what fraction of a 15o sample decays in 10 min

Daniel Parker

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25-02-2025

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Radioactive Decay: Calculating the Fraction of a 150 Sample Decaying in 10 Minutes

Understanding radioactive decay is crucial in various fields, from nuclear medicine to geology. This article will guide you through calculating the fraction of a 150-unit sample that decays within 10 minutes, given a specific decay constant or half-life. We'll explore the relevant formulas and provide a step-by-step solution.

Understanding Radioactive Decay

Radioactive decay is a first-order process, meaning the rate of decay is directly proportional to the amount of radioactive material present. This is described mathematically by the following equation:

N(t) = N₀ * e^(-λt)

Where:

• N(t) is the amount of the substance remaining after time t.
• N₀ is the initial amount of the substance.
• λ is the decay constant (related to the half-life).
• t is the time elapsed.
• e is the base of the natural logarithm (approximately 2.718).

To use this formula, we need either the decay constant (λ) or the half-life (t₁/₂). The half-life is the time it takes for half of the substance to decay. The relationship between the decay constant and half-life is:

λ = ln(2) / t₁/₂

Scenario: Decay of a 150-unit Sample

Let's assume we have a 150-unit sample of a radioactive substance. We need additional information to solve the problem: either the decay constant (λ) or the half-life (t₁/₂).

Example 1: Using the Decay Constant

Let's say the decay constant (λ) for our substance is 0.05 per minute. We want to find the fraction that decays in 10 minutes.

1. Calculate the remaining amount:

   N(10) = 150 * e^(-0.05 * 10) N(10) ≈ 150 * e^(-0.5) ≈ 150 * 0.6065 ≈ 90.98 units

2. Calculate the decayed amount:

   Decayed amount = N₀ - N(10) = 150 - 90.98 ≈ 59.02 units

3. Calculate the fraction decayed:

   Fraction decayed = (Decayed amount) / N₀ = 59.02 / 150 ≈ 0.393

Therefore, approximately 39.3% of the sample decays in 10 minutes.

Example 2: Using the Half-Life

Let's say the half-life (t₁/₂) of our substance is 13.86 minutes.

1. Calculate the decay constant:

   λ = ln(2) / 13.86 ≈ 0.05 per minute

2. Follow steps 1-3 from Example 1: This will yield the same result (approximately 39.3% decay).

Important Considerations

• Units: Ensure consistent units throughout your calculations (e.g., minutes for time, same units for initial and remaining amounts).
• Accuracy: The accuracy of your result depends on the accuracy of the provided decay constant or half-life.
• Complex Decay: Some substances undergo more complex decay schemes involving multiple isotopes or decay pathways. This requires more sophisticated calculations.

This article provides a foundational understanding of calculating radioactive decay. Remember to always carefully consider the specific decay parameters of the substance you are working with. Consulting relevant scientific literature and using appropriate tools will enhance the accuracy of your calculations.