# Question #bb66f

##### 2 Answers

Here's what I got.

#### Explanation:

A radioactive nuclide's **nuclear half-life** is defined as the *time* needed for **half** of an initial sample to undergo radioactive decay.

If you take

#1/2 * "N"_0 = "N"_0/2 -> # after the passing ofone half-life

#1/2 * "N"_0/2 = "N"_0/4 -># after the passing oftwo half-lives

#1/2 * "N"_0/4 = "N"_0/8 -># after the passing ofthree half-lives

#vdots#

and so on.

This means that you can express the amount of the nuclide that **remains undecayed** after a period of time

#color(blue)(|bar(ul(color(white)(a/a)"N"_t = "N"_0 * 1/2^ncolor(white)(a/a)|)))" " " "color(orange)("(*)")#

Here

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

All you have to do now is pick a period of time

For example, for

#"N"_t/"N"_0 = 1/2^n = 0.6484#

This is equivalent to

#2^n = 1/0.6484#

#ln(2^n) = ln(1/0.6484)#

#n * ln(2) = ln(1/0.6484) implies n = ln(1/0.6484)/ln(2) = 0.625#

So, you know that **half-lives** pass in **hours**, which means that the half-life of the nuclide,

#t_"1/2" = "5 h"/0.625 = color(green)(|bar(ul(color(white)(a/a)color(black)("8 h")color(white)(a/a)|)))#

I recommend using different time periods to find **must** come out the same in every case. That is the case because radioactive decay is a **first-order reaction**, meaning that its half-life is **constant** throughout the decay process.

For example, for

#"N"_t/"N"_0 = 0.2102#

This time you get

#n = ln(1/0.2102)/ln(2) = 2.25#

Once again, you have

#t_"1/2" = "18 h"/2.25 = "8 h"#

For the second part, use the half-life of the reaction to find the value of

#n = (64 color(red)(cancel(color(black)("h"))))/(8color(red)(cancel(color(black)("h")))) = 8#

then use equation **remains undecayed**

#"N"_t = "2.500 mg" * 1/2^8#

#"N"_t = "0.009766 mg"#

This means that the amount of the nuclide **that decayed** is

#"N"_"decayed" = "2.500 mg" - "0.009766 mg" = color(green)(|bar(ul(color(white)(a/a)color(black)("2.490 mg")color(white)(a/a)|)))#

I'll leave the answer rounded to four **sig figs**.

The half life is 8 hours.

#### Explanation:

O.K., this is actually one that requires almost no math to solve!

As you know, or are learning, a half-life is the amount of time it takes for half of a radioactive element to decay. So, in the language of your question Nt/N0 = 0.5 at one half life.

Looking at your chart, we see that that happened somewhere between 5 and 10 hours.

Now we also know that after two half lives Nt/N0 will be 0.25.

Woa!!!! THAT's ON THE CHART!!!! Awesome!

Two half-lives = 16 hours.. Therefore one half-life = 8 hours.

Wait! Does that check with the previous observation that the half-life will be between 5 and 10 hours? Yup! O.K.! We are in business!

On to part B! 64 hours is 64/8 half-lives = 8 half-lives. After 8 half-lives there will be