## Chemical Kinetics

# 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.

t_{1/2} ∝ [A]_{0}

\tt \Rightarrow t_{1/2}=\frac{[A_0]}{2k}

At t_{100,} [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}

∴ t_{1/2} = 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}

### Part1: View the Topic in this Video from 12:47 to 51:50

### Part2: View the Topic in this Video from 0:40 to 53:10

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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

1. **Half-life method:** This method is employed only when the rate law involve only one concentration term.

t_{1/2} ∝ a^{1-n} ; t_{1/2} = ka^{1-n }; log t_{1/2} = log k + (1 − n)log a