Research: Science and Education
Fugacity Examples Carl W. David Department of Chemistry, University of Connecticut, Storrs, CT 06269-3060;
[email protected] There are few literature or textbook examples of explicitly computing the fugacity of nonideal gases. As a result, it behooves us to note that the explicit evaluation of the fugacity of a van der Waals gas can be obtained in closed form (1) and herein, the fugacity for a pseudo gas constructed out of a one-dimensional square well gas can also be obtained in closed form. A Nontraditional Equation of State’s Fugacity For the one-dimensional square well gas, the equation of state can be obtained exactly (2). Here the well depth is W (in units of kT ) and the well extends from x = a to x = b, where x is the separation distance between particles. The quantity c is defined as b − a, the “length” of the well. The potential energy of interaction between two particles in this system is: ϕ( x
)
∞ , x < a = −W, a ≤ x < b 0, x ≥ b
where the first three terms in the square brackets come from eq 2 and the fourth term is from the ideal gas law. We have altered the notation slightly to emphasize the difference between the limits of integration and the variable being integrated. If u = 1 − χeax and du = +aχeax , integration yields,
R T ln
f f°
−
pdesired Γ
1 − χe R T − R T ln 1 − χ
= A pdesired
(4)
as may be verified by differentiation of the right-hand side of eq 4 leading to the integrand in eq 3. We plot f兾f versus pdesired (= p) in Figure 1. The van der Waals Fugacity For the van der Waals equation of state, we quote the result,
If one chooses to “generalize” the one-dimensional Walmsley equation of state (2) to a three-dimensional gas, that is, writing V for L in the original Walmsley equation of state (absent a derivation thereof ) one finds that the equation of state,
RT ln
f f°
=
RTb V pdesired − b − RT ln 1 −
−
2a V pdesired
a (V pdesired − b ) RT V ( pdesired )
(5)
2
3
qe −β p c V 1 = a3 + − c3 3 βp N 1 − qe −β pc
(1)
leads to a tractable integral form for the fugacity, where N is the number of particles, β ≡ 1兾kBT, and q ≡ 1 − eβW. The equation can be expressed in a “per mole form”, RT − Γ p
−
RT ln
f°
0
0.975 2.0
(V
)
− Videal gas dp =
V −
RT dp p
0
pdesired
RT x
− Γ
χe 1−
−
xΓ RT
xΓ − χe R T
0
www.JCE.DivCHED.org
•
−
2.5
3.0
3.5
4.0
4.5
5.0
p / bar
that is, the right-hand side becomes,
A +
0.980
p
p
=
0.985
(2)
pΓ RT
1 − χe _ where V = V兾(N兾NA), A ≡ NAa3, χ ≡ NAq, and Γ ≡ NAc3. We are interested in forming the standard textbook form of the natural logarithm of the fugacity,
f
0.990
f f °
V = A +
pΓ − χe R T
0.995
RT x
dx
(3)
Figure 1. A typical f兾f versus p (in bar) plot for the two gases at 298 K. In this plot, a = 4.17 bar/(L兾mol)2 and b = 0.0371 L兾mol for the van der Waals gas (shown as a line) and a Walmsley gas (shown as circles), with parameters a3 = 0.001 L兾mol, b3 = 0.0019 L兾mol, c3 = (b − a)3 L兾mol, and W = 200 (L bar)/mol. Notice that obtaining the p(V ) versus V plot for the Walmsley equation of state is nontrivial, although the V( p) versus p plot is simple to obtain (and turn on its side). The data shown here were obtained by computing pi(Vi) for the van der Waals gas over a discrete range of volumes, and using these same pi(Vi) values for the Walmsley gas, thereby forcing the abscissa to cover identical domains.
Vol. 81 No. 11 November 2004
•
Journal of Chemical Education
1653
Research: Science and Education
and plot the f兾f versus p curve in Figure _ 1, having first obtained the p values at the equivalent V values from the equation of state (van der Waals). Equation 5 can be _ _ expressed in a somewhat simpler form since a兾V 2 = [RT兾(V − b)] − p, R T ln
f f°
=
RTb V pdesired − b − R T ln
2a V pdesired
−
p (V pdesired − b )
(6)
RT
of the van der Waals gas at that same volume, so that the f versus p plot can be constructed point by point. Since both gases attempt to include long-range attractive forces and short-range repulsive forces, it is not surprising that their behaviors are similar, as shown in Figure 1. It is also noteworthy that since the graphs are quite similar, there is little reason to believe that experimental measurement of fugacities could distinguish between models for gases. Acknowledgment Thanks are due to an anonymous referee whose criticisms of this manuscript made me correct several errors.
however this form cannot be easily plotted. Literature Cited
Discussion The fugacity expression for the Walmsely gas is ideally set up to allow plotting of the fugacity versus the pressure. This contrasts with the difficulties in plotting the van der Waals fugacity against pressure, which requires obtaining the fugacity at a given volume and then obtaining the pressure
1654
Journal of Chemical Education
•
1. Winn, J. S. J. Chem. Educ. 1988, 65, 772. See also Jagannathan, S. J. Chem. Educ. 1987, 64, 677. Alessio, L. D. J. Chem. Educ. 1993, 70, 86. McQuarrie, D. A.; Simon, J. D. Physical Chemistry, A Molecular Approach; University Science Books: Sausalito, CA, 1997; p 917. 2. Walmsley R. H. J. Chem. Phys. 1988, 88, 4473.
Vol. 81 No. 11 November 2004
•
www.JCE.DivCHED.org
