# If we start with 1 mg of strontium, 0.953 mg will remain after 2.0 years. What is the half life of strontium-90?

##### 1 Answer

#### Explanation:

The **nuclear half-life** of a radioactive nuclide, **half** of its initial mass.

The first thing to notice here is that it took **years** for only

#"1 mg " - " 0.953 mg" = "0.047 mg"#

of strontium-90 to decay. This tells you that the half-life of the nuclide is **significantly longer** than **years**, since you need **half** of the original sample, the equivalent of *one half-life*.

Your tool of choice here will be the equation

#color(blue)(|bar(ul(color(white)(a/a)A_"t" = A_0 * 1/2^ncolor(white)(a/a)|)))#

Here

**remains undecayed** after a period of time

**initial mass** of the nuclide

**number of half-lives** that pass in the period of time

#color(purple)(|bar(ul(color(white)(a/a)color(black)(n = t/t_"1/2")color(white)(a/a)|)))#

Use this equation to find how many half-lives passed in **years**

#A_"t" = A_0 * 1/2^n#

#0.953 color(red)(cancel(color(black)("mg"))) = 1color(red)(cancel(color(black)("mg"))) * 1/2^n#

#2^n = 1/0.953#

This will be equivalent to

#ln(2^n) = ln(1/0.953)#

#n * ln(2) = ln(1/0.953) implies n = ln(1/0.953)/ln(2)#

This will get you

#n = 0.06945#

This means that only

#n = "2.0 years"/0.06945 = color(green)(|bar(ul(color(white)(a/a)color(black)("29 years")color(white)(a/a)|)))#

I'll leave the answer rounded to two **sig figs**, despite the fact that you only have one sig fig for the initial mass of the sample.