Question

Iodine-131 is used in the diagnosis and treatment of thyroid disease and has a half-life of 8.0 days. If a patient with thyroid disease consumes a sample of Na¹³¹I containing $\mathbf{10}\mathbf{}\mathit{\mu }\mathit{g}$of ¹³¹I, how long will it take for the amount of ¹³¹I to decrease to 1/100 of the original amount?

Short answer

It will take 53.54 days.

Step-by-step solution

Analyzing the given data

Initial mass =$10\mu g$

Mass after time t = 1/100^th amount =$0.1\mu g$

Rate constant

Rate constant depicts the relationship between a reaction’s rate and a substances’ concentration in the given reaction.

The rate constant is evaluated as:

$\begin{array}{c}\mathbf{k}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{693}}{{\mathbf{t}}_{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}}\\ =\frac{0.693}{8days}\\ =0.086{\text{days}}^{-1}\end{array}$

Time required for the amount of 131I to decrease to 1/100th of the original amount 

Using the integrated rate law for first-order reaction:

$\mathbf{ln}\frac{{\left[\mathbf{N}\right]}_{\mathbf{t}}}{{\left[\mathbf{N}\right]}_{\mathbf{0}}}\mathbf{=}\mathbf{-}\mathbf{kt}$

Where,

• k = Rate constant
• t = time required
• ${\left[\text{N}\right]}_t$= Concentration of N after time t
• ${\left[\text{N}\right]}_0$= Initial concentration of N

The known values are substituted.

$\begin{array}{c}\ln \left(\frac{0.1}{10}\right)=-0.086\times t\\ \ln \left(0.01\right)=-0.086\times t\\ -4.60=-0.086\times t\\ t=53.54\text{days}\end{array}$

Hence, the time required forthe amount of ¹³¹I to decrease to 1/100^th of the original amount is 53.54 days.