Half Life Formula: Calculate Decay Easily

This calculator works out the half-life of a radioactive substance using the standard exponential decay formula. You enter the initial quantity, the remaining quantity, and the elapsed time, and it returns the half-life in the same time unit you used. The result tells you how long it takes for half of the original substance to decay, which is the key constant for describing radioactive decay rates.

Formula

The half-life formula comes directly from the exponential decay law. If you have an initial amount N₀ and an amount N_t remaining after time t, the half-life T_1/2 is:

T_1/2 = t × ln(2) / ln(N₀ / N_t)

• t – elapsed time (any unit: seconds, minutes, hours, years)
• N₀ – initial quantity (mass, number of atoms, activity – any proportional measure)
• N_t – quantity remaining after time t (same unit as N₀)
• T_1/2 – half-life (same time unit as t)
• ln – natural logarithm (base e)

The decay constant λ (lambda) is related by λ = ln(2) / T_1/2. The calculator also outputs this value, which is useful when you want to plug it into the decay equation N_t = N₀e^−λt.

Worked Example

Using the default values from the calculator above:

1. Elapsed time t = 10 hours
2. Initial amount N₀ = 100 g
3. Remaining amount N_t = 25 g

Plug into the formula:

T_1/2 = 10 × ln(2) / ln(100 / 25)

First, 100 / 25 = 4. Then ln(4) ≈ 1.38629 and ln(2) ≈ 0.69315.

T_1/2 = 10 × 0.69315 / 1.38629 = 6.9315 / 1.38629 ≈ 5.00 hours

The half-life is 5 hours. That means after 5 hours, 50 g remains; after another 5 hours (10 total), 25 g remains – which matches the input. The decay constant λ = ln(2) / 5 ≈ 0.1386 per hour.

When to Use the Half-Life Formula

Half-life is used for any process that follows first-order exponential decay, not just nuclear physics. Common applications include:

• Radioactive isotopes in medicine, geology, and archaeology (e.g., carbon-14 dating)
• Pharmacokinetics – how long a drug stays in the body (biological half-life)
• Chemical reaction rates that are first-order
• Capacitor discharge in electronics (though usually expressed as time constant Ï„)
• Environmental persistence of pollutants

The formula above works whenever the decay rate is proportional to the amount present. If the decay is not exponential (e.g., zero-order or second-order), this formula does not apply.

How to Read the Result

The half-life is a constant for a given substance under fixed conditions – it does not change with how much material you start with. If you get 5 hours from the example, that means every 5 hours the amount halves, regardless of whether you started with 1 gram or 1 ton.

Note that the half-life is not the time until the substance is gone. After one half-life, 50% remains; after two, 25%; after three, 12.5%; and so on. It never reaches zero mathematically, though in practice it becomes negligible.

The decay constant λ tells you the probability per unit time that a single atom will decay. A larger λ means a shorter half-life and faster decay.

Common Mistakes and Unit Pitfalls

• Mixing units – If you enter time in hours, the half-life comes out in hours. Convert everything to the same time unit first (e.g., don’t mix minutes and hours).
• Using the wrong ratio – The formula uses N₀ / N_t, not N_t / N₀. Reversing the ratio gives a negative half-life, which is meaningless.
• Forgetting ln(2) – Some people use the natural log of the ratio alone, which gives the mean lifetime, not the half-life. The factor ln(2) ≈ 0.693 is essential.
• Rounding too early – Keep at least 4-5 significant figures during intermediate steps. Rounding ln(4) to 1.39 instead of 1.38629 introduces noticeable error.
• Assuming the remaining amount is exact – In real measurements, the remaining amount has uncertainty. Small errors in measuring N_t can cause large errors in the half-life, especially when the remaining fraction is close to 1 or very small.

Limitations and Real-World Considerations

The formula assumes pure exponential decay with a single decay constant. In reality:

• Some isotopes decay through multiple pathways (branching), each with its own constant – the observed half-life is a weighted average.
• If the substance is being produced or replenished while decaying (e.g., in a nuclear reactor), the simple formula does not apply.
• Environmental conditions (temperature, pressure, chemical state) can affect the half-life of some decay modes, like electron capture, though for most nuclear decays the effect is negligible.
• The formula gives the half-life from two measurements. If you have only one measurement and a known half-life, you would instead solve for t or N_t.

For carbon-14 dating, the half-life is about 5,730 years, but the calculation also requires knowing the initial carbon-14 ratio, which is estimated from atmospheric data. This calculator gives the half-life from your own data; it does not date objects.

Frequently Asked Questions

What is the half-life formula if I know the decay constant?

If you know the decay constant λ, the half-life is simply T_1/2 = ln(2) / λ. For example, if λ = 0.1386 per hour, then T_1/2 = 0.69315 / 0.1386 ≈ 5 hours.

Can I calculate half-life from just the initial and remaining amounts without time?

No. The half-life is a rate constant – it has units of time. Without an elapsed time, you cannot determine how fast the decay happens. You need at least two points in time (or one point plus the decay constant) to find the half-life.

Why is the half-life constant regardless of how much material I have?

Because radioactive decay is a first-order process: the rate of decay is proportional to the number of atoms present. If you double the amount, the decay rate doubles, so the time to halve remains the same. This is a statistical property of large numbers of atoms.

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

The mean lifetime (τ) is the average time an atom survives before decaying. It is related to the half-life by τ = T_1/2 / ln(2) ≈ 1.443 × T_1/2. The half-life is the median survival time – 50% of atoms decay by then, while the mean is longer because a few atoms survive for many half-lives.

Does the half-life formula work for drug elimination in the body?

Yes, for drugs that follow first-order elimination. You would use the same formula with plasma concentration instead of mass. However, many drugs have multi-phasic elimination (distribution phase then elimination phase), so a single half-life may not describe the whole curve. Also, the biological half-life can vary with age, kidney function, and other factors.

What if my remaining amount is greater than the initial amount?

That would mean the substance is growing, not decaying – for example, a sample being produced by a parent isotope. The formula would give a negative half-life, which is not physically meaningful for decay. In such cases, you are dealing with a growth or production scenario, not simple decay.

Related

• Expense Split Calculator: How It Works
• Experian Break Even Calculator: What It Does
• Exact Mass Calculator: Formula & How to Use

in t | Elapsed time | hours | 10 | 0
in n0 | Initial amount | g | 100 | 0
in nt | Remaining amount | g | 25 | 0
out hl | Half-life | hours | t * ln(2) / ln(n0 / nt) | 2
out k | Decay constant | per hour | ln(2) / hl | 4