Theory and Energetics

BY R. A. W. JOHNSTONE

This chapter covers theoretical developments in ion chemistry particularly with regard to application of theory to experimental work and the reflective effect of the latter on the development of theory. The last few years have seen renewed interest in the application of theory to the understanding of ion fragmentation and ion/neutral interactions. Frequently, developments in one area of mass spectrometry have spread over into others so that, although this chapter is divided into sections, extensive cross-references to other sections have had to be made to achieve overall coverage. For this reason, the division into sections has a strong flavour of convenience rather than strict logic but it is hoped the subject matter is more readily digestible and comprehensible treated this way. Photoelectron photoion coincidence spectroscopy is covered by the special review in Chapter 3 and is not dealt with as such here although results are referred to when necessary. The use of molecular orbital theory in mass spectrometry was comprehensively reviewed in Volume 4 of this series and, using criteria developed there, only the more significant applications and developments since then are reviewed in this Chapter.

1 Thermochemical Aspects

By well-known energy cycles, thermochemical data, such as heat of reaction and heat of formation, are interdependent in that one can be derived from others. Therefore, the division of this section into sub-sections describing for example, heats of formation, electron affinities, and proton affinities separately is artificial but has been done for convenience in dealing with the literature and for emphasizing particular points of interest.

A valuable compilation of thermochemical data for gaseous ions has appeared and a review on studies of metastable ions which lists advantages of their use as ions of low internal energy for determining thermochemical thresholds.

Free Energies of Reaction. — Total free energy changes in a reaction (ΔITL [Gθ) are dealt with here in discussion of the derivation of heat of reaction (ΔITL [Hθ) from equilibrium measurements on ion/molecule reactions. Fragmentation of isolated ions is dealt with in the section on RRKM theory. Free energy and heat of reaction are linked through the equation, ΔGθ = ΔHθ - TΔSθ, where T is temperature and ΔSθ the change in entropy. Also, - ΔGθ = RT ln K, in which K is the equilibrium constant for reaction. Usually, ΔSθ is very small and put equal to zero so that, - ΔGθ [congruent to] ΔHθ. An empirical correlation of exothermicity and activation energy is discussed later.

Equilibrium constants determined from ion/molecule reactions may be in error through competitive reaction, differential ion losses, or slow arrival at equilibrium. Suitable methods and precautions for obtaining equilibrium constants from measurements in ICR cells have been discussed. A further source of error is the extraction and trapping fields in many types of apparatus which give a non-Maxwell–Boltzmann distribution to the ions, i.e. the ions have an effective temperature greater than ambient. This point is discussed further for ICR and SIFf techniques in the section on ion/molecule reaction and for ion mobilities in the section of that name. After emphasizing this possible error, a satisfactory way of removing it from time-resolved experiments has been described. Equilibrium constants are measured for different values of the ratio, E/P, in which E is the strength of the electric extraction field and P, the pressure of gas in the apparatus. The value of K and therefore ΔGθ is obtained by extrapolation to E/P equal zero; this technique provided a heat of reaction for CO2H+(CH4, CO2)CH5+ in excellent agreement with other work using the flowing afterglow and ICR methods.

The common practice of putting ΔSθ = 0 for ion/molecule reactions has been examined and found satisfactory. Using the unimolecular reaction rate equation, K = ZP exp (E/RT), and making one or two assumptions, these authors showed that ΔSθ could be estimated through the expression, ΔSθ = R ln (Zf/Zp), where Zf, Zp are the collision rate constants for forward and back reactions; these collision rate constants can be calculated (see section on ion/molecule reactions). Estimates of ΔSθ for a number of reactions were shown to be small, of the right order of magnitude, and in the right direction.

Failure to ensure ions have been thermalized, i.e. have internal and kinetic energies corresponding to ambient temperatures, is a cause for concern when determining thermochemical quantities from equilibrium measurements. It has been shown that, at least for H+-transfer, the reaction, BH+ + B [??] [BHB+]* [??] B + BH+, is so efficient that the BH+ ions are rapidly relaxed.

The sign of ΔHθ can be inferred in ICR experiments from the variation of the double-resonance signal with variation in the irradiating field strength. Thus, by use of bracketing reactions, upper and lower limits can be set for ΔHθ; this point is illustrated in the later section on proton affinities.

All of these thermochemical quantities are determined in the dilute gas-phase in which there are no solvent effects (heat of solvation, dielectric, viscocity, and so on). However, solvent can greatly influence both the extent and nature of a reaction, with products changing and also rates by orders of magnitude from the gas-phase reaction. Further, ionic reactions in solution always have a gegenion which itself can modify the reaction. Bridging the gap between gas-phase and solution-phase has been attempted with considerable success. For example, the interaction of clusters of CH3CN molecules with Na+, K+, Rb+, or Cs+ has been examined, with ΔGθ, ΔHθ, and...