EMPIRICAL EQUATIONS
2.3 THE COMPRESSIBILITY FACTOR
The difference between ideal and real gaseous behavior can be made clearer if we define a compressibility factor Z, a way of indicating the degree of nonideality of a gas
Z = pVm
RT = pVm
( pVm)ideal
If Z is less than one, nonideality is largely due to attractive forces between molecules.
If Z is greater than one, the nonideal behavior can be ascribed to the volume taken up by individual molecules treated as hard spheres or to repulsive forces, or both.
An ideal gas would show a compressibility factor of 1.00 at all pressures. At high temperatures, the total volume is large for any selected pressure. Molecular crowding becomes less significant, and attractive or repulsive forces are weaker because they act over longer distances. The gas approaches ideal behavior and Z approaches the constant value of 1.00 as p approaches zero.
THE COMPRESSIBILITY FACTOR 21
Pressure
25 20
15 10
5 0
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 11.0C).
Dividing pVm by RT over a set of different pressures at a fixed temperature gives a series of Z values that can then be plotted against p. 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. The Rsqr (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 parameters a and b appear in a new and different context. Now,
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 11.0C).
a =B2[T ] is the second virial coefficient and b=B3[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 coefficient B2[T ]. The temperature variation of the second virial coefficients of helium, nitrogen, and carbon dioxide are shown schematically in Fig. 2.2. When B2[T ]=0, the slope of the virial equation for Z is zero and Z =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 N2is in between. This order is pretty much what we would expect from our qualitative knowledge of the three gases. Helium is a “noble”
gas, CO2is commonly available in the condensed state as “dry ice,” and atmospheric N2 is in between. Nitrogen is not as easily driven into the condensed state as CO2, but liquid nitrogen is far easier to produce than liquid helium.
0
300 He
N2
CO2 B2
Temperature [T]
0 0 6 0
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 of Z from Zidealover part of the temperature range.
THE COMPRESSIBILITY FACTOR 23 2.3.1 Corresponding States
An interesting comparison from among Z factors is shown schematically in Figs. 2.3 and 2.4 which might be Z = f ( p) for two different gases or Z = f ( p) for the same gas at two different temperatures. Simply by choosing the right two temperatures, one can make any two gases identical to each other in their degree of nonideality, that is, their Z factors. When different gases at different temperatures behave in the same way, they are said to be in corresponding states. To define the state of a real gas, we must describe it in such detail that a colleague can reproduce it in all of its physical properties from the description. Because of the essentially infinite number of physical properties one can measure, this would seem to be a tall order; but, if it is pure, the number of degrees of freedom for one mole of a real gas is 2 regardless of its degree of nonideality, so specifying any two properties is equivalent to specifying all of them. The equation of state is written with two independent variables, but it doesn’t matter which two.
To summarize up to this point: We are left with a van der Waals qualitative picture of nonideal gas behavior that is quite reasonable but gives an equation that doesn’t work very well outside of common laboratory conditions. Our alternative is to rely upon empirical equations that work quite well in most cases but are hard to interpret.
The term “empirical” as applied to the virial equation in this context has become somewhat of a misnomer over the years, however, because considerable progress has been made in theoretical interpretation of the virial equation and Fig. 2.3. Indeed, the statistical mechanics of these curves and others like them is an active research topic.
p (bar)
700 600
500 400
300 200
100 0
Z
0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
FIGURE 2.3 The Z= f ( p) curve for two different gases or for the same gas at two different temperatures. The unit—bar—is approximately one atmosphere.
Vm (dm3)
2 1
0
p (bar)
73.8
FIGURE 2.4 Three isotherms of a van der Waals gas. The top isotherm is above Tc, the middle isotherm is at the critical temperature Tcand the bottom curve is below Tc. The critical pressure is 73.8 bar.
The apparent paradox that there are only two degrees of freedom in the equation of state of a pure substance which may have an infinite number of terms in an equation of state is removed by noting that each term contains only the pressure, p, and an adjustable parameter (not a variable) that is a function of the temperature. Hence the only true variables in the equation are p and T.