Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
14. Some Special Topics: Reactions in Solutions
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
where A and B represents the reactants (R), and X and Y represents the products (P). The stoichiometric sum is defined by Σjνj = ΣPνP − ΣRνR. Thus, for Eq. (14.1) Σjνj becomes (νX + νY) − (νA + νB). Progress of the reaction as a function of time t is given by the extent of reaction ξ (t), defined by
where nj and νj denote the number of moles and the reaction coefficient for component j, respectively. The rate of reaction is defined by
So, for Eq. (14.1) we have (using the notation [Z] for the molarity of Z)
Often, the reaction rate r can be described by
with the (forward) rate constant kfor. 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)
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 rfor for the forward reaction νAA + νBB → νXX + νYY and the rate rrev for the reverse reaction νAA + νBB ← νXX + νYY; that is,
At equilibrium r = 0 or with Kc the equilibrium constant (in terms of molarities2)). For gas-phase reactions often the equilibrium constant KP in terms of pressure P is used. The partial pressure is Pj = (Nj/V)kT = cjkT, where k is Boltzmann's constant and cj is the molarity of component j. Hence, we have
The rate equations can be integrated, and this yields the concentration as a function of time explicitly. We have, defining cj,0 ≡ cj(t = 0) for component j,
Various methods exist to determine the order of the reaction from experimental data (see, e.g., Laidler , Connors , or Arnaut et al. ).
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 KP or Kc with temperature is given by the van't Hoff equation
where ΔH° and ΔU° are the enthalpy and energy under standard conditions, respectively. Empirically, the temperature dependence of the rate constant krea for limited temperature range can be described by the Arrhenius equation
where A is the pre-exponential factor, dependent on temperature T, and Eact is the activation energy. For a wider temperature range one often uses the modified Arrhenius equation
where A′ is a temperature-independent factor. The range of Eact 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 .