24Na Beta Decay Calculator Example and Explanation

A 24Na beta-decay calculator example predicts how much sodium-24 remains after a chosen time, using its half-life and the radioactive-decay law. Enter the starting amount and elapsed time; the calculator returns the remaining amount and the amount that has decayed, under an idealized decay-only model.

Formula

Radioactive decay follows an exponential law: N = N₀e^−λt = N₀(1/2)^t/T_1/2. Here, N is the number of sodium-24 nuclei remaining (nuclei), N₀ is the initial number (nuclei), t is elapsed time (hours), λ is the decay constant (h⁻¹), and T_1/2 is the half-life (hours). The calculator uses T_1/2 = 14.997 h, equivalently λ = ln(2)/14.997 h ≈ 0.04622 h⁻¹. The decayed amount is D = N₀ − N.

The half-life is a nuclear property, not an adjustable fit in this example. A calculator for a different nuclide would need that nuclide’s half-life and decay scheme. The model assumes the initial count is known and that decay is the only change to the population being tracked.

Worked Example

Use the calculator’s defaults: N₀ = 1,000,000 nuclei and t = 24 h. Substituting the 24Na half-life:

• N = 1,000,000 × exp(−ln(2) × 24 / 14.997)
• 24 / 14.997 ≈ 1.6003201 half-lives.
• N ≈ 1,000,000 × 0.32992 = 329,920 nuclei.
• D = 1,000,000 − 329,920 = 670,080 nuclei.

Rounded to the calculator’s three decimal places, the outputs are approximately 329,920.000 nuclei remaining and 670,080.000 nuclei decayed. The apparent decimal places are a display convention; the model does not imply that a physical sample contains a fractional nucleus. For real measurements, uncertainty in the initial count, counting statistics, and measurement conditions can matter.

Assumptions

• The decay is spontaneous and follows the standard exponential radioactive-decay model.
• The elapsed time and half-life use the same unit: hours.
• The sample is treated as a closed population for this calculation; no additions, removals, or chemical separation are modeled.
• “Decayed” means parent 24Na nuclei that have undergone decay, not the activity of a sample or the amount of a particular daughter product.

What Sodium-24 Beta Decay Means

Sodium-24, written ²⁴Na or Na-24, is a radioactive isotope of sodium with 11 protons and 13 neutrons. It undergoes beta-minus decay: a neutron in the nucleus changes into a proton, with an electron and an electron antineutrino emitted. The daughter nucleus is magnesium-24 (²⁴Mg), which is stable. The decay also involves gamma-ray emissions as the daughter nucleus releases excess energy.

This is a nuclear transformation. It is not ordinary chemical change, and chemical bonding does not determine the isotope’s half-life. The term “beta decay” describes a family of nuclear processes; beta-minus decay specifically emits an electron. It should not be confused with beta-plus decay, in which a proton converts to a neutron and a positron is emitted.

Half-life is the time in which, statistically, half the radioactive parent nuclei in a large population decay. It does not mean that every individual nucleus lasts exactly 14.997 hours. An individual decay is random; the exponential equation describes the expected population behavior.

How to Use the Calculator

1. Enter the initial quantity of 24Na as a number of nuclei. The default of one million is a convenient example, not a recommended sample size.
2. Enter elapsed time in hours. Convert other time units before using the result: for example, 1 day = 24 h.
3. Read the remaining-parent result first. The second result is the initial count minus the expected remaining count.

If your starting information is an activity in becquerels rather than a number of atoms, this calculator does not directly accept it. Activity is the decay rate, measured in decays per second, and is related to the number of parent nuclei by A = λN when λ is expressed in s⁻¹. Convert the half-life to seconds to use that relation: 14.997 h × 3,600 s/h = 53,989.2 s. Do not mix an activity value with a nuclei count as if they were the same quantity.

Reading the Result and Common Errors

The remaining amount is an expected count under the model. In a small sample, the actual number may differ from the expectation because individual decays are random. In a laboratory measurement, detector efficiency, background radiation, geometry, and calibration also affect what is observed; those measurement factors are not included here.

• Mixing units: The example uses hours for both t and T_1/2. Putting elapsed time in seconds while retaining 14.997 as though it were seconds produces a meaningless answer.
• Confusing remaining and decayed: The first output is the parent nuclei still present; the second is the difference from the initial amount.
• Assuming half-life is a deadline: After one half-life, half remains on average. After two, one quarter remains; the process continues rather than stopping.
• Rounding too early: Keep the exponent calculation at full precision until the final result. Display rounding can slightly affect subtraction if intermediate values are rounded first.
• Ignoring other changes: This model does not account for sample transfer, contamination, shielding or detector response, or daughter-product activity.

Limits, Activity, and Safety

The calculator answers a specific bookkeeping question: how many parent 24Na nuclei are expected to remain after a specified interval? It does not calculate absorbed dose, external exposure, shielding requirements, or the activity of the daughter. A fuller activity calculation can use A(t) = A₀e^−λt when the activity is measured consistently and the parent’s decay constant is known. In a decay chain, daughter nuclides can themselves be radioactive, so their activity needs a separate chain calculation.

Sodium-24 is radioactive, and this example is for educational decay arithmetic, not instructions for obtaining, preparing, handling, or disposing of radioactive material. Actual work involving radioactive sources is governed by applicable radiation-safety rules and qualified oversight. Follow your lab method, the safety data sheet (SDS) and your supervisor.

Frequently Asked Questions

What is the half-life of sodium-24?
For this example, the calculator uses 14.997 hours. Half-life values are reported with a particular precision; check a current authoritative nuclear-data reference when precision is important.

What does sodium-24 decay into?
Its beta-minus decay changes the element from sodium to magnesium, producing magnesium-24. The decay scheme also includes gamma emissions as the daughter nucleus de-excites.

How much 24Na remains after 24 hours?
Using a 14.997-hour half-life, about 32.992% remains after 24 hours. With an initial count of one million nuclei, the model gives approximately 329,920 remaining.

Is the amount after one half-life exactly half?
The expected fraction is one half. For a finite sample, the observed count can vary because decay events are random; measurement uncertainty adds further variation.

Can I enter becquerels instead of nuclei?
No. This calculator’s input is a number of parent nuclei. Becquerels measure decays per second; converting between activity and nuclei requires the decay constant, with time units handled consistently.

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in n0 | Initial amount of 24Na | nuclei | 1e6 | 0
in t | Elapsed time | h | 24 | 0
out n | 24Na remaining | nuclei | n0 * exp(-ln(2) * t / 14.997) | 3
out d | 24Na decayed | nuclei | n0 - n | 3