Understanding Radioactive Decay

What is the radioactive decay law and how is it expressed mathematically?

• N(t) = N0 + e^(λt)

• N(t) = N0 * λt

• N(t) = N0 / e^(λt)

• N(t) = N0 * e^(-λt)

Answer explanation

The radioactive decay law describes how the quantity of a radioactive substance decreases over time. It is mathematically expressed as N(t) = N0 * e^(-λt), where N0 is the initial quantity and λ is the decay constant.

Calculate the half-life of a substance if its initial activity is 80 Bq and after 3 hours it is 20 Bq.

• 2 hours

• 1 hour

• 1.5 hours

• 3 hours

Answer explanation

The activity decreases from 80 Bq to 20 Bq in 3 hours, indicating a reduction by a factor of 4. Since the activity halves every half-life, it takes 2 half-lives to reach 20 Bq. Therefore, the half-life is 3 hours / 2 = 1.5 hours.

Sketch a graph representing the exponential decay of a radioactive substance over time.

• A horizontal line above the x-axis.

• A downward-curving graph starting from a positive y-value and approaching the x-axis.

• A straight line increasing from the origin.

• A zigzag pattern oscillating between the x-axis and y-axis.

Answer explanation

The correct choice is a downward-curving graph starting from a positive y-value and approaching the x-axis, which accurately represents exponential decay, as the quantity decreases over time but never fully reaches zero.

Explain the relationship between the natural logarithm and the exponential decay function.

• Exponential decay functions are always positive and never reach zero.

• The natural logarithm is the inverse of the exponential decay function, allowing for the calculation of time or remaining quantity in decay processes.

• The natural logarithm is used to calculate growth rates in populations.

• The natural logarithm is a type of exponential function.

Answer explanation

The natural logarithm serves as the inverse of the exponential decay function, enabling calculations related to time or the remaining quantity in decay processes, making it essential for understanding decay dynamics.

If a sample has a half-life of 5 years, how much of a 100g sample remains after 15 years?

• 50g

• 75g

• 12.5g

• 25g

Answer explanation

After 15 years, which is three half-lives (5 years each), the remaining amount is 100g -> 50g -> 25g -> 12.5g. Thus, 12.5g of the original sample remains.

Define the term 'activity' in the context of radioactive decay.

• Activity refers to the total mass of a radioactive substance.

• Activity is the time it takes for half of a radioactive substance to decay.

• Activity is the rate of decay of a radioactive substance, measured in disintegrations per second.

• Activity is the energy released during radioactive decay.

Answer explanation

Activity in radioactive decay refers to the rate at which a radioactive substance decays, specifically measured in disintegrations per second. This distinguishes it from mass, half-life, and energy released.

What is the significance of the decay constant (λ) in radioactive decay?

• The decay constant (λ) measures the total mass of a radioactive substance.

• The decay constant (λ) indicates the energy released during radioactive decay.

• The decay constant (λ) determines the half-life of a radioactive isotope directly.

• The decay constant (λ) signifies the rate of radioactive decay, indicating the probability of decay per unit time.

Answer explanation

The decay constant (λ) is crucial as it signifies the rate of radioactive decay, representing the probability of decay per unit time. This directly influences the half-life of the isotope, making it a key parameter in understanding radioactive processes.

How does the count rate of a radioactive source change over time?

• The count rate decreases over time.

• The count rate remains constant over time.

• The count rate increases over time.

• The count rate fluctuates randomly over time.

Answer explanation

The count rate of a radioactive source decreases over time due to the decay of radioactive isotopes. As these isotopes decay, fewer particles are emitted, leading to a lower count rate.

If the initial number of undecayed nuclei is 1000, how many will remain after 2 half-lives?

• 250

• 1000

• 750

• 500

Answer explanation

After 1 half-life, 500 nuclei remain (1000/2). After 2 half-lives, half of 500 decays, leaving 250 nuclei (500/2). Thus, the correct answer is 250.

Discuss the concept of nuclear stability and how it relates to radioactive decay.

• Stable nuclei always emit radiation regardless of their composition.

• Radioactive decay occurs only in elements with high atomic numbers.

• Nuclear stability is solely determined by the number of protons.

• Nuclear stability is the balance of forces in a nucleus; instability leads to radioactive decay.

Answer explanation

Nuclear stability refers to the balance of attractive and repulsive forces within a nucleus. When this balance is disrupted, the nucleus becomes unstable, leading to radioactive decay as it seeks a more stable configuration.