Differential method (rate laws)

The differential method of determining the rate of a reaction makes use of the differential form of a rate law.

Just like the half-life method, the differential method is useful for reactions involving a single reactant, e.g.

$N_2O_5(g)\rightleftharpoons 2NO_2(g)+\frac{1}{2}O_2(g)$

where we can write the rate equation as

$-\frac{d[N_2O_5]}{dt}=k[N_2O_5]^i\; \; \; \; \; \; \; \; 23$

Taking the natural logarithm on both sides of eq23, we have

$ln\left [ -\frac{d[N_2O_5]}{dt} \right ]=iln[N_2O_5]+lnk\; \; \; \; \; \; \; \; 24$

To find the rate constant, k, and the order, i:

1. Run the experiment in a rigid vessel and record the concentrations of N₂O₅via spectroscopic methods at various times.
2. Plot a graph of [N₂O₅] versus time.
3. Find the gradients to the curve at selected concentrations, i.e. find $\frac{d[N_2O_5]}{dt}$ at various [N₂O₅].
4. With reference to eq24, plot a graph of $ln\left [- \frac{d[N_2O_5]}{dt} \right ]$ versus ln[N₂O₅] using the values determined in step 3.

The gradient and the vertical intercept of the line in the second graph give the values of i and lnk respectively. The differential method is commonly used together with the initial rate method and the isolation method in determining the unknowns of a rate equation with multiple reactants.

Differential method (rate laws)

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