Bateman Equation Calculator

To calculate the remaining number of radioactive atoms, multiply the initial number of atoms by Euler’s number raised to the negative product of the decay constant and time. This will give you the number of radioactive atoms left after the given time.

Formula:

$$N = N_0 \times e^{-\lambda \times t}$$

Where:

• N = Number of radioactive atoms remaining after time $t$
• N₀ = Initial number of radioactive atoms
• λ = Decay constant (rate of decay)
• t = Time (in seconds, minutes, or years)
• e = Euler’s number (approximately 2.718)

Example Calculation:

Let’s assume the initial number of atoms (N₀) is 1000 atoms, the decay constant (λ) is 0.02, and the time elapsed (t) is 10 years.

Using the formula:

$$N = 1000 \times e^{-(0.02 \times 10)} = 1000 \times e^{-0.2} \approx 1000 \times 0.8187 = 818.7 \, \text{atoms}$$

So, after 10 years, approximately 818.7 atoms remain.

Step-by-Step Calculation :

What is Bateman Equation Calculator ?

The Bateman Equation Calculator is a tool used to solve the equations related to radioactive decay chains, where a radioactive material decays into another radioactive substance, which in turn decays, continuing in a chain. The Bateman equation calculates the activity or the number of decays per second for each substance in the decay chain over time.

The general form of the Bateman equation is:

$$N(t) = N_0 \sum_{i=1}^{n} \left[ \frac{\prod_{j=1}^{i-1} \lambda_j}{\prod_{j=1}^{i} (\lambda_j – \lambda_i)} e^{-\lambda_i t} \right]$$

Where:

• N(t) is the number of atoms at time t,
• N₀ is the initial number of atoms,
• λ₁, λ₂, … λₙ are the decay constants for the isotopes in the chain.

This equation is crucial in nuclear physics for understanding how materials decay over time in a chain reaction. A Bateman equation calculator helps simplify this calculation, especially for long decay chains where manual calculation becomes complex.

Tools like a decay chain calculator or nuclear equation solver can automatically compute the number of remaining atoms at any time step, providing insights into the activity (rate of decay) of each isotope.

The Bateman equation also finds applications in chemistry, thermodynamics, and biology, where it helps in solving problems related to reaction rates, biological decay, or population dynamics. You can find Bateman equation calculators in PDF or app formats for easy access to these complex calculations.