How many half-lives must elapse until 78.0% of a radioactive sample of atoms has decayed? (Note that there is no unit for this answer.)

The half-life of a radioactive substance is the time it takes for half of the atoms in a sample to decay.

To find out how many half-lives have elapsed, we need to determine how much of the original sample remains.

In this case, we're told that 78.0% of the sample has decayed. This means that 22.0% of the original sample remains (100.0% - 78.0% = 22.0%).

The decay of a radioactive substance is a first-order process, which means that the amount of substance remaining after n half-lives can be calculated using the formula:

remaining = initial * (1/2)^n

We can rearrange this formula to solve for n:

n = log(remaining/initial) / log(1/2)

Substituting the given values:

n = log(0.22/1) / log(1/2)

n ≈ 2.18

So, approximately 2.18 half-lives must elapse until 78.0% of a radioactive sample of atoms has decayed.