## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 3. Basic Energetics: Intermolecular Interactions

### 3.8. The Virial Theorem*

Even without considering the details of the dynamics of a system of particles, we can connect the average kinetic energy with the average potential energy. This is achieved via a useful theorem, first introduced by Clausius [18], of which we present here a short derivation. To this purpose, it must be first noted that for a bounded function *G*(*t*) with time *t* as variable, the time average ⟨*g*⟩ of *g* = d*G*/d*t* is zero, since

(3.33)

Attention is now turned to the kinetic energy for a set of particles with mass *m _{i}*, position coordinates

*r**, velocity*

_{i}

*v**, and momentum*

_{i}

*p**=*

_{i}*m*

_{i}

*v**. Because*

_{i}*T*is a homogeneous function of the second degree in

_{i}

*v**, we have by Euler's theorem (see*

_{i}__Appendix B__)

(3.34)

Since ∂*T _{i}*/∂

*v**=*

_{i}

*p**, we further obtain*

_{i}(3.35)

If the system has a finite volume, *p** _{i}*·

*r**is bounded and thus we have ⟨d*

_{i}

*p**·*

_{i}

*r**/d*

_{i}*t*⟩ = 0. The kinetic energy

*T*is linked to the potential energy

*Φ*via Newton's Second Law reading

*f**=*

_{i}*m*

_{i}

*a**with force*

_{i}

*f**and acceleration*

_{i}

*a**. Using*

_{i}

*f**= −∂*

_{i}*Φ*/∂

*r**,*

_{i}*m*

_{i}

*a**= d*

_{i}

*p**/d*

_{i}*t*, and introducing the

*virial*(

*of force*) , we obtain the

*virial theorem*

(3.36)

where the second step can be made as long as . Frequently, *Φ* is a homogeneous function of degree *n* in ** r**, and from Euler's theorem we then have

(3.37)

Since the total energy *U* = ⟨*T*⟩ + ⟨*Φ*⟩, we have ⟨*Φ*⟩ = 2*U*/(*n* + 2) and ⟨*T*⟩ = *nU*/(*n* + 2). For harmonic potentials *n* = 2 resulting in ⟨*T*⟩ = ⟨*Φ*⟩ *=* ½*U*. For the Coulomb potential *n* = −1 and since *T* ≥ 0, this implies *U* ≤ 0. So, for an electrostatic system to be stable the energy must be negative. Finally, we note that the virial can be used in the derivation of the equation of state (see __Chapter 6__).

Notes

__1)__ Note again that ** r** is a vector and

**r**is a (column matrix of a) set of elements

*r*or

_{i}

*r**.*

_{i}__2)__ The unit of charge is coulomb [C].

__3)__ The unit of dipole moment is coulomb × meter [C m], but frequently (for historical reasons) the Debye unit [D] is used. 1[D] = 3.336 × 10^{−30} [C m].

__4)__ Because the molecule contains several charges we have to sum over them. We indicate here the charges by the subscript *i* and the molecule by the superscript (*i*). Only if we sum explicitly over charges is the molecule is indicated by the superscript; otherwise, we use a subscript.

__5)__ Isotropy implies equal response for all directions. When the polarization becomes anisotropic, these interactions become orientation-dependent and therefore temperature-dependent.

__6)__ The unit of polarizability is C^{2} m^{2} J^{−1}. Incorporating (4π*ε*_{0})^{−1} in *α* via *α′* = *α*/(4π*ε*_{0}) renders the dimension of *α′* to be m^{3}. Since, according to electrostatics, the polarizability of a perfectly conducting sphere of radius *r* in vacuum is given by 4π*ε*_{0}*r*^{3}, the molecular radius *r* can be estimated from *r*^{3} = *α′*. Hence, *α′* is known as the *polarizability volume*.

__7)__ For an overview of London's work on interactions, see Ref. [19]. The term “dispersion forces” was coined by London to indicate the analogy of the expressions derived to those which appear in the dispersion formula for the polarizability of a molecule when acted upon by an alternating field.

__8)__ We note that, since electromagnetic radiation has a finite velocity *c*, a second molecule feels the potential of the first molecule at a distance *r* only after a time *r*/*c*. This *retardation effect* becomes important only for distances larger than about 10 nm. The effect is important for the mutual interaction between macroscopic bodies and between a single particle and a solid surface.

