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

I calculated 22.9 minutes

Answer 2

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

Explanation:

Two important formulas in radioactive decay are

`(1) \qquad t_{(1)/(2)} = (\ln 2)/(k)\\\\(2) \qquad \ln \left((N_(0))/(N)\right) = kt`

1. Calculate the decay constant k

`\begin{array}{rcl}t_{(1)/(2)} &=& (\ln 2)/(k)\\\\k &= &\frac{\ln 2}{t_{(1)/(2)}}\\\\ & = & \frac{\ln 2}{\text{6 min}}\\\\& = & \text{0.1155 min}^(-1)\\\end{array}`

2. Calculate the time

`\begin{array}{rcl}\ln \left((N_(0))/(N)\right) &= &kt \\\\\ln \left((480)/(34)\right) &= &\text{ 0.1155 min}^(-1)* t \\\\\ln 14.12 &= & \text{ 0.1155 min}^(-1)* t \\t &= &\frac{\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}`