Question

What mass of a 60.0-gram sample of N-16 will remain unchanged after 28.8 seconds?

Answer 1

The answer is 3.75 g.

Half-life is the time required for the amount of a sample to half its value.
To calculate the fraction of the sample remained unchanged, we will use the following formulas:
1. $(1/2)^{n} = x$,
where:
n - a number of half-lives
x - a remained fraction of a sample

2. $t_{1/2} = \frac{t}{n}$
where:
$t_{1/2}$ - half-life
t - total time elapsed
n - a number of half-lives

The half-life of N-16 is 7.13.
So, we know:
t = 28.8 s
$t_{1/2}$ = 7.13 s

We need:
n = ?
x = ?

We could first use the second equation, to calculate n:
If:
$t_{1/2} = \frac{t}{n}$,
Then: 
$n = \frac{t}{ t_{1/2} }$
⇒ $n = \frac{28.8 s}{7.13 s}$
⇒ $n=4.04$
⇒ n ≈ 4

Now we can use the first equation to calculate the remained amount of the sample.
$(1/2)^{n} = x$
⇒ $x=(1/2)^4$
⇒$x= \frac{1}{16}$


If the fraction of the sample is 1/16 = 6.25%, then the mass of the sample could be calculated as:
x g : 6.25% = 60 g : 100% 
x =