Q.5.90 - An Introduction to General, Organic, and Biological Chemistry

Question

The radioisotope sodium-24 is used to determine the levels of electrolytes in the body. A 16$\mu g$ sample of sodium-24 decays to 2.0 $\mu g$ in 45 h. What is the half-life, in hours, of sodium-24? 

Step-by-Step Solution

Answer

 the half-life, in hours, is$15h$

Step 1: Given Information

A sample of sodium-24 = $16\mu g$

The time taken to decay to $2\mu g$ = $45h$

Step 2: Explanation

Given, 

The amount initially $A_o$ = $16\mu g$ and Final amount $A_f$= $2\mu g$

Time t = $45h$

Using,

$A_f=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}$

$2=16{\left(\frac{1}{2}\right)}^{\frac{45}{t_{1/2}}}=\frac{2}{16}={\left(\frac{1}{2}\right)}^{\frac{45}{t_{1/2}}}$

On applying log on both sides,

$\log \left(0.125\right)=\frac{45}{t_{\frac{1}{2}}}\log \left(0.5\right)$

$t_{1/2}=\frac{45\times (-0.3010)}{(-0.9031)}=15h$