Radioactive Decay and Activity

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Questions and Answers

What does the equation N/N0 = e–λt represent in the context of radioactive decay?

The total number of radioactive particles at time t

The fractional quantity of radionuclide remaining after time t (correct)

The rate of increase in activity over time

The initial mass of a radionuclide

How is the half-life T related to the decay constant λ?

T = 0.693 / λ (correct)

T = ln 2 / λ (correct)

T = e–λ/2

T = λ / ln 2

During the half-life period T, how does the activity A of a radionuclide behave?

It increases exponentially

It doubles

It remains constant

It decreases by a factor of one-half (correct)

What mathematical operation is used to derive the equation for half-life from the exponential decay law?

Taking the natural logarithm Signup and view all the answers

Which equation represents the activity of a radionuclide over time?

A = A0e–λt Signup and view all the answers

What happens to the dose rate at a fixed radionuclide source over time?

It decreases exponentially Signup and view all the answers

In the equation A/A0 = e–λt, what does the term A0 represent?

The initial activity at time t = 0 Signup and view all the answers

Which factor influences the rate of radioactive decay in the equations provided?

The decay constant λ Signup and view all the answers

What is the decay constant λ of the nuclide with a half-life of 15.0 hours?

0.0462 h–1 Signup and view all the answers

After 2.5 days, what is the activity of a 30-MBq source of the radionuclide mentioned?

1.88 MBq Signup and view all the answers

What is the total beta activity in a solution containing 0.10 µCi of 198Au and 0.04 µCi of 131I after 21 days?

7.02 × 10–3 µCi Signup and view all the answers

At what time will the total beta activity of the isotopes decay to half of its original value?

Both 2.70 and 8.05 days Signup and view all the answers

What is the exponent used in the equations to calculate decay from the half-life?

–0.693t Signup and view all the answers

Which of the following statements is true regarding the isotopes mentioned?

Both isotopes decay to stable daughters. Signup and view all the answers

What is the required condition for the exponential decay equation to be dimensionless?

Time units for λ and t must be the same. Signup and view all the answers

Which formula correctly relates the activity and decay constant?

A = A0 * e^(-λT) Signup and view all the answers

What is the unit of activity for radionuclides?

Bequerel (Bq) Signup and view all the answers

What relationship does the decay constant λ have with the number of atoms N in a sample?

λ is independent of N. Signup and view all the answers

What does the equation dN = –λN dt represent in radioactive decay?

The change in the number of atoms over time. Signup and view all the answers

How is the activity (A) of a radionuclide expressed mathematically?

A = –λN Signup and view all the answers

What does the negative sign in the equation dN = –λN dt indicate?

The number of atoms decreases over time. Signup and view all the answers

What is the significance of the constant c in the integrated equation ln N = –λt + c?

It depends on the initial amount of radionuclide. Signup and view all the answers

What does the integration of both sides of the equation dN/dt = –λN yield?

A logarithmic relationship between N and time. Signup and view all the answers

What transformation does the unit curie (Ci) represent in terms of becquerels (Bq)?

1 Ci = 3.7 x 10^10 Bq Signup and view all the answers

Study Notes

Activity

Activity measures the decay rate of a radionuclide, indicating the number of atoms decaying per unit time.
The unit of measurement for activity is the becquerel (Bq), defined as one disintegration per second (1 Bq = 1 s⁻¹).
The curie (Ci) is a traditional unit of activity, equivalent to 3.7 × 10¹⁰ Bq, based on 1 gram of 226Ra.

Exponential Decay

The quantity N represents the number of atoms of a radionuclide at a given time.
The change in atom number (dN) over time (dt) is proportional to the current quantity of atoms (N) multiplied by a decay constant (λ), expressed as dN = –λN dt.
The decay constant (λ) has units of inverse time (s⁻¹) and indicates the rate of decay of radionuclides.
The activity (A) of a sample relates to the decay:
- A = –(dN/dt) = λN.

Mathematical Foundations

Integration of the decay equation leads to ln N = –λt + c, with c determined by initial conditions, particularly if N0 atoms are present at t=0.
The exponential decay law can be expressed as:
- N/N0 = e⁻λt, indicating that the number of atoms decreases exponentially over time.

Activity and Half-Life

The relationship between activity at a specific time (A) and initial activity (A0) is given by A/A0 = e⁻λt.
The half-life (T) represents the time taken for the activity to reduce to half its original value, defined mathematically as:
- 1/2 = e⁻λT, leading to T = ln 2/λ (approximately T = 0.693/λ).

Examples of Decay Calculations

To calculate the activity of a radionuclide after a specified time using its decay constant:
- Activity (A) after time t can be calculated using A = A0 e⁻λt, ensuring time units are consistent.
Example given for a 30-MBq source of Na-24 after 2.5 days shows:
- λ calculated as 0.0462 h⁻¹ and resulting activity A = 1.88 MBq after time t.

Total Activity Calculation

For solutions containing multiple radionuclides, the total beta activity is the sum of individual activities at a given time.
Example calculation provided for a solution with 0.10 µCi of Au-198 and 0.04 µCi of I-131 at time t = 0:
- Activity calculated for both isotopes at t = 21 days gives a total beta activity of 7.02 × 10⁻³ µCi.

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Description

This quiz explores the concepts of radioactive decay, focusing on the activity of radionuclides. It examines the definitions of becquerel and curie, as well as exponential decay principles. Test your knowledge on the rates of transformation and units of measurement in radiation physics.