## 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

__(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* _{j}ν_{j}* = Σ

_{P}

*ν*

_{P}− Σ

_{R}

*ν*

_{R}. Thus, for

__Eq. (14.1)__Σ

*becomes (*

_{j}ν_{j}*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*denote the number of moles and the reaction coefficient for component

_{j}*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* = ∂ ln*r*/∂ 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 exceptions__ ^{1)}__) they have the same value.

The total reaction rate *r* is the difference between the rate *r*_{for} for the forward reaction *ν*_{A}A + *ν*_{B}B → *ν*_{X}X + *ν*_{Y}Y and the rate *r*_{rev} for the reverse reaction *ν*_{A}A + *ν*_{B}B ← *ν*_{X}X + *ν*_{Y}Y; that is,

(14.7)

At equilibrium *r* = 0 or with *K _{c}* the

*equilibrium constant*(in terms of molarities

__). For gas-phase reactions often the equilibrium constant__

^{2)}*K*in terms of pressure

_{P}*P*is used. The partial pressure is

*P*= (

_{j}*N*/

_{j}*V*)

*kT*=

*c*, where

_{j}kT*k*is Boltzmann's constant and

*c*is the molarity of component

_{j}*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*with temperature is given by the

_{c}*van*'

*t Hoff equation*

(14.9)

where Δ*H*° and Δ*U*° 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].