What is the Law of Radioactive Decay?

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Grade Level:

Class 12

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Definition

What is it?

The Law of Radioactive Decay states that the rate at which radioactive nuclei decay (break down) is directly proportional to the number of undecayed nuclei present at that moment. Simply put, the more radioactive atoms you have, the faster they will decay.

Simple Example

Quick Example

Imagine you have a big basket full of ladoos, and each ladoo has a small timer. When a ladoo's timer runs out, it magically disappears. If you start with 100 ladoos, many will disappear quickly. But as the number of ladoos decreases, fewer will disappear in the same amount of time because there are fewer ladoos left to disappear.

Worked Example

Step-by-Step

Let's say we have a radioactive sample with an initial number of 1000 atoms (N0 = 1000). The decay constant (lambda, λ) for this sample is 0.1 per hour. We want to find the number of undecayed atoms (N) after 5 hours (t = 5 hours).

Step 1: Recall the formula for radioactive decay: N = N0 * e^(-λt).

Step 2: Identify the given values: N0 = 1000, λ = 0.1/hour, t = 5 hours.

Step 3: Substitute the values into the formula: N = 1000 * e^(-0.1 * 5).

Step 4: Calculate the exponent: -0.1 * 5 = -0.5.

Step 5: Calculate e^(-0.5). Using a calculator, e^(-0.5) is approximately 0.6065.

Step 6: Multiply: N = 1000 * 0.6065 = 606.5.

Step 7: Since you can't have half an atom, round to the nearest whole number.

Answer: After 5 hours, approximately 607 undecayed atoms remain.

Why It Matters

Understanding radioactive decay is crucial for building safe nuclear power plants (Energy sector) and for creating medical imaging devices like PET scans (Medicine). Scientists at ISRO use it to understand how elements behave in space, and engineers use it to date ancient artifacts, helping us learn about history.

Common Mistakes

MISTAKE: Confusing decay constant (lambda) with half-life (T1/2). | CORRECTION: The decay constant is a measure of the probability of decay per unit time, while half-life is the time taken for half of the radioactive material to decay. They are related by T1/2 = ln(2)/λ.

MISTAKE: Assuming the decay rate is constant over time. | CORRECTION: The decay rate is NOT constant; it decreases over time because the number of undecayed nuclei decreases, as stated by the law.

MISTAKE: Forgetting that the decay law applies to a large number of atoms, not individual atoms. | CORRECTION: Radioactive decay is a statistical process. We cannot predict when a single atom will decay, only the average behavior of many atoms.

Practice Questions

Try It Yourself

QUESTION: If a sample has 800 radioactive atoms and its decay constant is 0.05 per minute, what is the initial rate of decay? | ANSWER: Rate of decay = λ * N = 0.05 * 800 = 40 atoms per minute.

QUESTION: A radioactive sample initially has 10,000 atoms. After 2 hours, 7,000 atoms remain. Calculate the decay constant (λ) for this sample. (Hint: Use N = N0 * e^(-λt)) | ANSWER: N = N0 * e^(-λt) => 7000 = 10000 * e^(-λ * 2) => 0.7 = e^(-2λ) => ln(0.7) = -2λ => -0.3567 = -2λ => λ = 0.178 per hour (approximately).

QUESTION: A radioactive isotope has a decay constant of 0.01 per day. If a sample initially contains 500 grams of this isotope, how much will remain after 100 days? | ANSWER: N = N0 * e^(-λt) = 500 * e^(-0.01 * 100) = 500 * e^(-1) = 500 * 0.3678 = 183.9 grams (approximately).

MCQ

Quick Quiz

Which statement correctly describes the Law of Radioactive Decay?

The rate of decay is constant over time.

The rate of decay is proportional to the number of undecayed nuclei present.

The rate of decay depends on temperature and pressure.

All radioactive atoms decay at the same exact time.

The Correct Answer Is:

Option B correctly states that the rate of decay is proportional to the number of undecayed nuclei, which is the core principle of the law. Other options are incorrect because the decay rate is not constant, it's independent of external conditions like temperature, and decay is a random process for individual atoms.

Real World Connection

In the Real World

This law is used in carbon dating, a technique used by archaeologists to determine the age of ancient artifacts like pottery or bones found in places like Harappa. It also helps doctors in oncology departments decide dosages for radiation therapy, ensuring precise treatment for patients.

Key Vocabulary

Key Terms

Radioactive Decay: The process by which an unstable atomic nucleus loses energy by emitting radiation. | Decay Constant (λ): A measure of the probability that an atom will decay per unit time. | Nucleus: The central part of an atom, containing protons and neutrons. | Undecayed Nuclei: Radioactive atoms that have not yet undergone decay.

What's Next

What to Learn Next

Next, you should learn about 'Half-Life' and 'Mean Life' in radioactive decay. These concepts build directly on the Law of Radioactive Decay and help us understand how quickly radioactive substances diminish over time.