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The half-life of a chemical reaction can be defined as the time taken for the concentration of a given reactant to reach 50% of its initial concentration (I.e., the time taken for the reactant to reach half of its initial value). It is denoted by the symbol ${\mathit{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}$and is usually expressed in seconds.

By definition, the half-life of an excited state is the time it would take for half the atoms to make a transition to a lower state. We use the formula for decay:

$N\left(t\right)=e-tl蟿N\left(0\right)$ (Equation-9.58 and 9.59).

$N_b\left(t\right)=N_b\left(0\right)e^{-At}$ (Equation-9.58).

$蟿=\frac{1}{A}$ (Equation-9.59).

The half-life of an excited state is the time it would take for half the atoms in a large sample to make the transition.

After one half-life,

$$\begin{gathered} N\left(t\right)=\frac{1}{2}N\left(0\right), \\ \mathrm{so}\frac{1}{2}={\mathrm{e}}^{-\mathrm{t}/\mathrm{蟿}},\mathrm{or}2={\mathrm{e}}^{\mathrm{t}/\mathrm{蟿}}, \\ \mathrm{so}\mathrm{t}/\mathrm{蟿}=\mathrm{In}2,\mathrm{or}{\mathrm{t}}_{1/2}=\mathrm{蟿滨苍}2 \end{gathered}$$

Thus, the relation betweenn role="math" localid="1658302506697" $t_{1/2}\&\mathrm{T}$is$t_{1/2}=\mathrm{蟿滨苍}2$