Half-Life & Exponential Decay: Radioactive Substance

What You Will Learn

• How to apply the half-life concept to calculate remaining quantities over time
• Understanding exponential decay as repeated multiplication by 1/2
• Converting time periods into half-life intervals for easier calculation
• Using the exponential decay formula A = A₀(1/2)^(t/T)
• Recognizing when to use sequential halving versus the general formula

Visualizing the Problem

Let's map out what happens to our radioactive tin over the 50-day period:

Notice how the amount halves every 10 days. After 50 days (which is 5 half-life periods), we need to find how much remains.

Solution: Method 1 — Sequential Halving

The most intuitive approach is to apply the half-life definition directly. Every 10 days, exactly half of the remaining tin decays.

Step 1 — Count the half-life periods

We need to find how many 10-day periods fit into 50 days:

Step 2 — Apply the halving repeatedly

Starting with 66,048 grams, we halve the amount 5 times:

Step 3 — Express as a single calculation

Since we're dividing by 2 five times, this is equivalent to:

Solution: Method 2 — Exponential Decay Formula

The general exponential decay formula gives us a direct path to the answer without sequential calculations.

Step 1 — Set up the decay formula

For radioactive decay with half-life T, the amount remaining after time t is:

Where A₀ = initial amount, t = elapsed time, T = half-life

Step 2 — Substitute our values

We have A₀ = 66,048 grams, t = 50 days, T = 10 days:

Step 3 — Calculate the exponential term

First, find (1/2)^5:

Step 4 — Complete the calculation

Verification

Let's check our answer by working backwards from the result:

Forward verification: If we start with 2,064 grams and let it grow by doubling 5 times (reverse of decay), we should get back to our starting amount:

Alternative calculation: We can also verify using the fraction of the original amount that remains:

Both checks confirm our answer is correct.

Does This Seem Reasonable?

Let's put our answer in context to see if it makes intuitive sense.

Magnitude check: We started with 66,048 grams and ended with 2,064 grams. That's about 3% of the original amount remaining after 50 days, which seems reasonable for 5 half-life periods.

Comparison with linear decay: If the tin decayed linearly (losing the same amount each day), we'd lose 66,048 ÷ 50 = 1,321 grams per day, leaving us with zero after 50 days. Exponential decay is much slower initially but never reaches zero.

Pattern recognition: Notice that 2,064 = 66,048 ÷ 32, and 32 = 2^5. This confirms we've correctly applied five halvings.

What Trips People Up

The General Formula

The exponential decay formula we used is part of a broader family of growth and decay models:

Where:

• Half-life formula: Best when the decay rate is given as a half-life period
• Continuous formula: Used in calculus and when decay happens continuously
• Discrete formula: Used when you know the percentage lost per period

Key insight: The exponent t/T tells you how many half-life periods have passed. This makes the calculation much more intuitive than working with decimal rates or natural logarithms.

Real Applications

Half-life calculations show up frequently outside of textbook problems:

Medical imaging: Radiologists use half-life formulas to calculate how long radioactive tracers remain active in patients. The same math determines safe waiting periods before repeat scans.

Nuclear power and waste: Engineers use these calculations to predict how long nuclear waste remains dangerous. Cesium-137, a common fission product, has a 30-year half-life—the same formula tells us how much will remain after decades of storage.

Carbon dating: Archaeologists use the 5,730-year half-life of carbon-14 to date organic artifacts. The exponential decay formula converts the remaining carbon-14 levels into the artifact's age.

What Comes After This

Once you're comfortable with half-life problems, the natural progressions include:

Inverse problems: Given the starting and ending amounts, find either the half-life or the elapsed time. These require logarithms to solve, since you're finding the exponent rather than calculating the result.

Continuous decay models: In calculus, you'll see the same decay pattern expressed as differential equations: dA/dt = -kA. The exponential solutions connect directly to what we've done here.

Multi-component systems: More advanced problems might involve multiple radioactive substances decaying simultaneously, each with different half-lives. You'd calculate each component separately and add the results.