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.