Suppose a sample of Cesium-137 with initial mass of 200 grams decays to a mass of 160 grams voer the course of 10 years.
Find the half-life of Cesium-137 (Round to the nearest year)
How much Cesium-137 is left in the sample after 60 years? Round to the nearest tenth of a gram.
If T is the half life,
(1/2)^(t/T) = 0.16/0.20 = 0.80
where t = 10 yr
(t/T) = 0.3219
T = 31 yr
After 60 yr, the fraction left is
(0.5)^(60/31) = 0.261
Amount left = 200*0.261 = 52.3 g
To find the half-life of Cesium-137, we can use the fact that the half-life is the time it takes for half of the substance to decay.
1. Start with the formula for exponential decay, which relates the mass of a radioactive substance at a given time (m) to the initial mass (m0), time (t), and the decay constant (λ):
m = m0 * e^(-λt)
2. Rearrange the formula to solve for the decay constant (λ):
λ = -ln(m/m0) / t
3. Substitute the given values into the rearranged formula:
λ = -ln(160/200) / 10
4. Evaluate the expression to find the decay constant (λ):
λ ≈ 0.032
5. To find the half-life (t1/2), use the formula:
t1/2 = ln(2) / λ
6. Substitute the known decay constant (λ) into the formula:
t1/2 ≈ ln(2) / 0.032
7. Calculate the half-life (t1/2):
t1/2 ≈ 21.6 years (rounded to the nearest year)
To find the amount of Cesium-137 left in the sample after 60 years, we can use the exponential decay formula:
m = m0 * e^(-λt)
1. Substitute the known values into the formula:
m = 200 * e^(-0.032 * 60)
2. Calculate the value:
m ≈ 102.4 grams (rounded to the nearest tenth of a gram)
Therefore, approximately 102.4 grams of Cesium-137 will be left in the sample after 60 years.