Activity & Decay Constant | CAIE 9702 Paper 4 Notes

Chapters

Paper 4

Physics9702

Activity is not the same as detector count rate

counts nuclear disintegrations in the sample. One becquerel means one disintegration per second.

Definition

1 mark

What is meant by the activity of a radioactive sample?

Model answer: Activity is the number of nuclear disintegrations per unit time.

(9702/41/M/J/25 Q9(a))

1\ \mathrm{Bq}=1\ \mathrm{s}^{-1}

Activity is positive. The number of undecayed nuclei decreases, so its rate of change is negative.

A=-\frac{\mathrm{d}N}{\mathrm{d}t}

Swipe left or right to read every column →

Quantity	What it measures	Why the values differ
Activity	All nuclear disintegrations in the source per unit time	Property of the source
Received count rate	Radiation events recorded by the detector per unit time	Some radiation misses the detector or is absorbed
Corrected count rate	Received count rate after background is subtracted	Used in the decay equation

If the detector geometry and efficiency stay fixed, corrected count rate is a constant fraction of activity. It therefore follows the same exponential decay law.

The decay constant describes one nucleus

Different nuclides have different probabilities of decay. The decay constant gives the probability per unit time for one undecayed nucleus.

Definition

1 mark

What is meant by the decay constant?

Model answer: The decay constant is the probability per unit time that one undecayed nucleus decays.

A=\lambda N

Symbols

$A$ = activity (Bq)
$\lambda$ = decay constant (s⁻¹)
$N$ = number of undecayed nuclei (none)

At fixed $\lambda$ , doubling $N$ doubles the activity because twice as many nuclei have a chance to decay.
At fixed $N$ , a larger $\lambda$ gives a larger activity because each nucleus is more likely to decay per second.
During decay, $\lambda$ remains constant for the nuclide while $N$ decreases. Therefore the activity decreases.
If both $N$ and $\lambda$ change, calculate their product before deciding how activity changes.

Your turnrelationship check

Two samples contain the same number of undecayed nuclei. Sample X has twice the decay constant of sample Y. Compare their activities.

Show worked answer

Sample X has twice the activity. The number of nuclei is fixed, so activity is directly proportional to the decay constant.

Your turncalculation check

A sample contains $3.2\times10^8$ undecayed nuclei. Its decay constant is $2.5\times10^{-4}\ \mathrm{s}^{-1}$ . Calculate its activity.

Show worked answer

A=(2.5\times10^{-4})(3.2\times10^8)

The activity is $8.0\times10^4\ \mathrm{Bq}$ .

Common mistake

Activity and the product below are both positive.

A=\lambda N

Do not place a minus sign in this equation. The minus sign belongs only in the rate-of-change form because $N$ decreases.

Activity is not the same as detector count rate

counts nuclear disintegrations in the sample. One becquerel means one disintegration per second.

Definition

1 mark

What is meant by the activity of a radioactive sample?

Model answer: Activity is the number of nuclear disintegrations per unit time.

(9702/41/M/J/25 Q9(a))

1\ \mathrm{Bq}=1\ \mathrm{s}^{-1}

Activity is positive. The number of undecayed nuclei decreases, so its rate of change is negative.

A=-\frac{\mathrm{d}N}{\mathrm{d}t}

Swipe left or right to read every column →

Quantity	What it measures	Why the values differ
Activity	All nuclear disintegrations in the source per unit time	Property of the source
Received count rate	Radiation events recorded by the detector per unit time	Some radiation misses the detector or is absorbed
Corrected count rate	Received count rate after background is subtracted	Used in the decay equation

If the detector geometry and efficiency stay fixed, corrected count rate is a constant fraction of activity. It therefore follows the same exponential decay law.

The decay constant describes one nucleus

Different nuclides have different probabilities of decay. The decay constant gives the probability per unit time for one undecayed nucleus.

Definition

1 mark

What is meant by the decay constant?

Model answer: The decay constant is the probability per unit time that one undecayed nucleus decays.

A=\lambda N

Symbols

$A$ = activity (Bq)
$\lambda$ = decay constant (s⁻¹)
$N$ = number of undecayed nuclei (none)

At fixed $\lambda$ , doubling $N$ doubles the activity because twice as many nuclei have a chance to decay.
At fixed $N$ , a larger $\lambda$ gives a larger activity because each nucleus is more likely to decay per second.
During decay, $\lambda$ remains constant for the nuclide while $N$ decreases. Therefore the activity decreases.
If both $N$ and $\lambda$ change, calculate their product before deciding how activity changes.

Your turnrelationship check

Two samples contain the same number of undecayed nuclei. Sample X has twice the decay constant of sample Y. Compare their activities.

Show worked answer

Sample X has twice the activity. The number of nuclei is fixed, so activity is directly proportional to the decay constant.

Your turncalculation check

A sample contains $3.2\times10^8$ undecayed nuclei. Its decay constant is $2.5\times10^{-4}\ \mathrm{s}^{-1}$ . Calculate its activity.

Show worked answer

A=(2.5\times10^{-4})(3.2\times10^8)

The activity is $8.0\times10^4\ \mathrm{Bq}$ .

Common mistake

Activity and the product below are both positive.

A=\lambda N

Do not place a minus sign in this equation. The minus sign belongs only in the rate-of-change form because $N$ decreases.