24Na Beta Decay Calculator Formula

A 24Na beta-decay calculator uses the radioactive-decay law to estimate how much sodium-24 remains after a chosen time, or how much has decayed. Enter the starting amount and elapsed time; the result depends on 24Na’s half-life, about 14.997 hours, and assumes a simple decay calculation for study purposes.

Formula

The radioactive-decay formula is N(t) = N₀e^−λt = N₀(1/2)^t/T_1/2. Here, N(t) is the number of 24Na nuclei remaining after time t, in nuclei; N₀ is the initial number of nuclei, in nuclei; t is elapsed time, in hours; λ is the decay constant, in inverse hours; and T_1/2 is the half-life, in hours. For 24Na, this calculator uses T_1/2 = 14.997 h, so λ = ln(2)/14.997 h ≈ 0.04622 h⁻¹. The second output is the fraction that has decayed, 1 − N(t)/N₀.

The calculator’s first output is the remaining number of nuclei, and the second is a dimensionless fraction from 0 to 1. Multiply that fraction by 100 to express it as a percentage. The model uses the exponential form; it does not calculate detector counts, radiation dose, sample mass, or activity unless those are separately derived using additional information.

Worked Example

Using the defaults, N₀ = 1,000,000 nuclei and t = 24 h. Convert elapsed time to half-lives: 24 h / 14.997 h = 1.600320 half-lives. Then calculate the remaining amount:

N(24 h) = 1,000,000 × (1/2)^24/14.997 ≈ 330,312.9 nuclei.

The calculator displays whole nuclei, so the result is 330,313 nuclei remaining. The decayed fraction is 1 − 330,312.9 / 1,000,000 ≈ 0.669687, or 66.9687%. Rounded to four decimal places, the calculator reports 0.6697. About two-thirds of the original nuclei have decayed during the 24-hour interval; the remaining amount is not zero because radioactive decay is exponential.

Assumptions

• The sample contains 24Na only for the calculation, and its decay follows a constant exponential decay law.
• The half-life is treated as exactly 14.997 hours for this estimate; published reference values and rounding may differ slightly.
• The starting number is a count of nuclei, not grams, becquerels, or detector events.
• The calculation ignores measurement uncertainty, background radiation, daughter-product behavior, and any physical loss of sample.

What 24Na Beta Decay Means

Sodium-24 is a radioactive isotope of sodium with 11 protons and 13 neutrons. In beta-minus decay, a neutron changes into a proton, emitting an electron (the beta particle) and an electron antineutrino. The nucleus therefore becomes magnesium-24: the mass number remains 24 while the atomic number increases from 11 to 12. In nuclear notation, the transformation is commonly written ²⁴Na → ²⁴Mg + e⁻ + ν̄_e, with energy carried by the emitted particles and recoil.

“Beta decay” describes the nuclear transformation; it does not mean that the beta particle carries away all released energy at one fixed value. In beta-minus decay the available energy is shared among the electron, antineutrino, and recoiling daughter nucleus, giving the beta particle a continuous energy spectrum. The calculator does not model that spectrum or individual decay events.

Using Half-Life and Decay Constant

Half-life is the interval in which the expected number of undecayed nuclei falls to half its previous value. After one half-life, 50% remains; after two, 25%; after three, 12.5%. Fractions of a half-life work through the same exponential relationship: after 24 hours, which is about 1.6003 half-lives for the value used here, the remaining fraction is about 0.3303.

The equivalent decay-constant form is useful when a problem gives λ rather than half-life. Since λ = ln(2)/T_1/2, its units must be the reciprocal of the time unit used for t. If time is entered in hours, λ must be in h⁻¹; if a reference gives the half-life in seconds, either convert it to hours or convert elapsed time to seconds. Mixing these units can produce a result that looks numerically plausible but is wrong.

What the Result Does—and Does Not—Tell You

The first output estimates the population of parent 24Na nuclei still present. It is generally fractional in the underlying mathematical model because the exponential law describes an expected population; this page rounds the displayed count to a whole nucleus. For a macroscopic sample containing enormous numbers of nuclei, the fractional expectation is practically useful. For a sample with very few nuclei, actual decay is stochastic and the exact count cannot be predicted from the mean formula.

The fraction decayed is not the same as activity. Activity is the number of decays per unit time, measured in becquerels (1 Bq = 1 decay per second), and for a single radionuclide it is A = λN when λ is expressed in s⁻¹. This calculator reports neither Bq nor dose. To calculate activity, convert the decay constant to inverse seconds and multiply by the remaining number of nuclei. Estimating dose also requires details such as emitted radiation, geometry, exposure duration, and absorption; the decay formula alone cannot supply it.

Common Mistakes and Limits

• Using the wrong isotope: 24Na is not stable sodium-23. Verify the mass number in the problem before choosing a half-life.
• Confusing the amount decayed with the amount remaining: the remaining fraction is (1/2)^t/T_1/2; the decayed fraction is one minus that value.
• Mixing time units: the input here is hours. Convert minutes or days to hours before entering a value.
• Assuming a finite-time sample reaches exactly zero: exponential decay approaches zero but does not reach it mathematically at a finite time.
• Over-interpreting precision: the half-life and initial count may be uncertain, and the output’s displayed digits do not eliminate that uncertainty.

This is a study and calculation aid, not a radiation-safety assessment or a measurement instrument. It cannot determine whether a sample or location is safe. For any laboratory or radiation-related work, follow the applicable lab method, the safety data sheet (SDS), radiation-safety rules, and your supervisor’s instructions.

Frequently Asked Questions

What is the formula for 24Na beta decay?
The remaining nuclei after time t are N(t) = N₀(1/2)^{t/14.997 h}, with t expressed in hours. Equivalently, use N(t) = N₀e^−λt, where λ = ln(2)/14.997 h.

What is the half-life of sodium-24?
A commonly used evaluated value is approximately 14.997 hours. Small differences in published rounded values do not usually change a classroom estimate substantially.

What does sodium-24 decay into?
In beta-minus decay, 24Na becomes magnesium-24. A neutron converts into a proton, and an electron and electron antineutrino are emitted.

How much 24Na remains after 24 hours?
Using a 14.997-hour half-life, about 33.03% remains after 24 hours. For an initial 1,000,000 nuclei, the model gives approximately 330,313 remaining.

Is the fraction decayed the same as activity?
No. The fraction decayed is the portion of the initial nuclei that have transformed. Activity is the decay rate at a particular time and is measured in becquerels; for a single isotope it is λN.

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in n0 | Initial number of 24Na nuclei | nuclei | 1000000 | 0
in t | Elapsed time | h | 24 | 0
out n | 24Na nuclei remaining | nuclei | n0 * exp(-ln(2) * t / 14.997) | 0
out dec | Fraction decayed | fraction | 1 - n / n0 | 4