Natural Uranium contains 15 percent Sm147. Sm147 is radioactive and decays by alpha decay. What will be the number of disintegrations per gram per second of the natural element? The half-life of Sm147 is 1.3x1010 years.
To determine the number of disintegrations per gram per second for Sm147, we need to use the radioactive decay equation. The equation is given by:
N(t) = N₀ * e^(-λt)
Where:
- N(t) is the number of radioactive atoms remaining at time t
- N₀ is the initial number of radioactive atoms
- λ is the decay constant
- t is the time elapsed
First, we need to calculate the decay constant (λ) using the half-life (T½) given for Sm147.
λ = ln(2) / T½
λ = ln(2) / 1.3x10^10 years
λ ≈ 5.35x10^(-21) 1/year
Next, we can calculate the number of disintegrations per gram per second (D) using the following equation:
D = (N₀ * λ) / M
Where:
- N₀ is the initial number of radioactive atoms
- λ is the decay constant
- M is the molar mass of Sm147
Now, we need to find the molar mass of Sm147. The molar mass of an element is the mass of one mole of the element, and it is given by the periodic table. In this case, the molar mass of Sm147 is approximately 146.92 grams/mole.
Finally, we can calculate the number of disintegrations per gram per second for Sm147.
D = (N₀ * λ) / M
Since we know that Natural Uranium contains 15 percent Sm147, we can assume that we have 15 grams of Sm147 in 100 grams of natural uranium.
N₀ = 0.15 grams (initial number of radioactive atoms)
D = (0.15 grams * 5.35x10^(-21) 1/year) / 146.92 grams/mole
D ≈ 5.15x10^(-24) disintegrations/gram/second
Therefore, the number of disintegrations per gram per second of Sm147 in natural uranium is approximately 5.15x10^(-24) disintegrations/gram/second.