Steady State Approximation

The steady-state-approximation is a mathematical technique used to simplify the analysis of a system that is in a steady state. The technique assumes that the system is in a steady state and that the only changes occurring in the system are the result of the input and output of the system. This allows the analyst to ignore the internal dynamics of the system and to focus on the system’s behavior over time. Steady State Approximation – Reaction Mechanism and Examples.

When a reaction mechanism has several steps of comparable rates, the rate-determining step is often not obvious. However, there is an intermediate in some of the steps. An intermediate is a species that is neither one of the reactants, nor one of the products. The steady-state approximation is a method used to derive a rate law. The method is based on the assumption that one intermediate in the reaction mechanism is consumed as quickly as it is generated. Its concentration remains the same in a duration of the reaction. Steady State Approximation – Reaction Mechanism and Examples.

The steady state approximation is applies to a consecutive reaction with a slow first step and a fast second step ( $k_{1} ≪ k_{2}$

k_{1} ≪ k_{2}

). If the first step is very slow in comparison to the second step, there is no accumulation of intermediate product, such as product B in the above example.

\begin{matrix} (3.2.6.3) & \frac{d [B]}{d t} = 0 = k_{1} [A] - k_{2} [B] \end{matrix}

Thus

\begin{matrix} (3.2.6.4) & [B] = \frac{k_{1} [A]}{k_{2}} \end{matrix}

From the mechanism:

\begin{matrix} (3.2.6.5) & \frac{d [C]}{d t} = k_{2} [B] = \frac{k_{2} k_{1} [A]}{k_{2}} = k_{1} [A] \end{matrix}

Solving for $[C]$

[C]

\begin{matrix} (3.2.6.6) & [C] = [A]_{0} (1 - e^{- k_{1} t}) \end{matrix}

Equation $3.2.6.6$

3.2.6.6

is much simpler to derive than Equation $3.2.6.2$

3.2.6.2

, especially with a more complicated multi-step reaction mechanisms.

Steady state approximation

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