 # Integrated Rate Equations and Half-life

Integrated rate law for zero order reaction:
\tt k = \frac{\left[A\right]_0-\left[A\right]}{t}
[A]0 = Initial concentration of reactant
[A] = concentration of reactant at a particular time.

Half life period: For zero order reaction half-life is directly proportional to the initial concentration and inversely proportional to the rate constant.
t1/2 ∝ [A]0
\tt \Rightarrow t_{1/2}=\frac{[A_0]}{2k}

At t100, [A]0 = 0
\tt At\ t_{100}=\frac{\left[A\right]_0}{k}
\tt t_{1/2}=\frac{t_{100}}{2}\Rightarrow t_{100}=2t_{1/2}

Integrated rate law of 1st order reaction:
\tt kt=2.303\ \log\frac{\left[A\right]_0}{\left[A\right]}
where k = rate constant
t = time, [A]0 = Initial concentration
[A] = Concentration of reactant at a particular time.

Graphs for 1st order reaction: Half-life period of 1st order:
(i) \tt t_{1/2}=\frac{0.693}{k}
∴ t1/2 = Independent of initial concentration
(ii) 1. For a reaction of nth order, \tt t_{1/2}\propto \left[A\right]_0^{1-n}
where [A]0 is initial concentration.
2. For 1st order reaction
(a) \tt t_{1/2}=\frac{1}{2}t_{75\%}=\frac{1}{3}t_{87.5\%}=\frac{1}{3.322}t_{90\%}
(b) \tt t_{90\%}:t_{99\%}:t_{99.9\%}:t_{99.99\%}=1:2:3:4
(c) \tt t_{93.75\%}=2t_{75\%}=4t_{50\%}
(3) Amount of substance left after n half lives in case of 1st order reaction = \tt \frac{\left[A\right]_0}{2^n}

### Part3: View the Topic in this Video from 0:40 to 54:20

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1. Half-life method: This method is employed only when the rate law involve only one concentration term.
t1/2 ∝ a1-n ; t1/2 = ka1-n ; log t1/2 = log k + (1 − n)log a