Question

A sample of sodium-24 chloride contains \(0.050 \mathrm{mg}\) of Na-24 to study the sodium balance of an animal. After \(24.9 \mathrm{~h}\), \(0.016 \mathrm{mg}\) of \(\mathrm{Na}-24\) is left. What is the half-life of \(\mathrm{Na}-24 ?\)

Short answer

Based on the given information, the half-life of sodium-24 is approximately 15.0652 hours.

Step-by-step solution

Identify given information and the required variable

We are given the following information: - Initial amount of Na-24: 0.050 mg - Remaining amount of Na-24 after 24.9 hours: 0.016 mg - Time passed: 24.9 h Our goal is to find the half-life of Na-24.

Write the radioactive decay formula

We will use the radioactive decay formula: N(t) = N0 * e^(-λt) Where N(t) is the remaining amount of the radioactive substance at time t, N0 is the initial amount, λ is the decay constant, and t is the time passed.

Solve for the decay constant (λ)

Plug the given values into the radioactive decay formula and solve for λ: 0.016 = 0.050 * e^(-λ * 24.9) Divide by 0.050: 0.32 = e^(-λ * 24.9) Take the natural logarithm of both sides: ln(0.32) = -λ * 24.9 Now, solve for λ: λ = -[ln(0.32) / 24.9] ≈ 0.0459655 h^(-1)

Calculate the half-life using the decay constant

We will now use the following formula to calculate half-life based on the decay constant: T_half = ln(2) / λ Plug the calculated decay constant into the formula: T_half = ln(2) / 0.0459655 ≈ 15.0652 h

Report the result

The half-life of sodium-24 is approximately 15.0652 hours.