The Virial Theorem, Perfect Gases, and the Second Virial Coefficient Mervin P. Hanson Humboldt State University, Arcata, California 95521
Most physical chemistry texts derive the perfect gas law using the approach of many first-year physics texts. The argument ( 1 )uses a Cartesian component of the velocity to calculate the time between collisions with a wall. Whenever this component appears squared the result is replaced with the time-average square of the speed of the particle. The argument is a t best semiquantitative and provides a third-year student with nothing new. The main motivations for this limited derivation are that the physics is correct and the effort required to derive i t rigorously (2)is beyond the value of the result. Because much of physical chemistry is the derivation (in this case, rederivation) of the equations of first-year chemistry, i t isappropriate that these armments he a t a more so~histicatedlevel. The main purpose of this paper is to recast an elegant and beautiful derivation of the gas law that uses the virial theorem (3)in a form easily understood by third-year students. This approach shows explicitly how one may extend the analysis to include particle interactions. An added hendit i s that such extension requtres the application of a I ' ~ n d a m e ~ ~postulate tal of atis is tical mecha~~ics: thc replacement of a-time average with a n ensemble or distribution average. The inclusion of pair interactions yields the second virial coefficient. Virial coefficient data for argon is used to justify the three parameters of a square-well po-
tential. and the results are used to oredict the JouleThompson coefficient. The agreementwith experimental data is imvressive. For comdeteness a standard derivation of the kirial theorem is presented. Although the dot product and vector notation have been introduced to most students, in my experience they are not comfortable with it, so I u s e simple scalar notation throughout this paper. The Virial Theorem
Many of the quantities of thermodynamics are time averages; the pressure is the time-average force per unit area, and the temperature is related to the time-average kinetic energy of the particles composing a system. The time-average value of any function of time At) is defined' as 'For example, 1 cos ".. lim -I cos (r)dr to =
sin(r) - 1 0
= lim -r-1-
f
The "se of tne d-mmy varrao e of ntegral on as a rn t to tnat megrat on 1s consdered ga-cne oy most matnemat clans. tne ommy bar aote d sappears aftertne ntegraton s compete
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Consider a particle of mass m located a t (x, y, z ) with respect to some origin. Evaluate the sum of the products of the Cartesian coordinate of the particle and the corresponding component of the force ( F , F , and F,) acting on it. The time average of this quantity is given by
where each force component has been replaced by the product of the mass and the corresponding component of the acceleration. This equation may be integrated by parts to yield
If eq 3 is to apply to a gas model, the model must contain a finite amount of energy and be confined to a restricted region of space. If all the energy of the system is taken up by one particle, the quantity
remains finite and the first term on the right-hand side of eq 3 goes to zero in the limit. Multiplying this limit by 112 to give the standard result produces
The motion of one panlcle demonstrates true chao;. Any &$wnparticle hiti 3 trajectory that moves from a collision with the wall back into the volume. to make! cdlisions with other particles before returning to collide again with the wall a t some other location. In the infinite time required for the average, the trajectory will pass through the entire volume available, spending equal times in equal volumes. More importantly, the trajectory will intersect the walls everywhere with uniform surface density. The contribution of the particle-wall collisions to the virial may be determined by integrating the contribution of some generalized small area dS over the entire surface. Let this small area be located a t (aI2, y, z ) , that is, somewhere on the right surface of the cube. Because this surface element is a t a fixed location, only the time average of the force is of importance. The pressure is the time-average force exerted by all the particles per unit area of surface, and this force is exerted normal to the surface. The contribution of one particle to this time-average force must be I I N of the total or (PINdS. If a single particle exerts this force component on the wall, Newton assures us that the particle experiences an equal but oppositely directed force of 4 P I N d S . The contribution of this particular infinitesimal surface (a12, y, z ) to the required time average is given by
because F, and F, average to zero a t this location. The Integration Adding up the contributions of all the infinitesimal surfaces on the right-hand wall gives the contribution of the right-hand wall to the required time average: We integrate over the right-hand surface. Everything on the left side of eq 5 is constant except dS, so integration of this differential surface area over the right surface gives a n area of a'. I n evaluating the contribution of the left-hand wall, x changes sign but so does the direction of the force; the lefthand side of eq 5 is valid for any of the cube's surfaces.
in which v represents the magnitude of the velocity. Summing eq 4 over all particles in the system produces the virial equation. The left side of the resulting equation is called the virial. In the analysis that follows, eq 4 will be used and referred to a s the one-particle virial. The derivation of eq 4 is so simple that it is easy to overlook the remarkable character of the result. The Perfect Gas Law Derivation
The model for the gas is taken to be a collection of N particles of mass m confined to a simple cubic volume of dimension a centered on the origin. The assumption of a cubic shape greatly simplifies the derivation but has no physical effect. I t is assumed t h a t the collection h a s reached thermal equilibrium a t some temperature T by collisions with the walls and with other particles. For any reasonable assumption concerning the nature of the forces acting on the particle, these forces may be separated into that part due to collisions with the walls and that part due to the other particles. 1' 1 is worth pointing out to students that the force of gravity is given by the gradient of the gravitational potential rngh. Hooke's law is given by the gradient of ( W 2 ) 9 , and Coulomb's force law as the gradient of qlr.
