# Question 188b6

Aug 8, 2017

$\text{23 g}$

#### Explanation:

All you have to do here is to use the following equation

${A}_{t} = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\textcolor{red}{n}}$

Here

• ${A}_{t}$ is the mass of the substance that remains undecayed after a period of time $t$
• ${A}_{0}$ is the initial mass of the substance
• $\textcolor{red}{n}$ represents the number of half-lives that pass in a given time period $t$

In your case, you know that the initial mass of the substance is equal to $\text{35 g}$.

Now, the number of half-lives that pass in a given time period $t$ can be calculated by dividing the period of time by the half-life of the substance, ${t}_{\text{1/2}}$.

$\textcolor{red}{n} = \frac{t}{t} _ \text{1/2}$

In your case, the number of half-lives that pass in $2100$ years is equal to

$\textcolor{red}{n} = \left(2100 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{years"))))/(3400color(red)(cancel(color(black)("years}}}}\right) = 0.617647$

Plug this into the aforementioned equation and solve for ${A}_{\text{2100 years}}$, the mass of the substance that remains undecayed after $2100$ years

A_ "2100 years" = "35 g" * (1/2)^0.617647 = color(darkgreen)(ul(color(black)("23 g")))#

The answer is rounded to two **[sig figs](