Half Life Formula Calculus: How to Solve Decay Problems

This calculator computes the remaining quantity of a substance after a given time using the calculus-based half-life formula, N(t) = N₀ × (1/2)^(t/t½). You enter the initial amount, the half-life, and the elapsed time, and you get back the remaining amount and the fraction left. It applies to radioactive decay, drug elimination, and any process that follows first-order exponential decay.

Formula

The half-life formula in calculus is derived from the differential equation for exponential decay:

dN/dt = -λN

where N is the quantity at time t, and λ (lambda) is the decay constant. Solving this differential equation gives:

N(t) = N₀ e^-λt

where N₀ is the initial quantity. The half-life (t½) is the time for N to drop to half of N₀. Setting N(t½) = N₀/2 gives:

t½ = ln(2)/λ, so λ = ln(2)/t½

Substituting λ back into the exponential solution yields the classic form:

N(t) = N₀ (1/2)^(t/t½)

This is the formula used in the calculator. Variables:

• Nâ‚€: initial amount (grams, milligrams, moles, etc.)
• N(t): amount remaining after time t (same unit as Nâ‚€)
• t½: half-life (any time unit, as long as t uses the same unit)
• t: elapsed time (same unit as t½)
• λ: decay constant (per unit time)

Worked Example

Using the default values: N₀ = 100 g, t½ = 10 hours, t = 25 hours.

Step 1: Compute the exponent: t/t½ = 25/10 = 2.5

Step 2: Compute 0.5^2.5 = 0.1767767 (since 0.5^2 = 0.25, and 0.5^0.5 = 0.7071, so 0.25 × 0.7071 = 0.1768)

Step 3: Multiply by N₀: 100 × 0.1768 = 17.68 g

The calculator shows 17.68 g (rounded to 2 decimals). The fraction remaining is 0.177 (rounded to 3 decimals).

When to Use This Formula

The half-life formula applies to any quantity that decays at a rate proportional to its current amount. This is called first-order kinetics. Common applications include:

• Radioactive decay (carbon-14 dating, medical isotopes)
• Drug elimination from the body (pharmacokinetics)
• Chemical reaction rates (first-order reactions)
• Capacitor discharge in electronics (though often written with time constant Ï„)

The key assumption is that the decay rate is constant—it does not change with time or concentration. If the process is second-order or zero-order, the half-life formula changes.

How to Read the Result

The remaining amount tells you how much of the original substance is left after the given time. The fraction remaining is the ratio N(t)/Nâ‚€, which is unitless and often more intuitive. For example, after 2 half-lives, 25% remains; after 3 half-lives, 12.5% remains. The calculator gives the exact fraction, not just an approximation.

If you are dealing with a radioactive sample, the remaining mass is proportional to the remaining activity (decays per second). If you are dealing with a drug, the remaining amount in the body determines the therapeutic effect and side effects.

Common Mistakes and Misconceptions

• Unit mismatch: The most common error is using different time units for t and t½. If t½ is in hours, t must be in hours, not minutes or days.
• Confusing half-life with mean life: The mean life (average lifetime) is Ï„ = 1/λ = t½/ln(2) ≈ 1.443 × t½. They are not the same.
• Assuming linear decay: Half-life decay is exponential, not linear. After two half-lives, you have 25%, not 0%.
• Forgetting the natural log: When solving for t½ from λ, you must use ln(2) ≈ 0.693, not log10(2) ≈ 0.301.

Limitations of the Formula

The simple half-life formula assumes:

• The decay process is first-order (rate proportional to amount)
• The half-life is constant (not affected by temperature, pressure, or concentration)
• No external factors change the decay rate (e.g., no new substance added)

For radioactive decay, the half-life is essentially constant under all normal conditions. For drug elimination, the half-life can vary with liver/kidney function, age, and interactions. For chemical reactions, a reaction may be first-order only under certain conditions. In those cases, the formula gives an approximation, not an exact prediction.

Relationship to the Differential Equation

If you are studying calculus, you might be asked to derive the half-life formula from the differential equation dN/dt = -λN. The solution is N(t) = N₀e^-λt. To find the half-life, set N(t½) = N₀/2, divide both sides by N₀, and take the natural log:

1/2 = e^-λt½

ln(1/2) = -λt½

-ln(2) = -λt½

t½ = ln(2)/λ

This is a classic first-order differential equation problem. The same approach works for other decay problems, such as Newton’s law of cooling or RC circuits.

Frequently Asked Questions

How do you calculate half-life from the decay constant?

Use t½ = ln(2)/λ. For example, if λ = 0.0693 per hour, then t½ = 0.693/0.0693 = 10 hours. The decay constant λ has units of inverse time (1/hour, 1/day, etc.).

What is the difference between half-life and decay constant?

The decay constant λ is the probability per unit time that a single atom will decay. The half-life t½ is the time for half the atoms to decay. They are inversely related: a larger λ means a shorter half-life. The relationship is t½ = ln(2)/λ.

Can I use this formula for any radioactive isotope?

Yes, for any isotope that undergoes exponential decay (which is most radioactive isotopes). You just need the correct half-life for that isotope. For example, carbon-14 has a half-life of about 5,730 years, while iodine-131 has a half-life of about 8 days.

How do I find the time when only a certain fraction remains?

Rearrange the formula: t = t½ × log₂(N₀/N). For instance, if you want to know when 10% remains, N/N₀ = 0.1, so t = t½ × log₂(10) ≈ 3.32 × t½. You can also use the natural log form: t = (t½/ln(2)) × ln(N₀/N).

Does the half-life formula work for drug elimination?

It works as an approximation for many drugs that follow first-order elimination. However, some drugs follow zero-order kinetics (like alcohol at high concentrations), where a fixed amount is eliminated per hour, not a fixed fraction. In those cases, the half-life concept is not directly applicable.

What is the difference between half-life and mean life?

Mean life (τ) is the average lifetime of a decaying particle. It is longer than the half-life: τ = 1/λ = t½/ln(2) ≈ 1.443 × t½. For example, if the half-life is 10 hours, the mean life is about 14.43 hours. Mean life is used in some physics equations, but half-life is more common in everyday contexts.

Related

• DC Cable Size Calculator: How to Size Wires
• Custody x Change Calculator: How It Works
• Dash Score Calculator: How to Use It

in N0 | Initial amount | g | 100 | 0
in t12 | Half-life | hours | 10 | 0
in t | Elapsed time | hours | 25 | 0
out N | Remaining amount | g | N0 * 0.5^(t/t12) | 2
out frac | Fraction remaining | - | 0.5^(t/t12) | 3