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Radioactive Decay Calculator

Calculate decay rates, half-lives, and remaining amounts using exponential decay laws

Parameters

yearsⓘ
atomsⓘ
yearsⓘ
ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Decay Constant:
0.13;1/years0.13;1/years
Remaining Amount:
268.40;atoms268.40;atoms
Decayed Amount:
731.60;atoms731.60;atoms
Activity:
35.30;decays/year35.30;decays/year
Half-Lives Elapsed:
1.90;1.90;

Examples

Example 1: Carbon-14 Dating

Calculate the remaining amount of carbon-14 after 5730 years (one half-life) from an initial amount of 1000 atoms.

  • Decay Constant: 0.000.00
  • Remaining Amount: 500.00500.00
  • Decayed Amount: 500.00500.00
  • Half-Lives Elapsed: 1.001.00

Example 2: Uranium-238 Decay

Calculate the remaining uranium-238 after 4.5 billion years (one half-life) from an initial amount of 10000 atoms.

  • Decay Constant: 0.000.00
  • Remaining Amount: 5000.005000.00
  • Decayed Amount: 5000.005000.00
  • Half-Lives Elapsed: 1.001.00

Example 3: Cobalt-60 Medical Use

Calculate the remaining cobalt-60 after 5.27 years (one half-life) from an initial amount of 1000 atoms.

  • Decay Constant: 0.130.13
  • Remaining Amount: 500.00500.00
  • Decayed Amount: 500.00500.00
  • Half-Lives Elapsed: 1.001.00

Visualization

Radioactive Decay

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one, accompanied by the emission of radiation. This process occurs randomly and cannot be predicted for individual atoms, but follows statistical laws for large numbers of atoms.

The half-life is the time required for half of the radioactive atoms in a sample to decay. It is a characteristic property of each radioactive isotope and is independent of the initial amount of material. The half-life can range from fractions of a second to billions of years.

The decay constant (λ) is related to the half-life by the equation λ = ln(2)/T₁/₂. The decay constant represents the probability per unit time that a nucleus will decay. The exponential decay law N(t) = N₀e^(-λt) describes how the number of radioactive atoms decreases over time.

Different types of radioactive decay include alpha decay (emission of helium nuclei), beta decay (emission of electrons or positrons), and gamma decay (emission of electromagnetic radiation). Each type has different characteristics and energy releases.

Radioactive decay has numerous applications including nuclear power generation, medical imaging and therapy, carbon dating in archaeology, and nuclear waste management. Understanding decay processes is crucial for nuclear safety and various scientific applications.

Key Concepts

  • Half-Life: Time for half of radioactive atoms to decay
  • Decay Constant: Probability per unit time of decay
  • Exponential Decay: N(t) = N₀e^(-λt)
  • Alpha Decay: Emission of helium nuclei (⁴He)
  • Beta Decay: Emission of electrons (β⁻) or positrons (β⁺)
  • Gamma Decay: Emission of electromagnetic radiation

Real-World Applications

  • Nuclear power generation and nuclear medicine
  • Carbon dating and archaeological dating
  • Medical imaging and radiation therapy
  • Nuclear waste management and safety
  • Geological dating and earth sciences

Explore Further

More nuclear physics tools

Physics Equations

Decay Constant:
λ=ln⁡(2)T1/2\lambda = \frac{\ln(2)}{T_{1/2}}
Exponential Decay Law:
N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}
Remaining Amount:
N(t)=N0(12)tT1/2N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
Decayed Amount:
ΔN=N0−N(t)\Delta N = N_0 - N(t)
Activity (Decay Rate):
A(t)=λN(t)A(t) = \lambda N(t)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Decay Constant

First, we calculate the decay constant using the half-life:

Equation:

λ=ln⁡(2)T1/2\lambda = \frac{\ln(2)}{T_{1/2}}

Calculation:

λ=0.6935.27=0.131527 1/years\lambda = \frac{0.693}{5.27} = 0.131527 \text{ 1/years}

Explanation:

The decay constant represents the probability per unit time that a nucleus will decay.

2

Step 2: Calculate Remaining Amount

Using the exponential decay law, we calculate the remaining amount:

Equation:

N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Calculation:

N(t)=1000×e−0.131527×10.00=268 atomsN(t) = 1000 \times e^{-0.131527 \times 10.00} = 268 \text{ atoms}

Explanation:

The exponential decay law describes how the number of radioactive atoms decreases over time.

3

Step 3: Calculate Decayed Amount

The decayed amount is the difference between initial and remaining amounts:

Equation:

ΔN=N0−N(t)\Delta N = N_0 - N(t)

Calculation:

ΔN=1000−268=732 atoms\Delta N = 1000 - 268 = 732 \text{ atoms}

Explanation:

This represents the number of atoms that have decayed during the elapsed time.

4

Step 4: Calculate Activity

The activity (decay rate) is the product of decay constant and remaining amount:

Equation:

A(t)=λN(t)A(t) = \lambda N(t)

Calculation:

A(t)=0.131527×268=35.30 decays/yearA(t) = 0.131527 \times 268 = 35.30 \text{ decays/year}

Explanation:

Activity represents the number of decays per unit time at the current moment.

Frequently Asked Questions (FAQ)

What is radioactive decay?

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one, accompanied by the emission of radiation such as alpha particles, beta particles, or gamma rays.

What is half-life?

Half-life is the time required for half of the radioactive atoms in a sample to decay. It is a characteristic property of each radioactive isotope and is independent of the initial amount of material.

How does exponential decay work?

Exponential decay follows the equation N(t) = N₀e^(-λt), where N(t) is the remaining amount at time t, N₀ is the initial amount, λ is the decay constant, and t is the elapsed time.

What are the different types of radioactive decay?

The main types are alpha decay (emission of helium nuclei), beta decay (emission of electrons or positrons), and gamma decay (emission of electromagnetic radiation).

How is radioactive decay used in dating?

Radioactive decay is used in radiometric dating to determine the age of materials. For example, carbon-14 dating is used for organic materials up to about 50,000 years old.

Practice MCQs

  1. What is the relationship between half-life and decay constant?
  2. After one half-life, what fraction of radioactive atoms remain?
  3. The decay constant represents:
  4. Which type of decay emits helium nuclei?
  5. The exponential decay law is:

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