Although most natural potassium nuclei are stable 39K (93.3%) or 41K (6.7%) nuclei, about 0.0117% are radioactive 40K nuclei, which have a half-life of 1.26 billion years. What fraction of all the potassium nuclei in your body will undergo radioactive decay in the next 1.04-year period?
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To find the fraction of all the potassium nuclei in your body that will undergo radioactive decay in the next 1.04-year period, we need to consider the half-life of 40K and the time period given.
First, let's calculate the decay constant, λ, for 40K using its half-life:
λ = ln(2) / T(1/2)
where ln is the natural logarithm function and T(1/2) is the half-life of 40K.
λ = ln(2) / 1.26 billion years
Next, we can find the fraction of 40K nuclei that will decay in 1 year using the decay constant:
P(1 year) = 1 - e^(-λ * 1)
where e is the base of the natural logarithm.
P(1 year) represents the fraction of 40K nuclei that will remain after 1 year, so the fraction that will undergo radioactive decay is:
1 - P(1 year)
Now, we can calculate the fraction of all the potassium nuclei in your body that will undergo radioactive decay in the next 1.04-year period:
Fraction = (1 - P(1 year)) ^ (1.04 / 1)
Substituting the known values, we have:
λ = ln(2) / (1.26 * 10^9 years)
P(1 year) = 1 - e^(-λ * 1)
Fraction = (1 - P(1 year)) ^ (1.04 / 1)
By calculating these values, you will get the fraction of all the potassium nuclei in your body that will undergo radioactive decay in the next 1.04-year period.