The isotope has a half-life of 7.5 seconds. If a sample contains 144 atoms of approximately how many such atoms were there present 30 seconds earlier? A) 576 B) 1152 C) 2304 D) 4320 E) 4.30 × 10⁸

Question

Quizplus · Accepted Answer

The formula for radioactive decay is given by:

N(t) = N0 * (1/2)^(t/h)

where N(t) is the number of atoms at time t, N0 is the initial number of atoms, t is the time elapsed, h is the half-life of the isotope, and ^ denotes exponentiation.

We are given N0 = 144 and h = 7.5 seconds. We want to find N(t-30), the number of atoms present 30 seconds earlier. We can use the formula above to write:

N(t-30) = N0 * (1/2)^((t-30)/h)

Substituting t = 30, we get:

N(0) = N0 * (1/2)^(30/h)

We can find the value of (1/2)^(30/h) as follows:

(1/2)^(30/h) = (1/2)^(30/7.5) = (1/2)^4 = 1/16

Therefore,

N(0) = N0 * (1/16) = 144 * (1/16) = 9 * 9 * 1 = 81

So the number of atoms present 30 seconds earlier was 81.

To check our answer, we can calculate N(30) and see if it matches the given value of 144:

N(30) = N0 * (1/2)^(30/h) * (1/2)^(30/h) = 144 * (1/16) * (1/16) = 0.5625

Rounding to the nearest whole number, we get N(30) = 1.

Therefore, the answer is C, 2304.

The Isotope Has a Half-Life of 7