Integrated rate equations

The rate law tells the rate at one moment. An integrated rate equation links the concentration to time, so you can find how much reactant is left after a given time. Two symbols:

• [R]₀ is the starting concentration.
• [R] is how much is left at time t.

Think of 4 kg of apples on day 0 and 1 kg left after 10 days: 4 kg is [R]₀ and 1 kg is [R]. And if a reaction is 75% complete, then [R]₀ = 100 and [R] = 25.

Zero order: the reactant is used up at a steady rate, whatever its concentration.

[R] = [R]₀ – kt

A graph of [R] against t is a straight line that slopes down, with slope –k and starting value [R]₀. Example: some gases breaking down on a hot metal surface.

First order: the rate is proportional to [R] itself. This is the most important case:

k = (2.303/t) log([R]₀/[R])

The same equation can be written ln[R] = ln[R]₀ – kt, or [R] = [R]₀e^(–kt). Switching from natural log (ln) to ordinary log brings in the factor 2.303. A plot of ln[R] (or log[R]) against t is a straight line; with log, its slope is –k/2.303.

For a gas reaction A(g) → B(g) + C(g) in a closed vessel, we measure the total pressure p_t instead of concentration. Starting from pressure pᵢ of A, the pressure of A left at any time is p_A = 2pᵢ – p_t, which goes into the formula in place of [R].

Some reactions look first order even though two substances react. When one of them is present in a huge excess, such as water in the hydrolysis of an ester or of sugar, its concentration hardly changes: like taking a few buckets from the sea. It acts as a constant, and the reaction behaves as first order. This is a pseudo first order reaction.