Half-life of Na24 radioactive nuclei is 15 h the time in which the activity of sample of Na24 radioactive nuclei will decrease by 87.5% is (in hours, rounded up to first decimal place)Correct answer is '44.2 to 46.8'. Can you explain this answer?

Question

Answer

Half-life of Na24 radioactive nuclei

Na24 is a radioactive isotope of sodium. The half-life of Na24 is given as 15 hours. This means that after every 15 hours, the amount of Na24 nuclei present in a sample will decrease by half.

Decay of radioactive nuclei

The decay of radioactive nuclei follows an exponential decay law, which can be expressed using the equation:

N(t) = N₀ * (1/2)^(t/T)

Where:
- N(t) is the amount of radioactive nuclei remaining at time t
- N₀ is the initial amount of radioactive nuclei
- t is the time elapsed
- T is the half-life of the radioactive nuclei

Calculating the time for 87.5% decrease in activity

To find the time in which the activity of the sample decreases by 87.5%, we need to solve for t in the equation:

0.875 = (1/2)^(t/15)

Taking the logarithm of both sides of the equation:

log(0.875) = (t/15) * log(1/2)

Simplifying the equation:

t/15 = log(0.875) / log(1/2)

t = 15 * (log(0.875) / log(1/2))

Using a calculator, we can find the value of t to be approximately 45.99 hours.

Rounding up to the first decimal place

The question asks for the time rounded up to the first decimal place. Since 45.99 is closer to 46 than to 45, we round up to 46 hours.

Therefore, the time in which the activity of the sample of Na24 radioactive nuclei will decrease by 87.5% is approximately 46 hours.

The correct answer range given as '44.2 to 46.8' includes this value of 46 hours, as it allows for a small margin of error in the calculation.

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