Half-Life Calculator

Half-Life Calculator

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Half-Life Calculator

Radioactive Decay Formula Solver

Decay Parameters

Radioactive Decay Formula:

$$N(t)=N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$$

Calculation Result

Enter values and click ‘Calculate’ to see the result here.

Formula Variables

  • $N(t)$: Remaining quantity after time $t$.
  • $N_0$: Initial quantity.
  • $T_{1/2}$: Half-life (the time required for a quantity to reduce to half its initial value).
  • $t$: Elapsed time.

What is Half-Life?

Half-life ($T_{1/2}$) is the time required for a quantity of a substance to reduce to half its initial value. This concept is most famously applied to radioactive decay, where unstable atomic nuclei spontaneously transform into a more stable state.

Biểu đồ suy giảm phóng xạ (Half-Life Decay Curve) $N_0$ 0 $T_{1/2}$ $N_0/2$ $2T_{1/2}$ $N_0/4$ $3T_{1/2}$ $N_0/8$ Elapsed Time ($t$) Quantity Remaining ($N(t)$)

The half-life of a radioactive isotope is a constant and intrinsic property of that isotope, completely independent of external conditions like temperature, pressure, or concentration. This property makes it invaluable in various fields:

  • Archaeology (Carbon Dating): The half-life of Carbon-14 is approximately $5,730$ years, allowing scientists to accurately date organic materials up to about $50,000$ years old.
  • Medicine: Radioactive tracers used in medical imaging (like PET scans) often have very short half-lives to ensure they decay quickly and minimize radiation exposure to the patient.
  • Nuclear Physics and Engineering: Understanding the half-lives of fissile materials is crucial for the safe design and management of nuclear reactors and waste storage.
  • Chemistry: It also applies to first-order chemical reactions, where the time taken for reactant concentration to halve is constant.

How to Use the Calculator

  1. Select the Variable to Solve For: Choose $N(t)$, $t$, or $T_{1/2}$ using the radio buttons. This will automatically hide the input field for the selected variable and require the other three values.
  2. Enter Input Values: Input the required numerical values for Initial Quantity ($N_0$), Half-Life ($T_{1/2}$), Elapsed Time ($t$), and/or Remaining Quantity ($N(t)$). Ensure all values are positive.
  3. Click ‘Calculate Result’: Press the button to run the calculation.
  4. Review Output: The final calculated value, formatted correctly, will be displayed below along with the full, substituted formula, rendered beautifully using LaTeX.

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