# Chemistry - General justification for steady state approximation

**No very satisfying justifications are available unless you do a case-by-case analysis. But PSSA does not come from common sense.** In general, pseudostationary state approximation (PSSA) is justified by the theory of differential equations.

In fact, the pseudostationary state concentrations represent the zeroth approximation to a set of particular solutions of the correct chemical kinetic equations, which are called principal solutions. After the initial period, the concentrations of the intermediates approach their principal solution rather than their pseudostationary state values.

Therefore, the accuracy or usefulness of the pseudostationary state assumption depends upon the magnitude of the differences between the principal solution and the pseudostationary state concentrations.

OK, that sounds useless. But after doing some maths (like finding the principle solutions), some (obviously very ambiguous) rules are indeed justified mathematically.

Induction period should be short. (As you can see PSSA solutions cannot satisfy the initial conditions)

Destruction of the intermediates should be fast.

For example, in $\ce{A ->[$k_1$] B ->[$k_2$] C}$ if $k_2\gg k_1$ then the conditions above are met while if $k_1\gg k_2$, they aren't.

By the way, PSSA originates from Michaelis-Menton theory for enzyme kinetics. PSSA in it is well justified while the conditions above are not necessarily met. Consider

$$ \ce{E + S <=>[$k_1,k_{-1}$] ES ->[$k_2$] P}. $$

Let $[\ce{S}]=s,[\ce{E}]=e,[\ce{ES}]=c,[\ce{P}]=p,e+c:=e_0,s+p+c:=s_0$. Furthermore, let $\tau=k_1e_0t,u(\tau)=s/s_0,v(\tau)=c/e_0$. The kinetic equations can be written as

$$ \begin{cases} \frac{\mathrm du}{\mathrm d\tau}=-u+(u+K-\lambda)v\\ \color{red}{\varepsilon}\frac{\mathrm dv}{\mathrm d\tau}=u-(u+K)v \end{cases} $$

where $\lambda=k_2/\left(k_1s_0\right),K=\left(k_{-1}+k_2\right)/\left(k_1s_0\right),\color{red}{\varepsilon=e_0/s_0\ll 1}$. Therefore, PSSA can be applied (i.e. setting $u-(u+K)v=0$, justified by perturbation methods) without loss of much accuracy. This example indicates that more often than not, a case-by-case analysis is required.

Finally, I think to use PSSA safely, one should do some experiments to check (numerical simulations, real experiments, etc.).