Question

Hydrogen-3 has a half-life of 12.32 years. A sample of H-3 weighing 3.02 grams is left for 15.0 years. What will the final weight of the H-3 sample be?

Answer 1

Answer:

$\large \boxed{\text{1.38 g}}$

Explanation:

Two important formulas in radioactive decay are

$(1) \qquad t_{\frac{1}{2}} = \dfrac{\ln 2}{k}\\\\(2) \qquad \ln \left(\dfrac{N_{0}}{N}\right) = kt$

1. Calculate the decay constant k

$\begin{array}{rcl}t_{\frac{1}{2}} &=& \dfrac{\ln 2}{k}\\\\\text{12.32 yr} &= &\dfrac{\ln 2}{k}\\\\k & = & \dfrac{\ln 2}{\text{12.32 yr}}\\\\& = & \text{0.056 26 yr}^{-1}\\\end{array}$

2. Calculate the mass remaining

$\begin{array}{rcl}\ln \left(\dfrac{A_{0}}{A}\right) &= &kt \\\\\ln \left(\dfrac{\text{3.20 g}}{A}\right) &= &\text{0.056 26 yr}^{-1}\times \text{15 yr} \\\\\ln \left(\dfrac{\text{3.20 g}}{A}\right) &= &0.8439 \\\\\dfrac{\text{3.20 g}}{A} &= &e^{0.8439} \\\\\dfrac{\text{3.20 g}}{A}&= &2.325 \\\\A &= &\dfrac{\text{3.20 g}}{2.325}\\\\&= & \textbf{1.38 g}\\\end{array}\\\text{The final mass of the sample will be \large \boxed{\textbf{1.38 g}} }$