Concise Physical Chemistry

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.