The relations Rate Bench uses
Every instrument uses the standard relations below, with least-squares straight-line fits. Nothing else goes into a result.
1. Rate laws
For a reaction whose rate depends on reactant concentrations, the rate law has the form
rate = k[A]m[B]n…
The orders m, n come from experiment, not from the balanced equation. The overall order is their sum, and the units of k follow from it: M1−order time−1.
2. Integrated rate laws (order finder)
| Order | Integrated law | Straight line | Slope | t½ |
|---|---|---|---|---|
| 0 | [A] = [A]0 − kt | [A] vs t | −k | [A]0/2k |
| 1 | ln[A] = ln[A]0 − kt | ln[A] vs t | −k | ln 2/k |
| 2 | 1/[A] = 1/[A]0 + kt | 1/[A] vs t | +k | 1/k[A]0 |
The order finder fits all three and reports the plot with the highest R². When even the best is below 0.98 it says so: the reaction may be of another order, or the data may need checking. These laws hold for rate = k[A]n with A the only concentration that changes (or the others in large excess, the “pseudo-order” case).
3. The method of initial rates
When two experiments change only [A], the ratio of their rates gives the order in A:
rate2/rate1 = ([A]2/[A]1)m so m = log(rate2/rate1) / log([A]2/[A]1)
Rate Bench shows that step wherever the table allows it, and also fits log(rate) = log k + m log[A] + n log[B] + … over every experiment at once, so tables without a clean pair still work. A fitted order within 0.15 of a whole or half number is rounded to it; otherwise it is kept to two decimals and flagged. k is then worked out from each experiment with the rounded orders and averaged.
4. Arrhenius
k = A e−Ea/RT ln k = ln A − (Ea/R)(1/T)
A plot of ln k against 1/T (in kelvin) is a straight line of slope −Ea/R. With two readings this is the two-point form ln(k2/k1) = (Ea/R)(1/T1 − 1/T2). R = 8.314462618 J mol−1 K−1; °C are converted with T = θ + 273.15.
5. Half-lives
First order: every half-life is the same length. Zero order: each is half as long as the one before (the reactant runs out at t = [A]0/k). Second order: each is twice as long as the one before. The half-life instrument counts down five of them.
6. Fits and R²
Straight lines are ordinary least-squares fits. R² = 1 − SSresidual/SStotal: 1 is a perfect line. Note that fitting ln[A] or 1/[A] weights the readings differently from fitting [A]; this is the standard classroom method, and the one Rate Bench uses.
7. Typing numbers
One reading or experiment per line, values separated by commas, tabs, semicolons or spaces (so a paste from a spreadsheet works). Scientific notation: 2.5e-5, 2.5E-5, 2.5x10^-5, 2.5×10^-5 or 2.5*10^-5. Use a point for decimals. A first line of column titles is skipped; lines starting with # are notes. Concentrations are in mol/L (M).