10.3
Reaction Mechanisms: The Steady-State Approximation
The steady-state approximation, also referred to as the quasi-steady-state approximation to differentiate it from a true steady state, is a widely use…
Many reactions proceed through multiple elementary steps involving reactive intermediates. Adding these steps gives the overall reaction without intermediates.
Consider a reaction where reactant, R, forms product, P, through an intermediate, I.
Here, the first step is fast and reversible. The second step is a slow and rate-determining step that controls the overall rate.
Initially, the concentration of I rises quickly. It reaches a small maximum, then decreases and stabilizes at a low, nearly constant value. So, its concentration remains negligible compared to R and P.
Note that I is formed during the first step. However, it is consumed during the reverse of the first step and the second step of the reaction.
The steady-state approximation assumes that the concentration of I remains constant. So, the rate of formation of I equals its rate of consumption.
Solving for I and substituting it into the overall rate law yields the final rate expression with no intermediates. This rate law is useful for reactions that proceed through multiple elementary steps.
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Q1: What is the steady-state approximation and why is it useful for complex reactions?
The steady-state approximation assumes that reactive intermediates maintain a low, nearly constant concentration during a reaction. This simplifies calculations for multi-step reactions involving reverse reactions, making them analytically solvable. It reduces mathematical complexity significantly compared to solving full differential equations for each elementary step.
Q2: How does an intermediate's concentration change during a reaction?
Initially, intermediate concentration rises quickly to a small maximum, then decreases and stabilizes at a low, nearly constant value. This occurs because the intermediate is formed in the first step but consumed by both the reverse reaction and subsequent steps. After the induction period, its concentration remains negligible compared to reactants and products.
Q3: What does it mean when d[I]/dt equals zero in the steady-state approximation?
Setting d[I]/dt = 0 means the rate of intermediate formation equals its rate of destruction. The intermediate reaches a quasi-steady state where its concentration changes negligibly. This mathematical assumption allows you to solve for the intermediate's concentration and substitute it into the overall rate law to eliminate intermediates.
Q4: How do you derive the overall rate law using the steady-state approximation?
Set the rate of formation of the intermediate equal to its rate of consumption, then solve for the intermediate's concentration. Substitute this expression into the rate law for the rate-determining step. The result is a final rate expression containing only reactants and products, with no intermediates appearing in the equation.
Q5: Why is the steady-state approximation called quasi-steady-state?
The term quasi-steady-state distinguishes this approximation from a true steady state. The intermediate concentration is not truly constant throughout the entire reaction; it rises during an initial induction period before stabilizing. The approximation assumes negligible change only after this induction period, when the intermediate reaches its low, stable concentration.
Q6: How does the steady-state approximation relate to equilibrium constants?
The steady-state approximation aligns with using equilibrium constants of the first elementary process. When you solve for intermediate concentration under steady-state conditions, the resulting rate law incorporates the equilibrium constant from the fast, reversible first step. This consistency shows that both approaches yield equivalent rate expressions for multi-step reactions.
Q7: What role does the rate-determining step play in the steady-state approximation?
The rate-determining step controls the overall reaction rate and determines which rate law you substitute the intermediate concentration into. The slow step's rate law, combined with the steady-state expression for the intermediate, produces the final overall rate law. This approach works for reaction mechanisms where the rate-determining step follows one or more fast, reversible steps.