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

Question

The radioscope sodium-$24$ is used to determine the levels of electrolytes in the body.$\mathrm{A}16$ - $\mu \mathrm{g}$ sample of sodium-$24$ decays to$2.0\mu \mathrm{g}$ in $45 \mathrm{h}$. What is the half-life, in hours, of sodium-$24$? $(5.4)$

Step-by-Step Solution

Answer

The sodium half-life-$24$ in hours is$t{y}_2\cong 15\mathrm{h}$

Step 1:Introduction

Sodium-$24$ radioactive decay  A radioactive sodium-$24$ (24  Na) nucleus decay scheme. It decays by beta decay to an excited magnesium-24   (24 Mg) nucleus with a half-life of $15$ hours. Two gamma rays are rapidly emitted, and the excitation energy is carried away, resulting in the stable ground state of magnesium-24 .

Step 2:Given data

Given data is the radioscope sodium- $24$is used to determine the levels of electrolytes in the body

To find half life sodium-$24$

Step 3:Explanation

Sodium $-24$ decay follows first orders kinetics

$k=\frac{2.303}{t}\log \frac{N_0}{N}$

$t{y}_2=\frac{0.693}{K}$

where $t{y}_2=$ Half life of N ${\mathrm{N}}_{\mathrm{C}}-24$

where $t=$time elapsed

$N_0=$ Initial amount

${\mathbf{N}}_{\mathbf{s}}=$Amount after time t

$k=$ Rate constant

$$\begin{gathered} \frac{0.693}{t{y}_2} \\ =\frac{2.303}{45h}\times \log \left(\frac{164g}{24g}\right) \end{gathered}$$

$t{y}_2\cong 15\mathrm{h}$