Half Life Calculator with Rate Constant

Half-Life & Decay Calculator

Calculate half-life from the rate constant, and determine the remaining quantity over time.

Calculation Results

• Calculated Half-Life ($t_{1/2}$): " + halfLife.toPrecision(5) + " time units

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• Remaining Quantity ($N_t$): " + remainingQty.toPrecision(5) + " initial units

In nuclear physics, chemistry, and pharmacology, the concepts of half-life and rate constants are fundamental to understanding how substances decay or are eliminated over time. This process typically follows first-order kinetics, meaning the rate of decay is proportional to the current amount of the substance.

What is the Decay Rate Constant ($k$ or $\lambda$)?

The decay rate constant, often denoted as $k$ or lambda ($\lambda$), defines the probability of a specific nucleus decaying per unit of time. It is an intrinsic property of a radioactive isotope or a chemical reactant in a first-order reaction. A larger rate constant means the substance decays faster.

What is Half-Life ($t_{1/2}$)?

Half-life is the time required for a quantity to reduce to exactly half of its initial value. It is inversely proportional to the rate constant.

Key Formulas Relationship

The relationship between the half-life and the rate constant is fixed for first-order processes. You can convert between them using the natural logarithm of 2 ($\ln(2) \approx 0.693$):

$t_{1/2} = \frac{\ln(2)}{k}$

$k = \frac{\ln(2)}{t_{1/2}}$

Calculating Remaining Quantity Over Time

To determine how much of a substance remains after a specific elapsed time ($t$), we use the exponential decay equation:

$N(t) = N_0 \cdot e^{-kt}$

Where:

• $N(t)$: The remaining quantity at time $t$.
• $N_0$: The initial quantity at time $t=0$.
• $e$: Euler's number (mathematical constant approx. 2.718).
• $k$: The decay rate constant.
• $t$: The elapsed time.

Realistic Example: Medical Isotope Decay

Consider Iodine-131 (I-131), commonly used in radiotherapy. It has a decay rate constant $k$ of approximately **0.0864 per day** (days⁻¹).

If a hospital starts with an initial sample of **100 mg** of I-131, how much remains after **16 days**, and what is the half-life?

1. Calculate Half-Life: $t_{1/2} = 0.693 / 0.0864 \approx 8.02$ days.
2. Calculate Remaining Amount: Using the calculator above, inputting $N_0 = 100$, $k = 0.0864$, and $t = 16$, the remaining amount is approximately **25.1 mg**.

This makes sense intuitively, as 16 days is roughly two half-lives (2 x 8.02 = 16.04). After two half-lives, you expect about a quarter ($1/2 \times 1/2 = 1/4$) of the original substance to remain.

Use the calculator at the top of this page to perform these calculations instantly for any substance following first-order exponential decay.