3.9. The quasi steady-state approximation (QSSA)

The quasi-steady-state approximation (QSSA) in chemical kinetics is a mathematical way of simplifying the differential equations describing some chemical kinetic systems (Seinfeld and Pandis, 2006). QSSA provides an assumption that there is no change in concentrations in time for all intermediates (). We will illustrate this concept using the Chapman mechanism (R3.18-21). The first step (Step I) is the determination of the reaction rates (3.4) for all elementary steps in the mechanism. In our example this will be:

(R3.18)

(R3.19)

(R3.20)

.

(R3.21)

The second step (Step II) of QSSA is the determination of the intermediates from the mechanism. In this example we can relatively easily find the intermediates; O and O3 are the intermediate species. In the next step (Step III) we should express the rate of the concentration change for all intermediates, to do this task we should use the definition of reaction rate (3.2). From reactions (R3.18-21) the rates of formation of O and O3 can be expressed as

,

(3.15)

.

(3.16)

We can use the steady-state approximation to solve for the concentration of O and O3 (Step IV). The steady state approximation assumes that after an initial time period, the concentration of the reaction intermediates remain a constant with time, i.e. the rate of change of the intermediate’s concentration with time is zero. Hence, using the steady state approximation:

,

(3.17)

thus

,

(3.18)

.

(3.19)

We can see that using the QSSA two ordinary differential equations can be reduced to two algebraic equations. Solving algebraic equations either numerically or analytically is much easier compared to ordinary differential equations. This fact is a power of the QSSA. We can solve for and :

,

(3.20)

.

(3.21)

Equation (3.21) can be rearranged:

.

(3.22)

These two equations (3.21) and (3.22) provide the steady state concentration for the intermediates. If we know that ~ 107 molecule/cm3 and ~ 1013 molecule/cm3 and , we find that . Hence the equation (31) can be simplified to

.

(3.23)

Equation (3.23) indicates that the concentration of O3 in the atmosphere depends proportionately on the rate of reaction (R.319) and inversely on k3, the rate constant for the photolysis of ozone by the UV light from the Sun.