Question

Explain half-life for zero-order and second-order reactions

Answer 1

The half-life of Zero-order:

• The zero-order reaction refers to a condition in which the rate of the reaction does not vary with the change in concentration of the reactant.
• Half-life refers to the amount of time required to reduce the amount of a substance to half of its initial value.

Let. $\mathrm{A}\to \mathrm{B}+\mathrm{C}$;

The rate law is given as:

Rate = $\mathrm{k}{\left[\mathrm{A}\right]}^{\mathrm{n}}$; where n is the order of the reaction. for zero-order n=0

The differential rate law is given as: $\mathrm{k}=-\frac{\mathrm{d}{\left[\mathrm{A}\right]}^0}{\mathrm{dt}}$

rearranging: $\mathrm{d}{\left[\mathrm{A}\right]}^0=-\mathrm{kdt}$

Integrating both side: $$\begin{gathered} {\int }_{{\left[A\right]}_0}^{\left[A\right]}d\left[A\right]=-{\int }_0^{t}kdt \\ \Rightarrow {\left[A\right]}_{{\left[A\right]}_0}^{\left[A\right]}=-kt \\ \Rightarrow \left[A\right]-{\left[A\right]}_0=-kt \\ \Rightarrow \left[A\right]={\left[A\right]}_0-kt----->\left(i\right) \end{gathered}$$

For half-life: at $\mathrm{t}={\mathrm{t}}_{\frac{1}{2}}$; $\left[A\right]=\frac{{\left[A\right]}_0}{2}$

Substituting in eq(i)

$$\begin{gathered} \Rightarrow \frac{{\left[A\right]}_0}{2}={\left[A\right]}_0-k{t}_{\frac{1}{2}} \\ \Rightarrow t_{\frac{1}{2}}=\frac{{\left[A\right]}_0}{2k} \end{gathered}$$.

The half-life for second-order:

Let, $2\mathrm{A}\to \mathrm{B}$

the differential rate law is given as: $-\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{dt}}=\mathrm{k}{\left[\mathrm{A}\right]}^2$

rearranging and integrating.

$$\begin{gathered} \frac{d\left[A\right]}{{\left[A\right]}^2}=-kdt \\ \Rightarrow {\int }_{{\left[A\right]}_0}^{\left[A\right]}\frac{d\left[A\right]}{{\left[A\right]}^2}=-{\int }_0^{t}kdt \\ \Rightarrow {[-\frac{1}{\left[A\right]}]}_{{\left[A\right]}_0}^{\left[A\right]}=-kt \\ \Rightarrow \frac{1}{\left[A\right]}-\frac{1}{{\left[A\right]}_0}=kt \\ \Rightarrow \frac{1}{\left[A\right]}=kt+\frac{1}{{\left[A\right]}_0} \end{gathered}$$

For half-life: at $\mathrm{t}={\mathrm{t}}_{\frac{1}{2}}$; $\left[\mathrm{A}\right]=\frac{{\left[\mathrm{A}\right]}_0}{2}$

substituting in the above equation

$$\begin{gathered} \Rightarrow \frac{1}{{\displaystyle \frac{{\left[A\right]}_0}{2}}}=k{t}_{\frac{1}{2}}+\frac{1}{{\left[A\right]}_0} \\ \Rightarrow t_{\frac{1}{2}}=\frac{1}{k{\left[A\right]}_0} \end{gathered}$$