Introduction Learning Objectives Make sure you thoroughly understand the following essential ideas: Describe some of the major differences between the kinetics of reactions in the gas phase, compared with those in liquid solutions. What role do solvent cages play in solution kinetics? Explain the distinction between diffusion-control and activation-control of reaction rates in solutions. Chemical reactions can occur in all phases of matter and at the interfaces between phases. Reactions in the liquid phase are particularly important in living organisms and the chemistry laboratory. Here we concentrate on a microscopic model of how reactions occur in liquid solutions. We will primarily compare them to reactions in the gas phase. What's different about kinetics in liquid solutions? Most of the added complications of kinetics and rate processes in liquid solutions arise from the much higher density of the liquid phase. In a typical gas at atmospheric pressure, the molecules occupy only about 0.2 per cent of the volume; the other 99.8 percent is empty space. In a liquid, molecules may take up more than half the volume, and the "empty" spaces are irregular and ever-changing as the solvent molecules undergo thermal motions of their own. In a typical liquid solution, the solvent molecules massively outnumber the reactant solute molecules, which tend to find themselves momentarily (~10 –11 sec) confined to a "hole" within the liquid. This trapping becomes especially important when the solvent is strongly hydrogen-bonded as is the case with water or alcohol. When thermal motions occasionally release a solute molecule from this trap, it will jump to a new location. The jumps are very fast (10 –12 - 10 –13 sec) and short (usually a few solvent-molecule diameters), and follow an entirely random pattern, very much as in Brownian motion . Consider a simple bimolecular process A + B → products. The reactant molecules will generally be jumping from hole to hole in the solvent matrix, only occasionally finding themselves in the same solvent cage (hole) where thermal motions are likely to bring them into contact. 0 1 2 3 NaN NaN NaN NaN A pair of reactants end up in the same solvent cage, where they bounce around randomly and exchange kinetic energy with the solvent molecules. Eventually the two reactants form an encounter pair. If they fail to react the first time, they have many more opportunities during the lifetime of the cage. The products form and begin to move away from each other. Finally, after about 10–11 sec, the solvent cage breaks up and the products diffuse away. The process can be represented as \[A + B \rightarrow \{AB\} → \text{products}\] in which the \(\{AB\}\) term represents the caged reactants including the encounter pair and the activated complex. Contrast this scenario with a similar reaction taking place in the gas phase; the molecules involved in the reaction will often be the only ones present, so a significant proportion of the collisions will be \(A\)-\(B\) encounters. However, if the collision should fail to be energetically or geometrically viable, the reactant molecules fly apart and are unlikely to meet again anytime soon. In a liquid, however, the solute molecules are effectively in a constant state of collision — if not with other reactants, then with solvent molecules which can exchange kinetic energy with the reactants. So once an A-B encounter pair forms, the two reactants get multiple whacks at each other, greatly increasing the probability that they will obtain the kinetic energy needed to kick them over the activation hump before the encounter pair disintegrates. Limiting Cases: Diffusion-Controlled and Activation-Controlled Reactions The encounter pair model introduces some new rate parameters: \[\ce{A + B <=>[k_1][k_{-1}] {AB} ->[k_2] P}\] The first step is an equilibrium between the reactants outside and inside the solvent cage. The rate constants \(k_1\) reflects the rate of A and B diffusing together into the solvent cage to become an encounter pair, and \(k_{-1}\) the rate they diffuse