__9)__ See Section 16.2 and Section 15.2, respectively.

References

1 Böttcher, C.J.F. (1973) *Theory of Dielectric Polarization*, vol. I, 2nd edn, Elsevier, Amsterdam. See also Hirschfelder *et al.* (1964).

2 Pitzer, K.S. (1959) *Adv. Chem. Phys.*, 1, 59.

3 Kihara, T. (1953) *Rev. Mod. Phys.*, 25, 831.

4 (a) Hammam, S.D. and Lambert, J.A. (1954) *Aust. J. Chem.*, 7, 1; (b) Hildebrand, J. and Scott, R.L. (1962) *Regular Solutions*, Prentice-Hall, Englewood Cliffs.

5 Morse, P.M. (1929) *Phys. Rev.*, 20, 57.

6 (a) Kohler, F. (1954) *Monatsh. Chem.*, 88, 857; (b) Kihara, T. (1976) *Intermolecular Forces*, John Wiley & Sons, Ltd, Chichester.

7 Rose, J.H., Smith, J.R., Guinea, F., and Ferrante, J. (1984) *Phys. Rev.*, B29, 2963.

8 See Pimentel and McLellan (1960).

9 See Grabowski (2006).

10 Scott, R.L. (1971) *Physical Chemistry*, vol. VIIIA, Academic Press, New York, Ch. 1, p. 11.

11 (a) Axilrod, B.M. and Teller, E. (1943) *J. Chem. Phys.*, 11, 299; (b) Axilrod, B.M. (1949) *J. Chem. Phys.* 17, 299 and 19, 71; (c) Midzuno, Y. and Kihara, T. (1956) *J. Phys. Soc. Jpn*, 11, 1045.

12 (a) Rowlinson, J.S. (1965) *Discuss. Faraday Soc.*, 40, 19; (b) Rowlinson, J.S. and Swinton, F.L. (1982) *Liquids and Liquid Mixtures*, 3rd edn, Butterworth, London.

13 Dymond, J.H. and Alder, B.J. (1968) *Chem. Phys. Lett.*, 2, 54.

14 Berendsen, H.J.C. (2007) *Simulating the Physical World*, Cambridge University Press, p. 187.

15 Sherwood, A.E. and Prausnitz, J.M. (1964) *J. Chem. Phys.*, 41, 429.

16 Parson, J.M., Siska, P.E., and Lee, Y.T. (1972) *J. Chem. Phys.*, 56, 1511.

17 Barker, J.A., Fisher, R.A., and Watts, R.O. (1971) *Mol. Phys.*, 21, 657.

18 Clausius, R.J.E. (1870) *Philos. Mag. Ser. 4*, 40, 122.

19 London, F. (1937) *Trans. Faraday Soc.*, 33, 8.

20 Atkins, P.W. (2002) *Physical Chemistry*, 7th edn, Oxford

21 Butt, H.-J., Graf, K., and Kappl, M. (2006) *Physics and Chemistry of Interfaces*, 2nd edn, Wiley-VCH.

22 Israelachvili, J. (1991) *Intermolecular and Surface Forces*, 2nd edn, Academic Press.

Further Reading

Grabowski, S.J. (2006) *Hydrogen Bonding – New Insights*, Springer, Dordrecht.

Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B. (1954) *Molecular Theory of Gases and Liquids*, John Wiley & Sons, Inc., New York (corrected edition, 1964).

Kaplan, I.G. (2006) *Intermolecular Interactions – Physical Picture, Computational Methods and Model Potentials*, John Wiley & Sons, Ltd, Chichester.

Maitland, G.C., Rigby, M., Smith, E.B., and Wakeham, W.A. (1981) *Intermolecular Forces – Their Origin and Determination*, Clarendon, Oxford.

Margenau, H. and Kestner, N.R. (1971) *Theory of Intermolecular Forces*, Pergamon, Oxford.

Parsegian, V.A. (2006) *Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers and Physicists*, Cambridge University Press, Cambridge.

Pimentel, G.C. and McLellan, A.L. (1960) *The Hydrogen Bond*, W.H. Freeman and Company, New York.

Stone, A.J. (1996) *The Theory of Intermolecular Forces*, Clarendon, Oxford.