The Rate Equation and Reaction Orders
\( \text{Rate} = k[A]^m[B]^n \)
\(k\) is the rate constant, \(m\) and \(n\) are individual reaction orders. Overall order \(= m + n\).
Order of Reaction
| Order | Effect on Rate | [A] vs Time Graph | Rate vs [A] Graph |
|---|---|---|---|
| 0 | No effect | Straight line (linear decrease) | Horizontal line |
| 1 | Rate ∝ [A] | Exponential decay (constant half-life) | Straight line through origin |
| 2 | Rate ∝ [A]² | Curve (steeper initial decrease) | Parabola through origin |
Deducing Reaction Orders from Initial Rates Data
Use the initial rates method: compare pairs of experiments where only one reactant concentration changes while all other conditions are kept constant:
- If doubling \([A]\) has no effect on rate \(\implies\) Zero order (\(m = 0\)).
- If doubling \([A]\) doubles the rate \(\implies\) First order (\(m = 1\)).
- If doubling \([A]\) quadruples the rate (\(\times 4\)) \(\implies\) Second order (\(m = 2\)).
The Arrhenius Equation
The Arrhenius Equation
\( k = A e^{-E_{\text{a}}/RT} \quad \Longleftrightarrow \quad \ln k = \ln A - \dfrac{E_{\text{a}}}{R}\left(\dfrac{1}{T}\right) \)
\(A\) = pre-exponential frequency factor, \(E_{\text{a}}\) = activation energy (\(\text{J mol}^{-1}\)), \(R = 8.314\text{ J K}^{-1}\text{mol}^{-1}\), \(T\) = temperature (\(\text{K}\)).
The linear form (for graphing):
\( \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \)
Plot ln k vs 1/T → straight line with gradient = −Ea/R
Deducing Overall Reaction Order
For \(\text{Rate} = k[\text{NO}]^2[\text{O}_2]\):
Order with respect to \(\text{NO} = 2\), order with respect to \(\text{O}_2 = 1\).
Overall order = 2 + 1 = 3 (third order overall). Doubling \([\text{NO}]\) quadruples rate; doubling \([\text{O}_2]\) doubles rate; doubling both increases rate by a factor of \(2^2 \times 2 = 8\).
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