Question

How is integrated rate equation for a first order reaction related to the rate constant?

Answer 1

For a first-order reaction

\[\ce{A -> products}\]

The integrated rate equation is

\[\ce{ln \frac{[A]_0}{[A]_t} = kt}\]

From this, the rate constant k is directly obtained as

\[\ce{k = \frac{1}{t} ln \frac{[A]_0}{[A]_t}}\]

Or, using base-10 logarithms

\[\ce{k = \frac{2.303}{t} log \frac{[A]_0}{[A]_t}}\]

The slope of a plot of ln⁡[A]_t is −k (straight line).

k is independent of the initial concentration for a first-order reaction.

The integrated form allows k to be calculated from concentration–time data.

So, the integrated rate equation is essentially the formula that links k with measurable quantities [A]₀ . [A]_t​, and t.