# What is the non-ideal equivalent of mass action kinetics?

Starting from the equations of mass action kinetics, one may show that (for suitably chosen numbers $A_i$), the function $\sum_i x_i(A_i + \log x_i)$ is a Lyapunov function, i.e. it decreases over time.

Apart from an irrelevant factor of $RT$, this is the expression for the free energy of an ideal solution (or ideal gas, etc). So it seems that mass action kinetics carries with it an assumption that the reactions are taking place in an ideal solution.

I'm interested in how to think about kinetics in non-ideal solutions, but I haven't been able to find any information about it. Is there a standard set of kinetic equations that remain valid when the chemical potential is given not by $\mu_i^0 + RT\log x_i$ but by $\mu_i^0 + RT\log \gamma_i x_i$, for activity coefficients $\gamma_i$ that depend on the concentrations?

There are several alternatives to mass action kinetics that I know about, such as Michaelis Menton kinetics for enzymes - but these are actually derived from mass action kinetics applied to elementary steps, so they don't seem to be able to escape from the assumption of ideality.

• I think you probably mean "non-unity activity coefficients" where you currently say "nonzero". Also, keep in mind that what makes the problem difficult is that $\gamma_i$ is a function of all of the $x_i$ and $x_j$. If the $\gamma_i$ where non-unity but constant, the problem easily reduces to mass action kinetics. The devil is in the details of the activity coefficient model you use to relate the $\gamma_i$ to the $x_i$ and $x_j$. – Curt F. Oct 8 '16 at 13:54
• @CurtF. thanks, I've updated that part of the question. – Nathaniel Oct 8 '16 at 14:38

According to Catalytic reaction rates in thermodynamically non-ideal systems Journal of Molecular Catalysis A: Chemical 163 (2000) 189–204 (citing to kinetics textbooks) non-ideal kinetics needs to consider the activity of the transition state in addition to the activities of the reactants:

For $\ce{A + B ->}$

rate $=\frac{k_BT}{h}K^{\ddagger}\frac{a_Aa_B}{\gamma_\ddagger}=k_0\frac{\gamma_A\gamma_B}{\gamma_\ddagger}C_AC_B$

Where $K^{\ddagger}$ is the equilibrium constant for the formation of the transition state, $a$ is activity, $\gamma$ is activity coefficient, $C$ is concentration, $k_B$ is Blotzmann constant, $T$ is temperature, $h$ is Planck's constant and $k_0$ is the thermodynamically ideal rate constant.

Another reference to look at for more info is The status of transition-state theory in non-ideal solutions and application of Kirkwood–Buff theory to the transition state J. Chem. Soc., Faraday Trans. 2, 1986,82, 1297-1303.

• Interesting, thank you. This makes a lot of sense. If you take the first equality and lump all the constants into a single constant $k$, it amounts to a mass action type equation, but using the activities in place of the concentrations, which is conceptually very striaghtfoward. – Nathaniel Oct 13 '16 at 2:20

You can formulate the kinetics using extent. The extent can be related to the chemical potentials (through the Gibbs free energy). The activity coefficients might have a concentration dependencies, and the changes of extent is describing the changes of concentration.

Doing so for ideal gasses, you will derive mass-action kinetics.

• I am familiar with the extent of reaction, but usually when formulating kinetics in terms of extent, we start by making a mass action assumption, no? (There is nothing on the wikipedia page you linked to that relates to kinetics.) – Nathaniel Oct 12 '16 at 9:24