How to Calculate Radioactive Decay Using the Exponential Decay Formula

Hello! Great question about radioactive decay - this is a fascinating topic that shows up everywhere from nuclear physics to carbon dating! Let me walk you through this step by step.

1. What We're Solving:

We need to find how many radioactive atoms remain after 5 years, starting with 100,000 atoms and knowing they decay at a rate of 0.05 per year.

2. The Approach:

Radioactive decay follows an exponential decay pattern - this means the number of atoms decreases by a constant percentage each year, not a constant amount. Think of it like compound interest, but in reverse! We'll use the exponential decay formula because the rate of decay is always proportional to how many atoms are currently present.

3. Step-by-Step Solution:

Step 1: Identify our exponential decay formula The formula is: N(t) = N₀ × e^(-λt) Where:

• N(t) = number of atoms after time t
• N₀ = initial number of atoms
• λ = decay constant
• t = time
• e = mathematical constant (≈ 2.718)

Step 2: Plug in our known values

• N₀ = 100,000 atoms (our starting amount)
• λ = 0.05 per year (the decay constant)
• t = 5 years (the time we're interested in)

So: N(5) = 100,000 × e^(-0.05 × 5)

Step 3: Simplify the exponent N(5) = 100,000 × e^(-0.25)

Step 4: Calculate e^(-0.25) e^(-0.25) ≈ 0.7788

Step 5: Multiply to get our final answer N(5) = 100,000 × 0.7788 = 77,880 atoms

4. The Answer:

After 5 years, approximately 77,880 atoms will remain.

5. Memory Tip:

Remember "Exponential Everything" - when something decays (or grows) at a rate proportional to its current amount, reach for the e formula! The negative sign in the exponent tells you it's shrinking over time. Notice how we still have most of our atoms after 5 years - that's because 0.05 is a relatively small decay constant, meaning this isotope decays slowly.

You're doing great tackling these exponential problems! They're super useful in real life - from understanding half-lives to modeling population changes. Keep up the excellent work! 🌟