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Section 5.6
5
3
2
¢b>1J>mol-K 2 0.11744, ¢c>1J>mol-K 2 9.8296 10 , Use of a Spreadsheet to Obtain
a Polynomial Fit
4
¢d>1J>mol-K 2 2.8049 10 8
With T 298.15 K, Example 5.5 gives H° 565.968 kJ/mol. We can thus
1 T 1
use the H° H° equation to find H° . Substitution of numerical values
T 2 T 1 T 2
gives at T 1200 K,
2
6
1
¢H° >1J>mol2 565968 39.871901.852 10.11744211.3511 10 2
1200
2
1
9
1 9.8296 10 5 211.7015 10 2
3
1
12
12.8049 10 8 212.0657 10 2
4
¢H° 1200 563.85 kJ>mol
The value 578.03 kJ/mol found in Example 5.5 with the approximation of tak-
ing C° as constant is greatly in error, as might be expected since the tempera-
P
ture interval from 298 to 1200 K is large. The C°-versus-T polynomial equa-
P
tion shows that C°/(J/mol-K) is 13 at 298 K, 7 at 400 K, and 8 at 1200 K
P
and is far from being constant.
Exercise
For O(g) in the range 298 to 1500 K, C° is given by the polynomial equa-
P,m
2
tion (5.20) with a 23.34 J/(mol K), b 0.006584 J/(mol K ), c 5.902
3
4
10 6 J/(mol K ), and d 1.757 10 9 J/(mol K ). Find H° for O (g) →
1000 2
2O(g). What is unusual about C° for O(g)? (Answer: 505.23 kJ/mol. It de-
P,m
creases with increasing T in this range.)
Note that H° in this example is not greatly changed from H° . Usually, H°
1200 298
and S° for reactions not in solution change slowly with T (provided no species un-
dergo phase changes in the temperature interval). The enthalpies and entropies of all
reactants and products increase with T (Sec. 4.4), but the increases of products tend to
cancel those of reactants, making H° and S°vary slowly with T.
5.6 USE OF A SPREADSHEET TO OBTAIN A POLYNOMIAL FIT
One often wants to fit a given set of data to a polynomial. This is easily done with a
spreadsheet such as Excel, Quattro Pro, or the free program Gnumeric (www.gnome.
org/projects/gnumeric/), which emulates the functionality of Excel. For example,
C° /(J/mol-K) values for CO(g) at 298.15, 400, 500, . . . , 1500 K are 29.142, 29.342,
P,m
29.794, 30.443, 31.171, 31.899, 32.577, 33.183, 33.710, 34.175, 34.572, 34.920, and
35.217. Suppose we want to find the coefficients in the cubic polynomial (5.20) that best
fits these values. The following directions are for the Excel spreadsheet, which is part of
the Microsoft Office suite of programs, and is widely available in student computer labs
of colleges. The directions are for Excel 2003; Excel 2007 directions are in parentheses.
Enter a title in cell A1. (To enter something in a cell, select the cell by clicking on
it with the mouse, type the entry, and press Enter.) Enter the label T/K in cell A2 and
the label Cp in cell B2. The temperatures are entered in cells A3 to A15 and the C°
P,m
values in cells B3 to B15 (Fig. 5.7). Select all the data by dragging the mouse over cells
A3 to B15. Click on the chart icon on the toolbar or chose Chart from the Insert menu.
Go through the Chart Wizard dialog boxes, choosing XY (Scatter) as the type of plot
and a plot showing data points only as the subtype. Choose Series in columns, omit
titles and a legend, and place the chart as an object in the sheet with the data.