Question

The half-life of $\left(t_{1/2}\right)$an excited state is the time it would take for half the atoms in a large sample to make a transition. Find the relation betweenrole="math" localid="1658300900358" ${\mathbf{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{}}\mathbf{and}\mathbf{}\mathbf{T}$(the “life time” of the state).

Short answer

The relation between $t_{1/2}\mathrm{and}T$is the$t_{1/2}=\mathrm{\tau In}2$

Step-by-step solution

Half-Life

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.

Finding the relation between t1/2 and T (the “life time” of the state)

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:

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

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

$\tau =\frac{1}{A}$ (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,

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

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