Steady state (chemistry)

For other uses of the term steady state, see steady state (disambiguation)

In Chemistry, steady state is an approximation, where, after an initial period where the concentration increases (or decreases), the concentration of an intermediate chemical species in a series of consecutive reactions is assumed to be constant. That is: $\frac{d$ ^*}{dt} \approx \; 0 . The steady state approximation is used in many chemical mechanisms, like Michaelis-Menten kinetics, to make calculations easier, otherwise many differential equations arising from rate equations lack an analytical solution.

Application

As an example, the steady state approximation will be applied to three consecutive first order reactions. This theoretical model corresponds, for example, to a series of nuclear decompositions like ${}^{239}U \rightarrow \; {}^{239}Np \rightarrow \; {}^{239}Pu\!$ .

If the rate constants for the following reaction are $k_1$ and $k_2$ ; $A \rightarrow \; B \rightarrow \; C$ , then the rate equations are:

Reaction rates

For reactant A: $\frac{d = -k_1 [A$

For reactant B: $\frac{d = k_1$ ^*

For reactant C: $\frac{d = k_2 [B$

Analytical Solutions

The analytical solutions for these equations (supposing that initial concentrations of every substance except A are zero) are

^*=^*_0 e^{-k_1 t}

^*=^*_0 \frac{k_1}{k_2 - k_1}\left ( e^{-k_1t}-e^{-k_2t} \right )

$= \frac{k_2 \left ( 1- e^{-k_1t} \right ) - k_1 \left (1- e^{-k_2t} \right ) \right \quad = [A" target="_blank" >$ ^*_0 \left (1 + \frac{k_1 e^{-k_2t}-k_2e^{-k_1t}}{k_2-k_1} \right )

Steady State

If we assume the steady state approximation then, the concentration of the intermediate is almost constant so its derivative is close to zero.

$\frac{d = 0 = k_1$ ^*" target="_blank" >\Rightarrow \; ^*" target="_blank" >therefore $\frac{d$ ^*" target="_blank" >so ^*_0 \left (1- e^{-k_1 t} \right ).

Validity

The analytical and approximated solutions should now be compared in order to decide when it is valid to use the steady state approximation. The analytical solution transforms into the approximated one when $k_2 >> k_1$ , because then $e^{-k_2t} << e^{-k_1t}$ and $k_2-k_1 \approx \; k_2$ .

Therefore it is only valid to apply the steady state approximation when the second reaction is much faster than the first ( $\frac{k_2}{k_1}>10$ is a right criterium)

Consecutive_reactions_rate_constants_1-10.JPG

Sources

P. W. Atkins "Physical Chemistry"