24Na Beta Decay Calculator: Physics & Half-Life

This calculator works out the remaining activity (in becquerels) of a sample of sodium-24 (24Na) after a time you enter, using the isotope’s known half-life of 14.959 hours. You type in the initial activity and the elapsed time, and you get back the activity that is left, along with the number of half-lives that have passed. It applies the standard exponential decay law used in nuclear physics for any beta-emitting isotope.

Formula

The decay of a radioactive isotope follows the exponential decay law:

A(t) = A₀ × (1/2)^(t / T½)

where:

• A(t) = activity remaining after time t (in becquerels, Bq)
• A₀ = initial activity (in Bq)
• t = elapsed time (in hours)
• T½ = half-life of 24Na = 14.959 hours (the time for half the nuclei to decay)

The exponent t / T½ tells you how many half-lives have passed. Because 24Na decays by beta emission (β⁻) to magnesium-24, the same formula applies whether you track the number of atoms or the activity, since activity is directly proportional to the number of undecayed nuclei.

Worked Example

Using the default values: A₀ = 1000 Bq, t = 24 hours.

1. Calculate the number of half-lives: n = 24 / 14.959 ≈ 1.6044
2. Apply the decay law: A = 1000 × (1/2)^1.6044
3. Compute (1/2)^1.6044 ≈ 0.3290
4. Multiply: A ≈ 1000 × 0.3290 = 329.0 Bq

So after 24 hours, about 329 Bq of the original 1000 Bq remains. The calculator shows 329.00 Bq and 1.60 half-lives.

What Is 24Na and Why Does It Decay?

Sodium-24 is a radioactive isotope of sodium with 11 protons and 13 neutrons. It is produced artificially, often by neutron activation of stable sodium-23 in a nuclear reactor or by certain nuclear reactions. It undergoes beta-minus decay: a neutron converts into a proton, emitting an electron (the beta particle) and an antineutrino. This changes the nucleus into magnesium-24, which is stable. The decay also releases gamma rays (at 1.369 MeV and 2.754 MeV), which is why 24Na is used in some medical and industrial applications despite its short half-life.

How to Use the Calculator

Enter the initial activity of your 24Na sample in becquerels (Bq). If you have a reading in curies (Ci), convert first: 1 Ci = 3.7 × 10^10 Bq. Then enter the elapsed time in hours. The calculator assumes you start at time zero with the given activity. If your sample was created at a known time, use that as your starting point. The result tells you how much activity remains after that time, assuming no additional 24Na is being produced.

Common Mistakes and How to Avoid Them

• Unit mix-ups: The half-life is in hours, so your time must be in hours. If you have minutes, divide by 60 first.
• Using the wrong half-life: 24Na has a half-life of 14.959 hours, not 15 hours exactly. The difference matters over many half-lives.
• Forgetting that activity is not the same as the number of atoms: The formula works for both, but if you start with a mass of 24Na, you must convert to activity first using the specific activity (which is about 3.22 × 10^17 Bq per gram).
• Rounding too early: Keep at least three significant figures in intermediate steps to avoid errors in the final answer.

Limitations of the Formula

The exponential decay law is exact for a pure sample of 24Na, but real-world measurements can deviate. If your sample contains other radioactive isotopes, the measured activity will include their contributions. Also, the half-life is a statistical average; individual nuclei decay randomly, but for a large sample the law holds very precisely. The calculator does not account for detector efficiency or background radiation—those are experimental factors you handle separately.

Safety and Handling Notes

24Na emits beta particles and gamma rays. Beta particles can penetrate skin but are stopped by a few millimeters of plastic or aluminum. Gamma rays are more penetrating and require shielding such as lead or concrete. Always follow local radiation safety regulations, use appropriate shielding, and minimize exposure time. This calculator is for educational purposes; it does not replace a qualified health physicist or radiation safety officer.

Frequently Asked Questions

How do I convert my activity from curies to becquerels?

Multiply the value in curies by 3.7 × 10^10 to get becquerels. For example, 1 µCi (microcurie) = 3.7 × 10^4 Bq.

What is the daughter isotope of 24Na?

The daughter is magnesium-24 (24Mg), which is stable. The beta decay changes a neutron to a proton, increasing the atomic number from 11 to 12.

Why is the half-life of 24Na given as 14.959 hours and not 15?

The value 14.959 hours is the currently accepted measured half-life from precision experiments. Using 15 hours introduces about a 0.3% error per half-life, which can become significant after several half-lives.

Can I use this calculator for other beta emitters?

Yes, if you replace the half-life with that of the other isotope. The formula is universal. For example, carbon-14 has a half-life of 5730 years, so you would enter time in years and change the half-life accordingly.

What does “activity” mean in becquerels?

One becquerel is one nuclear decay per second. It is a measure of how many atoms are disintegrating each second, not the number of atoms present.

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in a0 | Initial activity | Bq | 1000 | 0
in t | Elapsed time | h | 24 | 0
out a | Remaining activity | Bq | a0 * 0.5^(t / 14.959) | 2
out n | Half-lives elapsed | - | t / 14.959 | 2