Concise Physical Chemistry

c02 JWBS043-Rogers September 13, 2010 11:23 Printer Name: Yet to Come

THE COMPRESSIBILITY FACTOR 21

Pressure

0 5 10 15 20 25

Compressibility Factor,

Z

0.9840

0.9845

0.9850

0.9855

0.9860

0.9865

0.9870

FIGURE 2.1 A quadratic least-squares fit to an experimental data set for the compressibility

factor of nitrogen at 300 K and low pressures (sigmaplot 11.0©C).

DividingpVmbyRTover a set of different pressures at a fixed temperature

gives a series ofZvalues that can then be plotted againstp. This has been done

for nitrogen at 300 K to give Fig. 2.1. Commercial curve-fitting software can be

used to give the least-squares expression for a polynomial fit to the data points.

One needs to select the degree of the polynomial to be fit to the points. The data

set shown in Fig. 2.1 shows a little experimental scatter at the lower pressures and

(perhaps) some slight curvature. Therefore, we selected a simple quadratic fit to the

points.

Real gas law calculations like this one have considerable practical value. The

engineering literature contains data sets of a much more complicated nature, over a

much larger range than Fig. 2.1. The curve-fitting technique is the same, although one

might choose a cubic or quartic curve fit. The output for the simple nitrogen curve fit

is given in File 2.1. TheRsqr(square of the residual), being close to 1.0, indicates

a good fit, although the extrapolated intercept y0 is not as close to 1.0 as we would

like to see it.

The two virial coefficients are quite small,−0.0002 and 1. 69 × 10 −^6 ; nitrogen

is nearly ideal at 300 K over the short pressure range 1–10 bar. The second virial

coefficient is negative, reflecting the gentle downward slope away from ideal behav-

ior. Note that in File 2.1 the notation f=y0+a∗x+b∗x∧2 is used so that the some-

what overworked parametersaandbappear in a new and different context. Now,