Exponential Decay: Half-Life Applications

What This Problem Teaches

• Understanding exponential decay models in real-world contexts
• Applying the half-life formula with fractional exponents
• Converting between different exponential forms (base 1/2 vs. base e)
• Interpreting decay rates as percentages of remaining substance
• Connecting mathematical models to nuclear physics and chemistry

Solution: Method 1 — The Half-Life Formula

Half-life problems follow a specific pattern: after each half-life period, exactly half of the substance remains. We can capture this with an exponential formula.

Step 1 — Identify the given information

We have an initial amount of A₀ = 80 grams, a half-life of h = 1690 years, and we want to find the amount after t = 940 years.

Step 2 — Apply the half-life decay formula

The standard half-life formula is:

Where A(t) is the amount at time t, A₀ is the initial amount, and h is the half-life.

Step 3 — Substitute the known values

Plugging in our values:

Step 4 — Simplify the exponent

Calculate the fraction in the exponent:

So we have: A(940) = 80 × (1/2)^0.5562

Step 5 — Evaluate the exponential expression

Using a calculator:

Step 6 — Find the final amount

Solution: Method 2 — The Natural Exponential Approach

We can also express radioactive decay using the natural exponential function, which gives us insight into the continuous nature of the decay process.

Step 1 — Find the decay constant

First, we need to find the decay constant k from the half-life. When t = h, we have A(h) = A₀/2:

Step 2 — Set up the continuous decay formula

The continuous decay model is:

Step 3 — Substitute our values

Step 4 — Simplify the exponent

Step 5 — Evaluate the exponential

Notice this matches our first method exactly — both approaches describe the same physical reality.

Verification

Let's check our answer using the ratio method. In 940 years, we've gone through 940/1690 ≈ 0.556 half-lives. This means the substance should be reduced by a factor of (1/2)^0.556.

Starting amount: 80 grams
Decay factor: (1/2)^0.556 ≈ 0.6797
Remaining: 80 × 0.6797 = 54.4 grams ✓

We can also verify this makes physical sense: 940 years is about 56% of one half-life, so we expect more than half the original amount to remain, which matches our answer of 54.4 grams (68% of the original).

Watch Out For These

The Pattern Behind This

All half-life problems follow the same mathematical structure. The general formula is:

This formula works for any radioactive isotope, any initial amount, and any time period. The key insight is that radioactive decay is proportional to the amount present — more atoms mean more decay events per unit time.

The exponent t/h tells us how many half-life periods have elapsed. If this equals 1, we get (1/2)^1 = 1/2, meaning half remains. If it equals 2, we get (1/2)^2 = 1/4, and so on.

In our problem, t/h = 940/1690 ≈ 0.556, which is between 0 and 1, so we expect between 50% and 100% to remain. Our answer of 54.4 grams (68% of 80) fits this expectation perfectly.

Real Applications

Medical imaging: Radioactive tracers used in PET scans and other diagnostic procedures must be carefully timed based on their half-lives to ensure effective imaging while minimizing radiation exposure.

Nuclear waste management: Engineers use half-life calculations to determine how long radioactive waste must be stored before it becomes safe, planning storage facilities that can contain materials for multiple half-lives.

Archaeological dating: Carbon-14 dating relies on the 5,730-year half-life of carbon-14 to determine the age of organic artifacts, using the same exponential decay mathematics we used here.

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