112 Journal of Chemical Education  •  Vol. 86 No. 1 January 2009  • www.JCE.DivCHED.org  • © Division of Chemical Education

Research: Science and Education

One of the important aspects of a student’s experience in physical chemistry is the chance it offers to use the mathemati-cal tools that have been introduced in foundation mathematics classes. Among these tools are power series expansions, such as correcting the ideal gas expression for treatment of a real gases. Two forms of these expansions are standard in physical chemis-try (1), one in terms of the pressure of the gas

1P V R T B Pm C P D P2 3 … (1)

and one, which is oftentimes more convenient, that is in terms of the gas volume

P V R T

BV

CV

DV

mm m m

2 31 … (2)

Equations 1 and 2 use the familiar parameters of a gaseous state: P for pressure, V for volume, T for temperature, and their proportionality constant R—the ideal gas constant. They also include the intensive molar volume,

Vm

VVn (3)

where n represents the number of moles of gas sample.The expansion coefficients of eqs 1 and 2 are the virial coef-

ficients at each order. The first for each series is unity (A′ = A = 1), the second virial coefficients being B′ and B, respectively, and so forth. Virial coefficients can also be related to Z, the compression factor of a gas,

Z

P VR T

m (4)

Using eq 4 with eq 2 allows the virial expansion to be written in terms of the compression factor:

Z

BV

CV

DVm m m

1 2 3 … (5)

A familiar physical chemistry homework problem (1) in-volves expanding the van der Waals equation of state, followed by a collection of terms to equate its parameters to the virial coef-ficients of eq 5 at various orders. This article presents an alterna-tive approach to derivation of van der Waals virial coefficients, obtained by applying a direct differential method to the equa-tion of state. As a test of the methodology, the direct differential method is then applied to the gas model due to Dieterici.

Application to the van der Waals Equation

The van der Waals model for a gas is (2)

P

RT aVm

2bVm (6)

In addition to the previously mentioned parameters, eq 6 intro-duces the two familiar corrections to the ideal gas for volume ex-clusion, b, and molecular attraction, a. Substitution of eq 6 into

eq 4 gives the compression factor of the van der Waals gas:

Z

V aR T V

m

mbVm (7)

Virial coefficients are related to the compression factor by re-writing eq 7 in the form

11

Zb V

aR T Vm m/ (8)

Using x = b/Vm the first term of eq 8 is expanded using (1),

11

1 2 3

xx x x ... (9)

leading to

Zb

VbV

bV

aR T V

ba

R T VbV

m m m m

m m

1

11

2

2

3

3

2

2

…

bbVm

3

3 … (10)

Direct comparison of eq 10 with eq 5 leads to the van der Waals second virial coefficient,

B b

aR T (11)

third virial coefficient,

C b 2 (12)and fourth virial coefficient,

D b 3 (13)It is now demonstrated that the virial coefficients can be

obtained directly from the definition of a Taylor series (3). For example writing eq 1 in a form employing eq 4 and differentiat-ing gives

ZP

B C P D P2 3dd

22 .. . (14)

The second virial coefficient B′ is equivalent to dZ/dP when all higher-order terms vanish, or:

limB

ZP

dd0P

(15)

Similarly differentiating eq 5 gives,

2

2 3…B

CV

DVm m

Zd1/Vmd

(16)

and the second virial coefficient B is found from dZ/d(1/Vm) when all higher-order terms vanish

B

/V mm

ZV

dd 1

lim (17)

Second- and Higher-Order Virial Coefficients Derived from Equations of State for Real GasesWilliam A. ParkinsonDepartment of Chemistry and Physics, Southeastern Louisiana University, Hammond, LA 70403; wap@selu.edu

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Research: Science and Education

Limits on successive differentiations of eq 16 allow the determi-nation of the third virial coefficient,

/ mC

Z

V

d

d

12 1

2

2Vmlim (18)

and fourth virial coefficient,

D

1

6

Z3d

Vm1 3/dVmlim (19)

A direct evaluation of the Taylor series coefficients requires the differentiation of eq 7 with respect to 1/Vm. This is accom-plished by noting that

Vm

2Vm1/dVmd

(20)

which rearranges to

Vm1/d V Vm m2 d (21)

Substituting eq 21 into eq 17 gives

B V 2 Zd

VmdVmlim m (22)

Differentiating eq 7 with respect to Vm gives

1

Vmd

Zd a

R T Vm2

Vm2bVmbVm

(23)

The compression factor derivative with respect to 1/Vm is there-fore

Z V V a

R Tm

Vm

m2

2 3

1

d

d / bVm bVm (24)

so that eq 22 becomes

B

V2

aR T

2 3

Vmlim m Vm

bVmbVm (25)

The limiting value of eq 25 is found by noting that the first two terms its right-hand side can be combined via common denomi-nator so that it is written as

2B

b aR T

2

Vmlim

Vm

bVm (26)

However, this term is in an indeterminate form,

b2

2Vmlim

Vm

bVm (27)

Evaluating the limit of a quotient is performed using L’Hôpital’s rule (4):

f xg x

f xx

d //

dd d

xg x x

limxlim (28)

Successive applications of L’Hôpital’s rule results in

bb

22

Vmlim

Vmlim

Vmlim

b2

2Vm b2 Vm

bVm 2 bVm (29)

The van der Waals form of the second virial coefficient found previously in eq 11 follows directly upon substituting the result of eq 29 into eq 26.