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The integration over one surface (one-sixth ofthe surface area) produces one-sixth of the total surface contribution to the one-particle virial, or for all the surfaces,
If the particles of our model are of zero size so that there a r e no collisions between particles and we assume no forces act between particles, the one-particle virial is given exactly by eq 7. Because the volume Vof the cube i s a3,eqs 7 and 4 give the usual result ( 4 )for the perfect gas. -
Pairwise Additive Forces and the Second Virial Coefficient If the pressure is low enough, a gas particle spends most of its time far removed from other particles, making rare and brief encounters with one other particle and never with more than one a t a time. Because all particles are identical, in this pressure regime we may determine the contributions of the other particles to the one-particle virial as N times the contribution of one of them. Actually, we will determine the time average for two particles and multiply the result by N12. Consider two particles exerting forces on one another hut far removed from the walls. Let the particles (1 and 2) he located a t (XI, yl, zl) and (z2,y2, ZZ).Let the force particle 2 exerts on particle 1be F"', and let the force particle 1exerts on particle 2 be F"'. Again invoking Newton, we have Fi2'= -F1". In scalar notation we have the equalities
where p(r) is the number density of pairs of particles separated by the distance r, and the upper limit a is effectively infinity but is temporarily taken as finite. For large distances r the number density will approach the bulk density NIV, so p(r) is given by
where we have used the Boltzman factor e-U'kTto generate the relative probability of finding a pair of particles separated by the distance r. Integration
Integrating the right-hand side of eq 14 by parts, we get
because U goes to zero for large r. This equation gives our result a s the difference of two terms that go to infinity in opposite directions a s a + 0.. That the result of eq 16 remains finite i n this limit on a may be shown by replacing a3 with the integral of 3r2 from 0 to a giving Because the forces are central, the force Fn'and the vect o r r = (xl -x2, y l -y2, zl - 22) giving t h e separation between the two particles are either parallel or antiparallel. Taking the sign of F"' as positive when i t increases the separation distance between particles, we have Substitution of eq 17 into eq 13 gives The scalar magnitudes of the separation distance r and the force F a r e used because the two are parallel and the superscript on F has been suppressed. Equation 10 gives the time average for a pair of particles, so the contribution of one of them to the one-oarticle virial is half of this auantity. If we use the results from eqs 7 and 10, eq 4 becomes
Because the forces are conservative, they are.derivable from a scalar potential2 U a s
Use ot the Equiparfition Theorem and First Postulate of Statistical Mechanics
Invoking the equipartition theorem i n the form giving k T l 2 for each square term in the energy expression, eq 11 may he rearranged to
To proceed i t is necessary to invoke a modified form of the first postulate ( 5 ) of statistical mechanics. Thus, we replace the time average with the distribution average
where n = NINo; R = N&; NOis the Avogadro number; and the upper limit has been replaced by infinity without introduction of error. The second virial coefficient B follows from eq 18 as
A Square-Well Potential and the Joule-Thomson Coefficient for Argon An improvement to the hard-sphere model (6)is a square well with the potential given as
Thus, we have a hard core of radius cs with an attractive well of depth E and range go beyond the hard core. The potential is zero otherwise. Volume 72
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The substitution of this potential, eq 20, into eq 19 breaks naturally into three integrals giving the second virial coefficient a s
0-
~ ~appropriate ~values for Figure ~ ~ i ~ as a functioniof the absolute ~ temperature. ~Solid 1. The second virial coefficient of argon line given by eq 20 and experimental points ( -)from ref 8. the parameters is a nontrivial problem, so we use the recommendations of Hirsehfelder, Curtis, and Bird (7): a = 316.2 pm, E = 69.4 k, a n d g = 1.85. Figure 1 shows the model predictions using these values with ref 8 values. The agreement between the model and argon gas behavior is quite satisfacto~y.~ A standard problem i n thermodynamics i s t o show t h a t t h e JouleThomson coefficient is given by
Writing the virial equation of state in terms of the pressure and taking the limit of zero pressure in eq 22 gives the Joule-Thomson coefficient in this zeropressure limit a s
The result of substituting eq 21 into this expression for the Joule-Thomson coefficient is a bit cumbersome, but modern computer algebra systems make this straightforward. ~i~~~~ 2 Figure 2.The Joule-Thomson coefficient of argon as a function of the absolute temperature. comDaresthe oredictions of ea z3 with Solid line given by eq 22 and experimental points (*)fromref 9. literature values (9). Literature Cited I. For example. Ceatellan. G. W. Physiml Chemidry, 2nd ed.;Addimn-Wesley: Menlo Pavk, CA. 1971: p 53. 2. Kauzrnann, W. Kinelic Tkeov of C L I S ~Benjamin: S; New York, 1966: p 56. 3. Fowlel R. H. Statiaiicnl Mechonrcs. 2nd ed.; CamMdge Univenity: Cambridge, 1955; p 2R6. 4. R d I : p64.
%ee. however, the comment at the top of p 207 in ref 7.
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5 . ~ i aT., L. A~ ~ ~ ing, MA, 1960: p 4.
t to stotisticai ~ d ~
~h
e ~ ~ i ~~ ~dd
~d ~ i~ ~~R ~~ ~s ~~ .i - ~ ~ ~ ~ ;
6. Atkins. P. W. Phyricol Cksmisfry. 4th ed.: Freeman: New York, 1990: p 666. 7. Hirschfelder; C n h m ; and Bird. Moi#culor Theory of0oapsandLquidr; Wilcy: New Ynrk. 1954: p 160. 8. Hilrenrsth, J.; Berkett, C. W: Benedief, W. S.: Fmo, L.; Hoge, H. J.: Maw. J. F.; i of Nuttail. R. L.; Touloukian, Y S.: Woolley. H. W. ~ o b i e sof ~ h e r m aPropenisr Gases; National Bureau of Standards Cireviar 564. 195k p 136 9. Forsyfhe. W E. Smithsoninn Ph.picol Tnbles, 9th e d : Smithsonian institution: washington, 1954; p 279.
