path = +2 1
+2 1
reaction 1
Calculate ∆ rH ° for
SOLUTION. The matrix equation is reaction 1
C2H2 g O2 g CO2 g H2O C2H6 g H2 g
0 s1 0 s 2 = 1 +1 path
nunet
1 +1
Hence, the heat change of reaction 3 is
1 +1
242 286 = +242 – 286 = 44 kJ mol1
It was suggested to me by Alberty (R. A. Alberty, private communication) that this example is solved using the mathematical relation between the stoichiometric number matrix and the column matrix of ∆ fHo values ( standard enthalpies of formation ), that is, T∆ fH ° = ∆ rH ° (8) where T is the transposed stoichiometric number matrix, the dot indicates the matrix product, and ∆ f H ° represents the column matrix of standard enthalpies of formation. This matrix equation for example 3 is
Journal of Chemical Education • Vol. 77 No. 2 February 2000 • JChemEd.chem.wisc.edu
In the Classroom
1 1/2 1 1 1/2 0
0 1
0 242 0 = 286 242 286
The net reaction for this example is calculated by multiplying the numat by path column matrix, that is,
1 1 1/2 1/2 1 0 0 1
0 1 = 0 1 1 1
where the column matrix on the right is the net reaction. To obtain the standard enthalpy change for the net reaction, we use eq 8:
0 0 1 1
0 0 = 242 – 286 = 44 kJ mol1 242 286
standard enthalpy change for reactions is quicker, easier, and more direct than the conventional method. Literature Cited 1. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 2nd ed.; Wiley: New York, 1997; pp 157–162. 2. Atkins, P. W. General Chemistry, 2nd ed.; Scientific American Books: New York, 1992; pp 210–212. 3. Mortimer, C. E. Chemistry, 5th ed.; Wadsworth: Belmont, CA, 1983; pp 59–61. 4. Brady, J. E. General Chemistry—Principles and Structures, 5th ed.; Wiley: New York, 1990; pp 180–182. 5. Alberty, R. A. J. Chem. Educ. 1991, 68, 984; 1992, 69, 493; 1995, 72, 820. 6. Alberty, R. A. J. Phys. Chem. 1993, 97, 6226. 7. Pogliani, L.; Terenzi, M. J. Chem. Educ. 1992, 69, 278. 8. Alberty, R. A. Biophys. J. 1996, 71, 507–515. 9. Smith, W. R.; Missen, R. W. J. Chem. Educ. 1989, 66, 217– 218.
We believe that this matrix method for calculating the
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