back out, while k 2 is the rate at which the encounter pair becomes product. The values of the diffusion related constants are strongly dependent on the viscosity (and thus the temperature) of the solvent. (Note that k 1 is a second-order rate constant, while and k -1 and k 2 are first-order.) Diffusion is the transport of a substance through a concentration gradient; that is, from a region of higher concentration to one of lower concentration. Think of the way the color of tea spreads out when a tea bag is immersed in hot water. Diffusion occurs because random thermal motions (figure \(\PageIndex{2}\)) are statistically more likely to move molecules out of a region of higher concentration than in the reverse direction, simply because in the latter case fewer molecules are available to make the reverse trip. Eventually the concentrations become uniform and equilibrium is attained. This mechanism leads to the following rate laws for formation of the encounter pair and the products: \[\frac{d[\{AB\}]}{dt} = k_1 [A][B] -k_{-1}[\{AB\}] - k_2[\{AB\}]\label{Renc}\] \[\frac{d[P]}{dt} = k_2[\{AB\}]\label{Rprod}\] If we assume steady state in [{AB}] equation \(\ref{Renc}\) becomes: \[0 \approx k_1 [A][B] -k_{-1}[\{AB\}]_{ss} - k_2[\{AB\}]_{ss}\label{RencSS}\] Solving for \([\{AB\}]_{ss}\): \[[\{AB\}]_{ss} \approx \frac{k_1}{k_{-1} +k_2} [A][B]\label{EncSS}\] Substituting into equation \(\ref{Rprod}\) we get the dependence of the rate of product production on our rate constants and reactant concentrations: \[\frac{d[P]}{dt} = k_2\frac{k_1}{k_{-1} +k_2} [A][B]\label{Rprod2}\] There are two limiting cases: Diffusion Controlled ( \(k_2 \gg k_{-1}\) ): In this case the activation energy of the A+B reaction in the cage is very small or escape of molecules from the {AB} cage is difficult. This makes k -1 negligible in the denominator and equation \(\ref{Rprod2}\) simplifies: \[\frac{d[P]}{dt} = k_2\frac{k_1}{k_{-1} +k_2} [A][B] \approx \cancel{k_2}\frac{k_1}{\cancel{k_2}} [A][B] \approx {k_1} [A][B]\label{diffuslim}\] So, the kinetics will be dominated by k 1 , the rate the particles diffuse into the cage (by convention k 1 is often called k d , where the subscript stands for diffusion). Such a process is said to be diffusion controlled . This suggest there is an upper-limit for bimolecular reactions in solution set by the diffusion process. A careful analysis 1 of the rate of two spherical particles diffusing together provides an estimate for k 1 (or k d ): \[k_1 \approx \frac{8RT}{3\eta}\] where R = gas constant, T = temperature in K and \(\eta \) = the solvent viscosity. For water at 25 ˚C \(\eta \) = 0.891 x 10 - 3 kgm -1 s -1 (0.891 cP), which yields k 1 ≈ 7.4 x 10 6 m 3 mol -1 s -1 = 7.4 x 10 9 M -1 s -1 . So reactions with rate constants in water in the 10 10 M -1 s -1 to 10 11 M -1 s -1 are considered to be diffusion controlled, which in this model is the upper limit on the rate. An example is the acid-base reaction: \(H^+ + OH^– → H_2O \nonumber \) for which k = 1.4×10 11 dm 3 mol –1 s –1 at 298 K (reminder: 1 dm 3 = 1 L). Activation Controlled ( \(k_2 \ll k_{-1}\) ): Alternatively, if the activation energy of the A+B reaction in the cage dominates the kinetics, then k 2 in the denominator is negligible and equation \(\ref{Rprod2}\) simplifies: \[\frac{d[P]}{dt} = k_2\frac{k_1}{k_{-1} +k_2} [A][B] \approx k_2\frac{k_1}{k_{-1}} [A][B] \label{activlim}\] Thus the rate now depends on all three rate constants . These reactions have rates less than the diffusion limited reactions and are referred to as activation controlled . Keep in mind that the observed rate depends on the competition among the three processes. So the exact balance between the residence time of two particles in the cage (determined by k 1 and k -1 ) and the average time it takes the molecules in the cage to absorb enough energy to overcome the activation energy and come together in the right orientation to react (k 2 ) determines the obseved rate. References 1. P.W. Atkins, J. de Paula, Physical Chemistry , 10th Ed. (W.H. Freeman and Co. New York, 2014) P. 890.