Concise Physical Chemistry

(Tina Meador) #1

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.
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