Answer:
The half-life of a radioactive substance is the amount of time it takes for half of the substance to decay. We can use the given information to determine the half-life of potassium-44.
Let's start by finding the amount of potassium-44 remaining after one half-life. If a sample decays to half its original amount, then the remaining amount will be:
(1/2) x 20.00 g = 10.00 g
The time it takes for the sample to decay to this amount is the half-life, which we can call t₁/₂.
Now, let's use the given information to find t₁/₂. We can use the equation for radioactive decay:
N = N₀ e^(-λt)
where N is the amount of the substance remaining after time t, N₀ is the initial amount of the substance, λ is the decay constant, and e is the mathematical constant e.
If we divide both sides of this equation by N₀, we get:
N/N₀ = e^(-λt)
We can use this equation to find the decay constant λ, which is related to the half-life by the following equation:
t₁/₂ = ln(2) / λ
If we substitute the given values into the first equation, we get:
N/N₀ = 2.50 g / 20.00 g = 0.125
t = 100.0 minutes
N₀ = 20.00 g
Substituting these values into the second equation and solving for λ, we get:
0.125 = e^(-λ x 100.0)
ln(0.125) = -λ x 100.0
λ = ln(2) / t₁/₂
We can solve for t₁/₂ by substituting the value we just found for λ:
ln(0.125) = -ln(2) x 100.0 / t₁/₂
t₁/₂ = -100.0 ln(0.125) / ln(2)
t₁/₂ ≈ 20.7 minutes
Therefore, the half-life of potassium-44 is approximately 20.7 minutes.
Step-by-step explanation: