Half lives?

Question

A 100-mg sample of technetium-99m is used in a medical study.
How much of the technetium-99m sample remains after 24hr?
The half life is 6.0 hrs?
Can you please explain how to get the answer?

Answer 1

After each half-life, 1/2 of the sample is lost.

After 6 hours, 1/2 of the 100 mg remains....50 mg,
after another 6 hours (12 hours total), 1/2 of the 50 mg remains, ... 25 mg,
after 18 hours, 1/2 of 25 mg remains, ... 12.5 mg,
after 24 hours, 1/2 of 12.5 mg remains, ... 6.25 mg

When dealing with just a few number of half-lives it is not too difficult to work it out 'by hand' to calculate the amount remaining, however in other questions when far greater number of half lives have passed, this may get tedious.
A mathematical equation can be formulated to speed the process.

The amount remaining = The initial amount * (1/2)^x
where x is the number of half-lives which have passed since the initial amount was measured.
x = time which has passed / length of half life

In your question,
x = 24 hours / 6 hours = 4
The amount remaining = 100 mg * (1/2)^4 = 6.25 mg

This equation also becomes very useful when one wants to solve more difficult problems involving half-lives.

Answer 2

You can do this two different ways, the best does not require the luck of having the time (24 hours) being an integer multiple of the half life (6 hours)

the simple way(but not general for real life cases)

You have 4 half-lives, therefore after each 6 hours 1/2 of what you started with a half-life before is gone:

original amount*(1/2*1/2*1/2*1/2)

The other way requires you to calculate k, the decay constant, but if you are given the half-life it is easy, let A be the original amount, then after a half-life of time has elapsed you have A/2 left.

A/2 = Ae^(-k*half_life)
Then:
ln(1/2) = -k*half_life
Then
k = [-ln(1/2)]/half_life

the general equation now is:

A(t) = Ae^(-k*t), where at time t=0 it is true that A(t) = A

Note I wrote A(t), showing that the amount left is a function of t.

Answer 3

Half life is simply the term for how long it takes for half of the original substance to decay into its daughter substance (element)

to find the half life

1) Determine the half life (Usually given to you)
2) Find out How many half lives will pass
3) Solve the problem

In your problem there is a 24hr time period with 6hr half lives, that means that there will be 4 half lives in that time period which
means:

100/2 = 50 (1 HL)
50/2 = 25 (2 HL)
25/2 = 12.5 (3 HL)
17.5/2 = 6.25 (4HL)

To do this more rapidly you do

Original amount / 2^(number of half lives)

In this case 100/2^4 which is 100/16 which = 6.25

Hope that helps