Concise Physical Chemistry

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

22 REAL GASES: EMPIRICAL EQUATIONS

Data Source: Data 1 in N2 molar density

Equation: Polynomial, Quadratic

f=y0+a∗x+b∗x∧ 2

R Rsqr Adj Rsqr Standard Error of Estimate

0.9977 0.9954 0.9932 6.7995E-005

Coefficient Std. Error t P

y0 0.9869 6.0263E-005 16377.0857 <0.0001

a −0.0002 1.6559E-005 −9.6859 0.0006

b 1.6977E-006 7.5741E-007 2.2415 0.0885

FILE 2.1 Partial output from a quadratic least-squares curve fit to the compressibility factor

of nitrogen at 300 K (SigmaPlot 11.0©C).

a=B 2 [T]is the second virial coefficient andb=B 3 [T]is the third. The first

virial coefficient (term rarely used) is 1.0 by definition. In File 2.1, the third virial

coefficient is positive, indicating a slight upward curvature. (Experiments at higher

pressures confirm the curvature.)

Nonideal gas behavior is nearly linear at low pressures. That is why the slope of the

linear function is a measure of the second virial coefficientB 2 [T]. The temperature

variation of the second virial coefficients of helium, nitrogen, and carbon dioxide are

shown schematically in Fig. 2.2. WhenB 2 [T]=0, the slope of the virial equation

forZis zero andZ=1 over the range. If this is true, the gas is ideal. Helium shows

ideal or nearly ideal behavior over most of the temperature range. Carbon dioxide is

very nonideal over the range, and N 2 is in between. This order is pretty much what we

would expect from our qualitative knowledge of the three gases. Helium is a “noble”

gas, CO 2 is commonly available in the condensed state as “dry ice,” and atmospheric

N 2 is in between. Nitrogen is not as easily driven into the condensed state as CO 2 ,

but liquid nitrogen is far easier to produce than liquid helium.

0

300

He

N 2

CO 2

B 2

Temperature

[T]

0 600

FIGURE 2.2 The second virial coefficient of three gases as a function of temperature. Notice

the slight maximum in the curve for helium. It is not a computational error, helium really does

that. Intermolecular repulsion brings about a small positive deviation ofZfromZidealover part

of the temperature range.