Half Life Formula: Which of Three Forms Fits Your Question

• Pick the formula from what the question gives. Times, or a fraction left that is a whole power of ½, mean N = N₀ ÷ 2^(t÷T). Two activity readings or a decay constant mean A = A₀e^(−λt) and T = 0.693 ÷ λ.
• For two samples, compare activity, A = λN = 0.693 × N ÷ T. Comparing half lives alone gives the wrong ratio whenever the nuclei counts differ.
• Run four checks on every answer, including matching units and T × λ ≈ 0.693.
• Use the half life the question gives. Measured values differ slightly between sources: for carbon-14 the IAEA lists 5700 years and a standard textbook uses 5730 years, which moves the age in a textbook example by about 0.5%.

A half life formula needs three inputs: the initial amount N₀, the half life T and the elapsed time t. They give the number of half lives and the amount left. Keep t and T in the same unit. N comes out in whatever unit N₀ uses.

Key terms:

• Half life (T): the time for half of the radioactive nuclei in a sample to decay.
• Decay constant (λ): the chance per unit time that one nucleus decays, written in "per hour", "per minute" and so on.
• Activity (A): the number of decays per unit time. One becquerel (Bq) is one decay per second.

The default half life is sodium-24, which the IAEA (International Atomic Energy Agency) lists as 14.956 hours in data queried on 29 October 2026. We round it to 15 hours for exam-style work.

The decay law N = N₀e^(−λt), the link λ = ln 2 ÷ T ≈ 0.693 ÷ T and the activity relation A = λN are set out in OpenStax Physics, section 22.3. Substituting λ = ln 2 ÷ T gives N = N₀ × 2^(−t÷T), which is the halving form in the table. Our arithmetic for the defaults: 45 ÷ 15 = 3, then 100 ÷ 2³ = 12.5.

Choose the formula from what the question hands you, not from what it asks for. The given quantities decide whether you can simply halve, need a logarithm, or must bring in activity.

Our library of previous year questions (PYQs) shows where to practise them. The full list of Class 12 formulas sits on our HSC Physics formula sheet, so we do not repeat it here.

Work each kind with its own formula and cross-check it against the halving method. The three examples below are our own calculations, using real measured half lives.

Times: how much sodium-24 is left after 45 hours?

Start with 100 mg, T = 15 hours, t = 45 hours. The number of half lives is 45 ÷ 15 = 3, so N = 100 ÷ 8 = 12.5 mg left and 87.5 mg has decayed. With the exact IAEA value of 14.956 hours, n is 3.009 and 12.4 mg is left, a gap of under 0.1 mg.

If the fraction is not a power of ½, use n = log₂(N₀ ÷ N) = ln(N₀ ÷ N) ÷ ln 2. For 25% decayed (75% left) that gives n = 0.415, so about 6.2 hours for a 15-hour half life.

Rates: what is the decay constant if activity falls from 8000 to 2000 per minute in 4 minutes?

Activity falls from 8000 to 2000 disintegrations per minute in 4 minutes. Then λ = (1 ÷ 4) × ln(8000 ÷ 2000) = ln 4 ÷ 4 = 0.347 per minute, and T = 0.693 ÷ 0.347 = 2.0 minutes.

Halving confirms it: 8000 → 4000 → 2000 is two half lives in 4 minutes, so T = 2 minutes.

Change the numbers to 9000 falling to 3000 in 2 minutes and halving fails, because 3 is not a power of 2. The logarithm still works: λ = ln 3 ÷ 2 = 0.549 per minute and T = 1.26 minutes.

Two samples: which is more active, phosphorus-32 or iodine-131?

Take equal numbers of nuclei of phosphorus-32 (14.266 days) and iodine-131 (8.0252 days). Since A = 0.693 × N ÷ T, the activity ratio is A(I-131) ÷ A(P-32) = 14.266 ÷ 8.0252 = 1.78.

The shorter half life means more decays per second from the same number of nuclei. If the phosphorus sample had twice as many nuclei, the ratio would fall to 1.78 ÷ 2 = 0.89, so the half life comparison alone would mislead.

Six slips cause many wrong answers in numericals (calculation problems), and each has a quick test that catches it before you write the answer.

The slips:

1. Halving when the ratio is not a power of 2. A fall from 9000 to 3000 needs ln.
2. Using the given time as the half life. If activity falls to ⅛ in 4 days, three half lives fit in 4 days, so T = 4 ÷ 3 days. In 16 days that is 12 half lives, and the activity is 1 ÷ 2¹² = 1 ÷ 4096 of the original, about 0.024%.
3. Putting a minus sign on λ. λ is positive. The minus sits in the exponent of e^(−λt).
4. Multiplying instead of dividing. T = 0.693 ÷ λ, not 0.693 × λ.
5. Expecting zero after two half lives. 25% is left then. The test is below.
6. Mixing units. Hours with minutes, or a λ in "per minute" with a time in seconds. See our note on units and dimensions for the habit of checking units first.

Four tests to run on every answer:

• Units: t and T are in the same unit. Pass: both say "hours" or both say "minutes". Fail: any mismatch.
• Bracket: the amount left lies between the two whole-half-life values on either side of n. Pass: for n between 2 and 3, the fraction left is between 12.5% and 25%.
• Product: T × λ is about 0.693. Pass: 15 hours × 0.0462 per hour = 0.693. Fail: a product near 1 means mean life was used as the half life. A product near 1.44 means 0.693 was inverted (T × λ = 1 ÷ 0.693). Any other value means a unit or arithmetic slip.
• Direction: Pass: λ > 0, N ≤ N₀ and τ ≈ 1.44 × T. Fail: a negative λ, N larger than N₀, or τ smaller than T.

If you want the full routine for any numerical, our how to solve physics numericals page gives four steps for HSC and MHT-CET.

Practise with measured half lives from the IAEA, because they show what realistic numbers look like. Half lives are measured, not derived: the formula converts between T and λ, but cannot produce T on its own. IAEA data queried 29 October 2026, from the IAEA LiveChart.

Carbon-14 shows why you use the question's own value. OpenStax uses 5730 years and works out that a cloth with 92% of its original carbon-14 is about 690 years old. Our check gives 689 years at 5730 and 686 years at 5700, about 0.5% apart.

We teach MHT-CET Physics through our MHTCET 2027 Long Term Batch, taught by RG Sir (Rahul Jewaani). As of October 2026 it costs ₹1,632. It covers the full MHT-CET Physics syllabus, with live classes and recordings, notes, PYQs, mock tests and Class 11 fundamentals.

The decision table on this page is the habit we want every numerical to start with: write what the question gives, then choose the formula. The batch's PYQs and mock tests are where you test that habit on exam-style questions, and the recordings let you replay a method until it sticks.