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

An important driving force for discussing liquids, apart from their intrinsic scientific interest, is their use as solvent for reactions. In this chapter reactions in solution are dealt with. First, we briefly review the basics of kinetics, and thereafter deal with transition state theory for gas-phase reactions. In liquid media, the transport of reactants is important, and thus we briefly review the effects of diffusion and viscosity. The transition from diffusion-controlled to chemically controlled reactions is indicated. Finally, the physical effect of solvents on reactions is discussed.

14.1. Kinetics Basics

For the description of reactions some terms and concepts are required which we briefly iterate here. A typical reaction may be written as

(14.1)

where A and B represents the reactants (R), and X and Y represents the products (P). The stoichiometric sum is defined by Σ_jν_j = Σ_Pν_P − Σ_Rν_R. Thus, for Eq. (14.1) Σ_jν_j becomes (ν_X + ν_Y) − (ν_A + ν_B). Progress of the reaction as a function of time t is given by the extent of reaction ξ (t), defined by

(14.2)

where n_j and ν_j denote the number of moles and the reaction coefficient for component j, respectively. The rate of reaction is defined by

(14.3)

So, for Eq. (14.1) we have (using the notation [Z] for the molarity of Z)

(14.4)

Often, the reaction rate r can be described by

(14.5)

with the (forward) rate constant k_for. The reaction order n is given by n = α + β, in which α and β are the partial reaction order. It must be stated that r is not necessarily described by a power law. For example, it may also be described by (with P and Q constants)

(14.6)

If [S] << Q we obtain r = P[S]/Q and the apparent order n = 1. On the other hand, if Q << [S] we have r = P and the apparent order n = 0. For an expression like Eq. (14.6) the reaction order is defined by α = ∂ lnr/∂ ln[S]. In many cases a reaction is discussed in terms of elementary reactions such as A + BC ↔ AB + C. More complex reaction schemes, involving two or more elementary reactions, are accordingly denoted as composite reactions. Finally, with molecularity the number of molecules participating in the reaction is indicated. Clearly, for the reaction as given above the molecularity is two. Molecularity and reaction order generally have different values, but for elementary reactions (with certain exceptions¹⁾) they have the same value.

The total reaction rate r is the difference between the rate r_for for the forward reaction ν_AA + ν_BB → ν_XX + ν_YY and the rate r_rev for the reverse reaction ν_AA + ν_BB ← ν_XX + ν_YY; that is,

(14.7)

At equilibrium r = 0 or with K_c the equilibrium constant (in terms of molarities²⁾). For gas-phase reactions often the equilibrium constant K_P in terms of pressure P is used. The partial pressure is P_j = (N_j/V)kT = c_jkT, where k is Boltzmann's constant and c_j is the molarity of component j. Hence, we have

(14.8)

The rate equations can be integrated, and this yields the concentration as a function of time explicitly. We have, defining c_j,0 ≡ c_j(t = 0) for component j,

Various methods exist to determine the order of the reaction from experimental data (see, e.g., Laidler [1], Connors [2], or Arnaut et al. [3]).

The rate of a reaction is influenced by several variables. Apart from the molarity [X] of component X, the temperature T is important. The change of equilibrium constants K_P or K_c with temperature is given by the van't Hoff equation

(14.9)

where ΔH° and ΔU° are the enthalpy and energy under standard conditions, respectively. Empirically, the temperature dependence of the rate constant k_rea for limited temperature range can be described by the Arrhenius equation

(14.10)

where A is the pre-exponential factor, dependent on temperature T, and E_act is the activation energy. For a wider temperature range one often uses the modified Arrhenius equation

(14.11)

where A′ is a temperature-independent factor. The range of E_act for reactions in liquids is typically 40 to 120 kJ mol⁻¹, which implies an increase in rate by a factor of 2 to 3 for every 10 K. The value of the parameter m can be positive as well as negative, but typically |m| < 2.

Furthermore, the reaction medium is relevant – that is, the type of solvent and ionic strength. We discuss these effects in Section 14.3. Finally, light and catalysts are important, but for the effect of these we refer to the literature [3].