Question

If 250 mg of a radioactive element decays to 240 mg in 36 hours, find the half-life of the element.

Answer 1

The half-life of the radioactive element is approximately 500.7 hours.

Step-by-step explanation:

The half-life of a radioactive element is the time it takes for half of the initial amount of the element to decay. To determine the half-life of the element in this question, we can use the formula:

Half-life = t / log2(N0/N)

where t is the time period and N0 and N are the initial and final amounts of the element, respectively.

In this case, the initial amount N0 is 250 mg and the final amount N after 36 hours is 240 mg. Therefore, we can substitute these values into the formula:

Half-life = 36 / log2(250/240) = 36 / log2(1.0417) = 36 / 0.0718 = 500.7 hours.

Therefore, the half-life of the element is approximately 500.7 hours.

Answer 2

The half-life of the radioactive element is approximately 288.4 hours.

Define the half-life: The half-life is the time it takes for half of the original amount of the element to decay. In this case, we need to find the time it takes for 250 mg to decay to 125 mg (half of the original amount).

Use the exponential decay formula: Radioactive decay follows an exponential model. The amount of the element remaining after time t is given by:

Remaining amount = Initial amount * (0.5)^(t / half-life)

In our case, the initial amount is 250 mg, the remaining amount is 240 mg, and the time is 36 hours. We need to solve for the half-life:

240 = 250 * (0.5)^(36 / half-life)

Solve for the half-life: Take the natural logarithm of both sides:

ln(240 / 250) = (36 / half-life) * ln(0.5)

-0.0408 = (36 / half-life) * -0.6931

half-life = 36 * -0.6931 / -0.0408

half-life ≈ 288.4 hours