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 N2O5 via 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[N2O5using 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.


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