Question

Iodine-131 has a half-life of 8.0 days. How many days will it take for 174 g of${}^{\mathbf{131}}\mathbf{\text{I}}$ to decay to 83g of ${}^{\mathbf{131}}\mathbf{\text{I}}$?

Short answer

The time elapsed for the particle is 8.55 days.

Step-by-step solution

Decay constant

The reciprocal of the time during which, the count of atoms left in the sample reduces to $\frac{\mathbf{1}}{\mathbf{e}}$ times the original number of atoms in the sample is called the decay constant of a radioactive element.

Rate constant

• The half-life of ${}^{131}\text{I}$is 8.0 days.
• The initial amount of ${}^{131}\text{I}$is $N_0=174\text{g}$.
• The final amount of${}^{131}\text{l}\to \text{N}=83\text{g}$.

Now the rate constant is evaluated as:

$\begin{array}{rcl}\text{Rate constant (k)}& =& \frac{\ln \left(2\right)}{{\text{Half - life (t}}_{1/2})}\\ & =& \frac{0.693}{8.0\text{days}}\\ & =& 0.0866{\text{days}}^{-1}\end{array}$

Time elapsed

The integrated rate law of first order reaction is used to find the time.

$\begin{array}{rcl}\ln \left(\frac{\text{N}}{{\text{N}}_0}\right)& =& -\text{kt}\\ \ln \left(\frac{83\text{g}}{174\text{g}}\right)& =& -\text{kt}\\ -0.7402& =& -0.0866{\text{days}}^{-1}\times \text{t}\\ & & \text{t =}8.55\text{days}\end{array}$

Thus, time elapsed for the particle is 8.55 days.