# AP Physics 2 : Radioactive Nuclear Decay

## Example Questions

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### Example Question #1 : Radioactive Nuclear Decay

Compound X is found to radioactively decay with a rate constant equal to . If a time of 20min passes by, what percentage of Compound X has decayed within this period?

Explanation:

To begin with, we'll have to make use of the radioactive decay equation:

Where:

= the amount of Compound X after an amount of time, , has passed

= the amount of Compound X at the start

= rate constant for the decay process

= the amount of time that has passed

Rearranging, we can see that:

Furthermore, we can convert minutes into seconds.

Next, we can plug the values we have into the above equation to obtain:

From the above expression, we see that:

This means that after a time of 20min has elapsed, there will be  as much of Compound X as there was when at the beginning. Since there is  remaining after this time period, then we can conclude that  of Compound X has degraded within this amount of time.

### Example Question #2 : Radioactive Nuclear Decay

Which of the following types of decay particles is known as a positron?

None of these particles are known as positrons

Explanation:

Positron decay (positive beta decay) is where a proton is converted into a neutron, a positron, and an electron neutrino.

Positrons are called  particles (beta plus particles). They are the antiparticle to  particles (the antimatter version of electrons). They are used in the medical field for PET (Positron Emission Tomography) scans by highlighting a radioactive substance (called a tracer) ingested earlier to show how organs and tissues are working. In nuclear reactors, they cause the water coolant to give a blue glow called Cherenkov radiation. This is because the positrons move faster than light does through water.

### Example Question #3 : Radioactive Nuclear Decay

The half-life of carbon-14 is 5730 years.

Rex the dog died in 1750. What percentage of his original carbon-14 remained in 1975 when he was found by scientists?

Explanation:

225 years have passed since Rex died. Find the number of half-lives that have elapsed.

To find the proportion of a substance that remains after a certain number of half-lives, use the following equation:

Here,  is the number of half lives that have elapsed.

### Example Question #4 : Radioactive Nuclear Decay

You measure the beta decay activity of an unknown substance to be . 48 hours later, the activity is .

What is the half life in hours?

Explanation:

Use the following equation:

The decay constant is defined as:

From the first equation, we find:

Plug this value into the second equation above and solve.

### Example Question #5 : Radioactive Nuclear Decay

A particular sample of a newly discovered isotope has an activity of . 10 minutes later, it has an activity of .

Determine the radioactive decay constant of this isotope.

None of these

Explanation:

Use the relationship:

Here, is the activity at a given time, is the initial activity, is the radioactive decay constant, and is the time passed since the initial reading.

Rearrange the equation to solve for .

Convert minutes to seconds and plug in values to solve for the radioactive decay constant.

### Example Question #6 : Radioactive Nuclear Decay

A scientist tests a radioactive sample which has an activity of . 15 minutes later, it has an activity of .

Determine the number of radioactive nuclei in the initial sample.

None of these

Explanation:

Use the relationship:

Where is the activity at a given time, is the initial activity, is the radioactive decay constant and is the time passed since the initial reading.

Rearrange to solve for .

Convert minutes to seconds and plug in values to find the decay constant.

It is then necessary to use the relationship:

Where is the activity, is the decay constant and is the number of atoms.

Use the initial activity and the calculated decay constant to solve for the number of atoms:

### Example Question #7 : Radioactive Nuclear Decay

A scientist tests a radioactive sample which has an activity of . 15 minutes later, it has an activity of .

Determine the activity 18 minutes after the initial reading.

None of these

Explanation:

Use the relationship:

Where is the activity at a given time, is the initial activity, is the radioactive decay constant and is the time passed since the initial reading.

Rearrange to solve for

Convert minutes to seconds and plug in values.

Again use the relationship:

Use the new , which is equal to  to plug in and solve for the activity.

### Example Question #8 : Radioactive Nuclear Decay

A scientist test a radioactive sample which has an activity of . 15 minutes later, it has an activity of .

Determine the half life of this isotope.

None of these

Explanation:

Use the relationship:

Here, is the activity at a given time, is the initial activity, is the radioactive decay constant, and is the time passed since the initial reading.

Rearrange the equation to solve for .

Convert minutes to seconds and plug in values.

Use the relationship:

Plug in the calculated value for :

### Example Question #9 : Radioactive Nuclear Decay

A scientist test a radioactive sample which has an activity of . 15 minutes later, it has an activity of .

Determine the nuclear decay constant of this isotope.

None of these

Explanation:

Use the relationship:

Here, is the activity at a given time, is the initial activity, is the radioactive decay constant, and is the time passed since the initial reading.

Rearrange the equation to solve for .

Convert minutes to seconds and plug in values.

### Example Question #10 : Radioactive Nuclear Decay

A scientist takes a reading of a radioactive material, which has an activity of . 15 minutes later, it has an activity of .

Determine the number of radioactive atoms in the initial sample.

None of these

Explanation:

Use the relationship:

Where is the activity at a given time, is the initial activity, is the radioactive decay constant and is the time passed since the initial reading.

Rearrange to solve for .

Convert minutes to seconds and plug in values.

Use the relationship:

Where is the activity, is the decay constant and is the number of nuclei.

Use the initial activity and the calculated decay constant to solve for the number of nuclei:

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