# What is the half-life of (Na^24) if a research assistant made 160 mg of radioactive sodium (Na^24) and found that there was only 20 mg left 45 h later?

##### 2 Answers

#### Explanation:

We need to find an equation of the form:

Where:

We are given:

We need to solve for

Divide by 160:

Taking natural logarithms of both sides:

Hence:

Dividing by 45:

Since by definition the half life is the time period when we have half of the starting amount:

So we need to solve for t in:

Taking natural logarithms:

The half life is 15 hours.

15 hours

#### Explanation:

**Quick Way**

As the amount of a decaying substance halves over **each** half-life (hence the name), halving the amount in steps requires 3 steps to get from 160 to 20:

#160 to 80 to 40 to 20#

And

So the half-life is 15 years.

**More formal way**

For half-life

So:

Plugging the values that are given into