Question

a. Find an expression in terms of $t$for the half-life t1/2 of a sample of excited atoms. The half-life is the time at which half of the excited atoms have undergone a quantum jump and emitted a photon. b. What is the half-life of the $3p$ state of sodium?

Short answer

a) The expression in terms of$t$for the half-life $t_{1/2}$ of a sample of excited atoms is$0.693\tau$

b) The half-life of the $3p$state of sodium is$11.781ns$

Step-by-step solution

Part(a) Step 1: Given information 

We are given that The half-life is the time at which half of the excited atoms have undergone a quantum jump and emitted a photon .

We need to find expression in terms of$t$for the half-life $t_{1/2}$of a sample of excited atoms ,

Part(a) Step 2: Simplify

We will use exponential decay law which is , $N_e\left(t_{1/2}\right)=\frac{1}{2}{N}_0$where $t_{1/2}$is half life .Let this be equation $1$.

We will write this as $N_e\left(t_{1/2}\right)=N_0{e}^{-\raisebox{1ex}{$t_{1/2}$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.}$.

Comparing both the equation and taking logarithm on both the sides

, role="math" localid="1650441078255" $$\begin{gathered} e^{\raisebox{1ex}{$-t_{1/2}$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.}=\frac{1}{2} \\ {t}_{1/2}=\tau \ln 2 \end{gathered}$$

Substituting ,$t_{1/2}=0.693\tau$

Part(b) Step 1: Given information 

We need to find out half-life of the $3p$ state of sodium.

Part(b) Step 2: Simplify

We need to find half life , We will use this equation for half life ,$t_{1/2}=\ln 2\tau$

Simplifying , we get$t_{1/2}=\left(0.693\right)\left(17ns\right)=11.781ns$.