The Virial Theorem - Basic Energetics: Intermolecular Interactions - Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

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 = dG/dt 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_i, velocity v_i, and momentum p_i = m_iv_i. Because T_i is a homogeneous function of the second degree in v_i, we have by Euler's theorem (see Appendix B)

(3.34)

Since ∂T_i/∂v_i = p_i, we further obtain

(3.35)

If the system has a finite volume, p_i·r_i is bounded and thus we have ⟨dp_i·r_i/dt⟩ = 0. The kinetic energy T is linked to the potential energy Φ via Newton's Second Law reading f_i = m_ia_i with force f_i and acceleration a_i. Using f_i = −∂Φ/∂r_i, m_ia_i = dp_i/dt, 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⟩ = 2U/(n + 2) and ⟨T⟩ = nU/(n + 2). For harmonic potentials n = 2 resulting in ⟨T⟩ = ⟨Φ⟩ = ½U. 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_i or 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 C² m² J⁻¹. Incorporating (4πε₀)⁻¹ in α via α′ = α/(4πε₀) renders the dimension of α′ to be m³. Since, according to electrostatics, the polarizability of a perfectly conducting sphere of radius r in vacuum is given by 4πε₀r³, the molecular radius r can be estimated from r³ = 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.