Reactivity 2.2.9 – Rate Equations

Rate of Reaction:
The rate of reaction is defined as the change in concentration of a reactant or product per unit time. It gives a measure of how quickly reactants are converted into products in a chemical reaction.

Mathematically:

\( \text{Rate} = \frac{-\Delta [\text{Reactant}]}{\Delta t} = \frac{+\Delta [\text{Product}]}{\Delta t} \)

Negative sign for reactants indicates their concentration is decreasing; positive for products indicates increase.

Units of Rate: mol·dm⁻³·s⁻¹ (concentration per unit time)

Average vs Instantaneous Rate:

• Average rate: Calculated over a time interval.

• Instantaneous rate: The rate at a specific moment, found using the tangent to the curve on a graph of concentration vs time.

How to Measure Rate:

• Change in mass (for reactions releasing a gas)
• Change in volume of gas produced (using a gas syringe)
• Change in conductivity (for ionic reactions)
• Change in colour (colorimetric methods)
• Precipitate formation (time taken for cross to disappear in reactions like sodium thiosulfate + HCl)

Definition of Rate Equation:

The rate equation (or rate law) is an experimentally determined expression that relates the rate of a chemical reaction to the concentration of the reactants.

General form:

Rate = \( k [A]^m [B]^n \)

• \( k \) = rate constant (depends on temperature and reaction conditions)

• \( [A], [B] \) = concentrations of reactants A and B
• \( m, n \) = reaction orders with respect to A and B (must be found experimentally)

Key Concepts:

• The rate equation reflects the actual mechanism of the reaction, not just the stoichiometry of the balanced chemical equation.
• It indicates which reactants are involved in the rate-determining step and how they influence the rate.
• The reaction order (e.g. first, second, zero) tells how the rate depends on the concentration of each species.
• Only the species in the slow (rate-determining) step appear in the rate law.

Important Distinctions:

• Reaction stoichiometry shows the proportions of reactants and products overall.
• Reaction mechanism breaks it into elementary steps, revealing the actual path and molecularity.
• The rate equation must be determined by experiment — it cannot be predicted from the overall equation alone unless the mechanism is fully known.

Why Experimental Determination Is Necessary:

• In some reactions, a reactant that appears in the overall equation may not influence the rate (zero order).
• Sometimes intermediates or catalysts are involved that affect the rate but don’t appear in the net equation.
• Different mechanisms can lead to the same overall reaction, but have different rate laws.

Reaction Orders:

Units of the Rate Constant:

The units of \( k \) depend on the overall order of the reaction:

Overall Order of Reaction: Sum of the powers of the concentration terms in the rate law:

\( \text{Overall order} = m + n \)

Common Experimental Techniques to Determine Rate Laws:

• Initial rate method: Compare how changing the initial concentrations affects the initial rate.
• Clock reactions: Measure time taken for a visible change and relate it to initial concentrations.
• Graphical analysis: Plot concentration vs time and examine the shape/slope to infer order.

Example of Importance:

For the reaction:

\( 2\text{NO} + \text{O}_2 \rightarrow 2\text{NO}_2 \)

Rate law: Rate = \( k[\text{NO}]^2[\text{O}_2] \)

This suggests the rate-determining step involves two NO molecules and one O₂ molecule, even though the balanced equation doesn’t show how those species interact mechanistically.

Deduce the Rate Equation from Experimental Data

Steps to Deduce the Rate Equation:

1. 1. Identify experiments where the concentration of one reactant changes while the others stay constant.
   2. Observe how the rate changes when this concentration changes.
   3. Determine the order with respect to that reactant by comparing the factor change in concentration to the factor change in rate:

If doubling [A] doubles rate → first order in A

If doubling [A] quadruples rate → second order in A

If changing [A] does not affect rate → zero order in A

1. Repeat for other reactants.
2. Write the complete rate equation using determined orders.
3. Substitute values into the rate equation to find the rate constant \( k \).

Note: The rate equation can help infer aspects of the reaction mechanism, especially the rate-determining step, but it does not always match the overall balanced chemical equation.