Question

A sample of radioactive iodine-131 atoms has a half-life of about 8 days. Suppose that 1,000,000 iodine-131 atoms are initially present. a) Determine a formula for the number of atoms at time t, where t represents number of days. b) How long will it take for the sample to reach 180,000 atoms? Answer correct to 2 decimal places.

Answer 1

a) To determine the formula for the number of atoms at time t, we can use the half-life formula for exponential decay:

N(t) = N₀ * (1/2)^(t / T)

where:

N(t) = number of atoms at time t

N₀ = initial number of atoms

t = time (in this case, measured in days)

T = half-life of the radioactive substance

Given that the half-life of iodine-131 is 8 days and there are initially 1,000,000 atoms (N₀ = 1,000,000), the formula becomes:

N(t) = 1,000,000 * (1/2)^(t / 8)

b) Now, we need to find out how long it will take for the sample to reach 180,000 atoms (N(t) = 180,000). We'll set up the equation as follows:

180,000 = 1,000,000 * (1/2)^(t / 8)

To solve for t, we'll first divide both sides by 1,000,000:

(1/2)^(t / 8) = 180,000 / 1,000,000

Next, take the logarithm (base 1/2) of both sides to isolate t:

t / 8 = log₁/₂ (180,000 / 1,000,000)

Now, solve for t:

t = 8 * log₁/₂ (0.18) (rounded to two decimal places)

Using a calculator:

t ≈ 8 * (-2.515)

t ≈ -20.12

Since time cannot be negative, we can ignore the negative solution. The correct answer is approximately 20.12 days (rounded to two decimal places). So, it will take about 20.12 days for the sample to reach 180,000 atoms.