The cesium-131 nuclide has a half life of 30 years. After 90 years, about 6g remain. What is the original mass of the cesium-131 sample?
3 half-lives
m * (1/2)^3 = 6
To find the original mass of the cesium-131 sample, we can use the formula for exponential decay:
N = N₀ * (1/2)^(t / T₁/₂)
Where:
N = Final mass
N₀ = Initial mass (unknown)
t = Time elapsed (90 years)
T₁/₂ = Half-life (30 years)
We are given that N = 6g and T₁/₂ = 30 years. Let's substitute those values into the formula and solve for N₀:
6g = N₀ * (1/2)^(90 / 30)
Now let's simplify the right side of the equation:
6g = N₀ * (1/2)^3
Since (1/2)^3 = 1/8, we can rewrite the equation as:
6g = N₀ / 8
Now, we can solve for N₀ by multiplying both sides of the equation by 8:
8 * 6g = N₀
48g = N₀
Therefore, the original mass of the cesium-131 sample is 48 grams.
To find the original mass of the cesium-131 sample, we can use the formula for exponential decay:
N(t) = N0 * (1/2)^(t / T)
Where:
- N(t) is the remaining quantity after time t
- N0 is the initial quantity
- t is the time that has passed
- T is the half-life of the substance
In this case, we know that after 90 years, only 6g of cesium-131 remain, and the half-life is 30 years. Let's substitute these values into the equation:
6g = N0 * (1/2)^(90 / 30)
Now let's solve for N0:
Dividing both sides of the equation by (1/2)^(90/30):
6g / (1/2)^(90 / 30) = N0
Simplifying the exponential term:
6g / (1/2)^3 = N0
Applying the exponentiation:
6g / (1/8) = N0
Simplifying the expression:
6g * 8 = N0
48g = N0
Therefore, the original mass of the cesium-131 sample was 48 grams.