Extending the methodology to an analysis of the third virial coefficient C begins with a second differentiation of eq 7 with respect to 1/Vm, which is done as before by first differentiating eq 24 with respect to Vm,

mV

2

3

2

2d Zd

/1d2

2

3

32 Vm

Vmd Vm

Vm Vm

bVm bVm

bVmbVm

(30)

followed by combining terms and then applying eq 21,

2

3 4

1

2 4Zd

d

2

/

Vm Vm 2 5Vm

Vm bVm3bVm

2bVm

(31)

Finding the common denominator and simplifying gives

1

Zd

d

2 2 3 2bVm2/Vm

3bVm (32)

The van der Waals third virial coefficient may be found by ap-plying the limiting condition to eq 32, which again is in an indeterminate form,

Vm

lim2 3 2bVm

3bVm (33)

Successive applications of L’Hôpital’s rule results in

b

bb

6

3

126

2

2 2

22

Vmlim

Vmlim

Vmlim

Vmlim

2 3 2bVm

1

Zd

d

2

2/Vm

Vm

Vm

3bVm

bVm

2bVm (34)

The van der Waals third virial coefficient (eq 12) follows directly upon substituting the result of eq 34 into eq 18.

For the fourth virial coefficient, application of eq 21 to eq 32 and simplifying leads to

3

33

4 3

46 bVm

1

Zd

d /Vm bVm (35)Four applications of L’Hôpital’s rule leads to

3

3

4 3

1

6Z

V

V b

m

m

/

d

d 44 6 3bVm

limVm

limbVm

(36)

The van der Waals fourth virial coefficient (eq 13) follows di-rectly upon substituting the result of eq 36 into eq 19.

Application to the Dieterici Equation

The Dieterici model of the gaseous state provides a further interesting application of the direct differential method. Diet-erici’s equation for a real gas is (5)

P

R T ea

R T Vm

bVm

(37)

114 Journal of Chemical Education  •  Vol. 86 No. 1 January 2009  • www.JCE.DivCHED.org  • © Division of Chemical Education

Research: Science and Education

The Dieterici parameters a and b have the same physical inter-pretations and dimensionality as those for the van der Waals gas, but numerically differ from them by proportionality constants. Further details on this are provided in Appendix I. For simplic-ity, the same symbolic representation for these parameters is used in the present discussion. The Dieterici gas has a compression factor in the form

Z

V em

aR T Vm e

aR T Vm

1bVm b /Vm

(38)

To find a virial expansion of the Dieterici gas, the denomi-nator of eq 38 is again expressed in the form of eq 9. The expo-nential is also expanded using (3)

e xx x

x

x 12 3

2 3

! !…

(39)

Equation 38 becomes

Zb

Vb

m

Vm Vm

Vm1

1

2

2 ...

aR T

aR T

12

2

2 ...

(40)

which rearranges to

Vm

VmZ b

aR T

ba b

R

11

2

TaR T

2

2 22

1…

(41)

Comparing eq 41 with eqs 2 and 10 reveals that the second virial coefficient of a Dieterici gas has the same parametric form as that of a van der Waals gas. According to eqs 2 and 18, the third virial coefficient in the Dieterici case is recognized to be

VmC

Z

ba bR

Vm

2

2

2

1

2 1lim

/

d

d

TaR T

2

22

(42)

Equation 21 allows one to obtain Dieterici virial coef-ficients from the direct differential method. Using previous techniques, the first derivative of eq 38 is

Vm

Vm

Vm

VmVm

Vm

Zb

aR T b

Vm b

2 3

21d

d /

ea

R T Vm

(43)

which rearranges and simplifies to the form

Vm Vm

Vm Vm VmZ R T b a a bR T b

e1

2 2

2d

d /

aR T Vm

(44)

In the limit that Vm → ∞, e–a/RTVm → 1. However,

VmV

Vm m m

m

m

Z

R T b V a V a b VR T b

1

2 2

lim/

lim

dd

2 1Vm

(45)

Evaluating with L’Hôpital’s rule it is found that

2 2

2R T b V a V a b V

R T V bVm m m

mmlim b

aR T (46)

as expected. The Dieterici third virial coefficient is found by applying

the direct differential method to eq 44 followed by combining terms and simplifying. The second derivative is

1

2 Z/

d

d VVm2

ea

R T Vm

R T b V a VR T a R T a b Vb V

a b V a b V

m m

m m

m m

2 2 3 2 3

3 2 2

2 2 2 2

22 2

2

RT V bm2 3

(47)

Three applications of L’Hôpital’s rule gives

ZV

R T b

V mm

2

2

2 2

1

12

lim/

d

d

6 12

6

2 2

2

2

22

2

a R T

R T

a b

b aR T

a bR T

(48)

Inserting eq 48 into eq 42, the Dieterici third virial coefficient follows directly.

