Lecture 6 The Steady State Approximation

6.1 Introduction

In our previous lectures, we explored how complex reaction mechanisms can be analyzed by breaking them down into elementary steps and deriving their rate laws. However, for many reactions, the complete kinetic analysis becomes mathematically intractable. In cases where the reaction intermediates are highly reactive and are quickly converted into products, we can apply a simplifying approximation called the Steady State Approximation, which often allows us to simplify the differential rate equations to derive an approximate mathematical description of the overall reaction rate.

Consider a reaction pathway involving a reactive intermediate B:

\[\mathrm{A} \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} \mathrm{B} \xrightarrow{k_2} \mathrm{C}\]

The concentration of intermediate B changes through three competing processes: - Formation from reactant A (rate \(k_1[\mathrm{A}]\)) - Back-reaction to A (rate \(k_{-1}[\mathrm{B}]\)) - Forward reaction to product C (rate \(k_2[\mathrm{B}]\))

# Figure showing competing processes affecting [B]
# - Blue arrow showing formation from A 
# - Red arrow showing back-reaction to A
# - Green arrow showing conversion to C

Under steady state conditions, the intermediate’s concentration remains approximately constant because its rate of formation equals its rate of consumption. Mathematically, this means:

\[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} \approx 0\]

This approximation doesn’t imply that [B] = 0, but rather that [B] rapidly reaches and maintains a nearly constant value throughout most of the reaction.

6.2 Analysis of Pre-equilibrium Systems

Let’s examine a mechanism involving both equilibrium and steady state behavior:

\[\mathrm{A} + \mathrm{B} \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} \mathrm{C} \xrightarrow{k_2} \mathrm{D}\]

The complete set of rate equations for this system is:

\[\begin{align*} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} &= -k_1[\mathrm{A}][\mathrm{B}] + k_{-1}[\mathrm{C}] \\ \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} &= -k_1[\mathrm{A}][\mathrm{B}] + k_{-1}[\mathrm{C}] \\ \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{d}t} &= k_1[\mathrm{A}][\mathrm{B}] - k_{-1}[\mathrm{C}] - k_2[\mathrm{C}] \\ \frac{\mathrm{d}[\mathrm{D}]}{\mathrm{d}t} &= k_2[\mathrm{C}] \end{align*}\]

# Plot showing concentration profiles over time
# - [A] and [B] decreasing
# - [C] reaching steady state rapidly
# - [D] increasing as product forms

6.2.1 Application of the Steady State Approximation

We apply the steady state approximation to intermediate C:

\[\frac{\mathrm{d}[\mathrm{C}]}{\mathrm{d}t} = k_1[\mathrm{A}][\mathrm{B}] - k_{-1}[\mathrm{C}] - k_2[\mathrm{C}] \approx 0\]

This equation can be rearranged:

\[k_1[\mathrm{A}][\mathrm{B}] = (k_{-1} + k_2)[\mathrm{C}]\]

Solving for [C]:

\[[\mathrm{C}] = \frac{k_1[\mathrm{A}][\mathrm{B}]}{k_{-1} + k_2}\]

The overall reaction rate becomes:

\[\begin{align*} \nu &= \frac{\mathrm{d}[\mathrm{D}]}{\mathrm{d}t} = k_2[\mathrm{C}] \\ &= k_2\left(\frac{k_1[\mathrm{A}][\mathrm{B}]}{k_{-1} + k_2}\right) \\ &= \frac{k_2k_1[\mathrm{A}][\mathrm{B}]}{k_{-1} + k_2} \end{align*}\]

6.2.2 Analysis of Limiting Cases

Two important limiting behaviors emerge from this rate law:

Pre-equilibrium Limit (\(k_{-1} \gg k_2\)): \[\nu = \frac{k_2k_1}{k_{-1}}[\mathrm{A}][\mathrm{B}]\] The reaction exhibits second-order kinetics with an effective rate constant \(k_\mathrm{eff} = k_2k_1/k_{-1}\).
Rate-determining Step Limit (\(k_2[\mathrm{C}] \gg k_{-1}\)): \[\nu = k_1[\mathrm{A}]\] The reaction shows first-order kinetics with rate constant \(k_1\).

# Side-by-side plots comparing concentration profiles
# Left: Pre-equilibrium case
# Right: Rate-determining step case

6.3 Mathematical Foundation of Steady State Behavior

Consider a general reaction system where an intermediate species I is: - Produced with rate \(\nu_f\) - Consumed with rate proportional to [I]

The rate equation is:

\[\frac{\mathrm{d}[\mathrm{I}]}{\mathrm{d}t} = \nu_f - k[\mathrm{I}]\]

This linear first-order differential equation has the solution:

\[[\mathrm{I}]_t = \frac{\nu_f}{k}(1 - e^{-kt})\]

This solution shows that: 1. [I] approaches steady state value \(\nu_f/k\) exponentially 2. Characteristic time to reach steady state is \(1/k\) 3. Steady state approximation is valid when \(t \gg 1/k\)

# Plot showing:
# - Exponential approach to steady state
# - Characteristic time marked
# - Steady state value indicated

6.4 Validation Criteria for Steady State

The steady state approximation is valid when: 1. The intermediate concentration remains small throughout the reaction 2. The intermediate reaches steady state rapidly compared to overall reaction time 3. The steady state persists for most of the reaction duration

# Side-by-side plots:
# Left: Valid steady state case
# Right: Invalid case where approximation fails

These criteria can be assessed by comparing characteristic timescales: - \(\tau_\mathrm{SS} = 1/k\) for reaching steady state - \(\tau_\mathrm{rxn}\) for overall reaction progress

The approximation is valid when \(\tau_\mathrm{SS} \ll \tau_\mathrm{rxn}\).