# soln-l27

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```					Linear regression analysis to determine Antoine equation constants                          W.R. Wilcox, Clarkson University

Correlations of data have great utility for comparison with theory and for summarizing
the data in a more convenient form for engineering calculations. One parameter highly
useful for chemical engineers is the vapor pressure of a liquid versus temperature. For
pure liquids, the Antoine equation is often used to correlate vapor pressure data. This
equation is given by ln(p) = A - B/(T + C), where p is the vapor pressure, T is absolute
temperature, and A, B & C are constants to be determined. If linear regression analysis is
to be used to find these constants, we must put this equation into a linear form. This may
be done by multiplying the Antoine equation by (T + C)/T and rearranging to obtain
ln(p) = A + (AC - B)/T - C ln(p)/T. In this linear form, ln(p) is the dependent variable, 1/T and
ln(p)/T are the independent variables, A is the intercept, (AC - B) is the coefficient for 1/T
and -C is the coefficient for ln(p)/T. Use the Regression tool in Data Analysis to find A, B & C
for the carbon monoxide vapor pressure data via the tab below. Plot the experimental
and predicted values of p versus T. Convert to a log scale for p in order to show the
smaller values. Show the experimental values as points and the predicted values
as a curve. Change the scales to reduce wasted space on the graph.

As is customary, put the dependent variable on the vertical (y) axis
and the independent variable on the horizontal (x) axis.
W.R. Wilcox, Clarkson University, spring 2004
A             B                C        D          E           F           G            H         I   J
1   Solution to Laboratory Assignment 27
2   Analysis of data on vapor pressure of carbon monoxide from Perry's Chemical Engineers Handbook
3   Fit to Antoine equation (ln(p) = A - B/(T + C)).          Where T is absolute temperature and A, B, C
4   Solution: One linear form is ln(p) = A + (AC-B)/T - C ln(p)/T       are constants to be determined.
5   Click on cells to see the formulas used.
6
7          Original Data             Modified units     Parameters for regression p (Pa) from
8   p (torr)        T (C)            p (Pa)   T (K)      ln(p)   1/T      ln(p)/T correlation
9         1                 -222.0       133     51.2     4.89       0.0196        0.096           132
10         5                 -217.2       667     56.0     6.50       0.0179        0.116           734
11        10                 -215.0      1333     58.2     7.20       0.0172        0.124          1412
12        20                 -212.8      2666     60.4     7.89       0.0166        0.131          2544
13        40                 -210.0      5333     63.2     8.58       0.0158        0.136          4971
14        60                 -208.1      7999     65.1     8.99       0.0154        0.138          7497
15       100                 -205.7     13332     67.5     9.50       0.0148        0.141         12062
16       200                 -201.3     26664     71.9    10.19       0.0139        0.142         25931
17       400                 -196.3     53329     76.9    10.88       0.0130        0.142         54084
18       760                 -191.3    101325     81.9    11.53       0.0122        0.141        100861
19                                     202650     89.7    12.22       0.0112        0.136        225131
20    T (C)        p (atm)             506625    102.5    13.14       0.0098        0.128        613903
21    -191.3                     1    1013250    112.2    13.83       0.0089        0.123       1098707
22    -183.5                     2    2026500    123.5    14.52       0.0081        0.118       1895541
23    -170.7                     5    3039750    131.3    14.93       0.0076        0.114       2594755
24    -161.0                    10
25    -149.7                    20
26    -141.9                    30
27
28   Using Analysis Tools, Regression we get the output below.
29          SUMMARY OUTPUT
30
31          Regression Statistics
32          Multiple R            0.999691
33      R = R Square
2
0.999382
35          Standard Error        0.081747
36          Observations                15
37
38          ANOVA
39                              df         SS       MS            F            Significance F
40          Regression                   2 129.617 64.809         9698.15 5.587E-20
41          Residual                    12 0.08019 0.0067
42          Total                       14 129.697
43
44          Parameter                      Standard Error
Coefficients        t Stat        P-value Lower 95% Upper 95%
45          Intercept            19.08881 0.23499 81.231          8.1E-18 18.576807  19.600822
46          1/T                  -805.322 5.87716       -137      1.5E-20 -818.1275 -792.51707
47          ln(p)/T              16.13833 1.65059 9.7773          4.6E-07 12.54199 19.7346626
48
49          From this we see:
50                          A = 19.08881
51                          C = -16.1383
52                          B = 497.2608
53 Interpretation of results produced by Regression
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Antoine correlation for CO

10000000

1000000
Vapor pressure, Pa

100000

Experimental
Correlation
10000

To get a smooth line for the correlation, right click on
a correlation point, select Format Data Series,
1000                Patterns, Line Custom Smoothed, Marker None. Do
not try to use Trendline as none of the equations
built into this tool is the Antoine equation.

100
50   70     90                110                130               150
Temperature, K

```
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