Results and Discussion

The expressions for virial coefficients in terms of correction parameters give insight into both the similarities and differences of the two model gas equations. Over pressure ranges where the series converge to the extent that only the second virial coef-ficient is important, the van der Waals and Dieterici models predict the same parametric form of a gaseous state. At these pressures, the equations also predict the same Boyle tempera-ture (1). The Boyle temperature is that where the second virial coefficient vanishes and the gas maintains ideal behavior. This temperature is determined from eq 11 to be TB = a/Rb. Para-metric differences between the models arise at pressures where the third virial coefficient becomes important. Equation 12 shows that the van der Waals third virial coefficient (and, indeed any higher-order coefficient) is temperature-independent and

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Research: Science and Education

only depends upon the volume exclusion parameter b. In fact, there is a temperature at which the van der Waals and Dieterici models coincide in parametric form through third order. This value is found where the temperature-dependent terms of eq 42 cancel one another to be T = a/2Rb; exactly one-half the Boyle temperature.

In conclusion, a direct differential method has been applied to the derivation of the second virial coefficient of a gas obeying the van der Waals equation of state and extended to the evalu-ation of higher-order viral coefficients. The approach has been verified using a gas represented by the Dieterici equation of state. The coefficients, expressed in terms of correction parameters for these model gases provide an intuitive look at their similarities and differences. The direct differential method offers the oppor-tunity to introduce chemistry students to applications of Taylor series expansions, differential transformation, and evaluating indeterminate quotients.

Appendix I. Numerical Differences between van der Waals and Dieterici Parameters

As discussed above, the molecular attraction, a, and volume exclusion, b, parameters for the van der Waals and Dieterici gases have the same respective dimensionality, but numerically differ. Parameters for each model can be empirically fit for a series of gases, but it is more insightful here to examine their relation-ships to critical temperatures, Tc, and critical pressures, Pc. The processes are again standard physical chemistry homework problems in which the critical points are characterized by the slope (dP/dVm = 0) and curvature (dP2/dVm

2 = 0) of the models along their critical isotherms (1).

The results for the van der Waal gas are a critical pressure of

27 2a

bPc

vdW

vdW (49)

and critical temperature of

avdW

b RTc

vdW

827 (50)

The same technique shows the Dieterici gas has critical pres-sure

P

ae b

cD

D4 2 2 (51)

and critical temperature of

T

ab Rc

D

D4 (52)

Squaring both sides of eq 50, eliminating b2 using eq 49, and solving gives

a

RvdW

27 2

cPTc

2

64 (53)

Similarly treating eqs 52 and 51

D

ca

Re P

Tc22

24

(54)

Equations 54 and 53 show that the molecular attraction param-eters for any gas have the proportionality

D

vdW

aa

1 28. (55)

The volume exclusion parameter of the van der Waals gas is found by rearranging eq 50 for a and eliminating a in eq 49

vdW

c

cb R T

P8 (56)

Similarly treating eqs 52 and 51

b R

eD

Tc

Pc2 (57)

Equations 57 and 56 show that the volume exclusion parameters for any real gas have the proportionality

1 08.

bD

vdWb (58)

Equations 53, 54, 56, and 57 can be used to find Boyle temperatures, TB, for the gas models in terms of their critical temperatures. For the van der Waals gas this value is

T TB cvdW

278, (59)

and for the Dieterici gas

T TB D c4, (60)The proportionality of Boyle temperatures for any gas within the two models is

TT

B D

B1 19,

,.

vdW (61)

Literature Cited

1. Atkins, P.; DePaula, J. Physical Chemistry, 8th ed.; W. H. Freeman and Company: New York, 2006; pp 15–26.

2. van der Waals, J. D. Over de Continuïteit van den Gas—en Vloeistoftoestand (On the Continuity of the Gas and the Liquid State). Ph.D. Thesis, Leiden University, Leiden, the Netherlands, 1873.

3. Jeffreys, H.; Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed.; Cambridge University Press: Cambridge, 1988; pp 50–51.

4. de l’Hôpital, G. Analyse des Infiniment Petits pour L’intelligence des Lignes Courbes (Analysis of the Infinitely Small to Understand Curves), 2nd ed.; Lefèvre: Paris, 1781.

5. Dieterici, C. Ann. Phys. Chem. Wiedemanns Ann. 1899, 69, 685.

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