Question

Element X decays radioactively with a half life of 6 minutes. If there are 480 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 34 grams?

Answer 1

Answer:

I calculated 22.9 minutes

Answer 2

Answer:

$\large \boxed{\text{22.9 min}}$

Step-by-step 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

$\begin{array}{rcl}t_{\frac{1}{2}} &=& \dfrac{\ln 2}{k}\\\\k &= &\dfrac{\ln 2}{t_{\frac{1}{2}}}\\\\ & = & \dfrac{\ln 2}{\text{6 min}}\\\\& = & \text{0.1155 min}^{-1}\\\end{array}$

2. Calculate the time

$\begin{array}{rcl}\ln \left(\dfrac{N_{0}}{N}\right) &= &kt \\\\\ln \left(\dfrac{480}{34}\right) &= &\text{ 0.1155 min}^{-1}\times t \\\\\ln 14.12 &= & \text{ 0.1155 min}^{-1}\times t \\t &= &\dfrac{\ln 14.12}{\text{0.1155 min}^{-1}}\\\\& = & \textbf{22.9 min}\\\end{array}\\\text{It would take \large \boxed{\textbf{22.9 min}} for the mass to decrease to 34 g}$