c02 JWBS043-Rogers September 13, 2010 11:23 Printer Name: Yet to Come
PROBLEMS AND EXERCISES 31
TABLE 2.2 Observed Real Gas Behavior
Expressed as (p,pVm).
p(bar) pVm(dm^3 bar)
10.0000 24.6940
20.0000 24.6100
30.0000 24.5400
40.0000 24.4820
50.0000 24.4380
60.0000 24.4070
70.0000 24.3980
80.0000 24.3990
90.0000 24.4180
100.0000 24.4600
Solution 2.2 This problem and its solution are expressed using SigmaPlot 11.0©C
plotting software. Load the data set in the form of Table 2.2. Click on statistics→
nonlinear regression→regression wizard→quadratic,→next, specify columns as
xandyvariables, and click finish.
The graph in Fig. 2.10 is shown with its fitted quadratic curve. The curve param-
eters are, as they should be, the parameters we started with in the previous problem:
y0 24.7906
a −0.0103
b 6.5341E-005
The parameters are expressed in the formy=y 0 +ax+bx^2 +cx^3 + ···. Please
notice the slight change in notation: Theyintercept is calledy 0 , the slope isa, and the
quadratic coefficient isb. The general independent variable isx, which is the pressure
pin our case. Translated to the terms of the problem, we have
pV=RT+B[T]p+C[T]p^2 +D[T]p^3 + ···
= 24. 7906 − 0. 0103 p+ 6. 5341 × 10 −^5 p^2
where we have truncated the series at the quadratic term.
These two exercises make the important point that any data set can be expressed
as a collection of numbers (table of observations), a graph, or an analytical equation.
Tables, graphs, and empirical equations are merely different ways of saying the
same thing. Difficulties may be encountered (imaginaries, singular points, multiple
real roots, discontinuities, etc.), but they often point to interesting phenomena and
their explanation may lead to new concepts in science. Larger data sets and more
complicated functional behavior can be treated in the same way as this quadratic,
except that the curve fit may be cubic, quartic, and so on.