Rate Equations

• Experimental Basis: Unlike stoichiometry, which is derived from a balanced equation, rate equations can only be determined through experimental observation. The coefficients in a balanced chemical equation do not necessarily match the orders in the rate equation.

• Temperature Dependence: The Arrhenius equation, $k = Ae^{-E_a/RT}$, explains why the rate constant changes with temperature. As temperature increases, a higher fraction of molecules possess energy exceeding the activation energy ($E_a$), leading to more frequent successful collisions.

• Molecularity and Mechanism: The rate equation reflects the species involved in the rate-determining step (the slowest step of a multi-step mechanism). If a reactant appears in the rate equation, it must be involved in or before this critical bottleneck step.

• Initial Rates Method: This involves measuring the reaction rate at the very start ($t=0$) for several trials with varying starting concentrations. By comparing how the rate changes when one reactant's concentration is doubled (while others are held constant), the order for that reactant can be deduced.

• Continuous Monitoring: Data is collected throughout the entire reaction to plot concentration-time graphs. The gradient of a tangent to the curve at any point gives the instantaneous rate, allowing for the determination of the rate equation from a single experiment.

• Deducing Units of $k$: Units are found by rearranging the rate equation for $k$ and substituting the units for rate ($mol \, dm^{-3}s^{-1}$) and concentration ($mol \, dm^{-3}$). For example, a second-order reaction results in units of $dm^3 \, mol^{-1}s^{-1}$.

Comparison of Reaction Orders

• $S_N1$ vs $S_N2$ Mechanisms: In organic chemistry, $S_N1$ reactions are first-order because the rate-determining step involves only the halogenoalkane. In contrast, $S_N2$ reactions are second-order because both the nucleophile and the halogenoalkane collide in a single, concerted rate-determining step.

• Check the Units: Examiners frequently award marks for the correct units of the rate constant. Always perform a dimensional analysis after calculating the numerical value of $k$ to ensure the units match the overall order.

• Standard Form Precision: When working with initial rates data, ensure you account for powers of ten (e.g., $10^{-4}$ vs $10^{-5}$). A common mistake is misidentifying a rate change because the decimal place in the table was overlooked.

• Mechanism Validation: If asked to propose a mechanism, ensure the sum of your elementary steps equals the overall balanced equation. Furthermore, the reactants in your 'slow' step must match the species found in the experimentally determined rate equation.

• Stoichiometry Trap: Students often assume the coefficients in a balanced equation are the orders in the rate equation. This is only true for simple one-step reactions; for most reactions, the order must be found experimentally.

• Intermediates in Rate Equations: Reaction intermediates (species produced in one step and consumed in another) should never appear in the final rate equation. If an intermediate is part of the rate-determining step, it must be replaced by the reactants that formed it.

• Catalysts: It is a misconception that catalysts do not appear in rate equations. Because they affect the mechanism and the rate-determining step, they can indeed be part of the rate expression even though they aren't consumed.