Integrated Rate Equations: Definition, Derivation, Types, and Many More

Integrated Rate Equations are mathematical equations which are used to determine how the concentration of reactants change in a chemical reaction over a certain duration of time. These equations can vary depending upon the order of reactions and the rate of a reaction. The general representation of an integrated rate equation as per the rate law can be given by:

-d[A]/dt = k[A]^n

Where n = order of the reaction (zero, first or second)

Now, let us study these orders of a reaction one by one.

In such a type of reaction, the value of k(reaction aret) remains constant irrespective of the reactant's concentration. Mathematical representation of zero order reactions can be given by:

−d[A]​/dt=k

Where, A = Concentration of the reactant A

And k = rate constant for zero order reaction

After integration of the above equation, we get

[A] = [A]` - kt

Where,

[A]` = Initial concentration of the reactant

[A] = concentration of the reactant at time t

k = rate constant

For a reaction  $R⟶P$ , the rate law states that:

$-\frac{\mathrm{d}\left[\mathrm{R}\right]}{\mathrm{d}t}=k$

d[A]=−kdt

Now integrate both sides of the equation,

$\begin{array}{rr}{\int }_{[\mathrm{R}{]}_0}^{\left[\mathrm{R}\right]} \mathrm{d}\left[\mathrm{R}\right]& =-{\int }_0^t k\mathrm{d}t\\ \left[\mathrm{R}\right]-[\mathrm{R}{]}_0=-kt& \Rightarrow \left[\mathrm{R}\right]=[\mathrm{R}{]}_0-kt\end{array}$

Hence, we get the final equation as [R]=[R]0​−kt

where  $[R{]}_0$ is the initial concentration at  $t=0$ .

And [R] is the concentration at time t.

The Half-Life of a reaction is referred to as the amount of time in which the concentration becomes half of what its original value was.

Mathematical formula for Half Life is given by:

t₁/₂ = [R]₀ / 2k

These are the type of reactions where the rate of reaction is proportional to the concentration of only 1 single reactant. Mathematical representation for a First order reaction can be given by:

−d[A]/ dt​=k[A]

Where, [A] = Concentration of the reactant A at time t

And k = rate constant for first order reaction

Integrating the above equation, we get

[A] = [A]'e^-kt

For a reaction  $\mathrm{R}⟶\mathrm{P}$ , the rate law is depicted by:

$\begin{array}{rr}-\frac{\mathrm{d}\left[\mathrm{R}\right]}{\mathrm{d}t}& =k\left[\mathrm{R}\right]\\ -\frac{\mathrm{d}\left[\mathrm{R}\right]}{\left[\mathrm{R}\right]}& =k\mathrm{d}t\end{array}$

Further integrating both sides of the equation, we get:

∫[A]0​[A]​[A]d[A]​=−k∫0t​dt

ln[A]−ln[A]0​=−kt

Combining the log values, we get:

ln([A]0​[A]​)=−kt

[A]=[A]0​e−kt

Hence, proved

Half Life of a first order reaction is depicted by:

ln([A]0​/2[A]0​​)=kt1/2​

ln2=kt1/2​

t1/2​=0.693​/k