Iodine-131 has a half life of 2.2 × 10-2 year. How long will it take to reduce a 130 g sample of 131I to 74 g 131I?
To find out how long it will take to reduce a 130 g sample of iodine-131 (131I) to 74 g, we can use the concept of half-life.
First, let's calculate the number of half-lives needed to reach 74 g from 130 g.
Initial mass of 131I = 130 g
Final mass of 131I = 74 g
First, we need to find the difference in mass:
Mass difference = Initial mass - Final mass
= 130 g - 74 g
= 56 g
Now, let's find the number of half-lives:
Number of half-lives = Mass difference / Half-life
Substituting the values:
Number of half-lives = 56 g / (2.2 × 10^(-2) year)
Calculating:
Number of half-lives ≈ 2545.45
Since each half-life is equal to 2.2 × 10^(-2) years, we can find the total time it takes for 131I to reduce from 130 g to 74 g by multiplying the number of half-lives by the length of one half-life.
Total time = Number of half-lives × Half-life
= 2545.45 × 2.2 × 10^(-2) year
Calculating:
Total time ≈ 55.99 years
Therefore, it will take approximately 55.99 years to reduce a 130 g sample of iodine-131 to 74 g.
To solve this problem, we can use the formula for radioactive decay:
N(t) = N₀ * (1/2)^(t / half-life)
Where:
N(t) is the amount of a substance remaining at time t
N₀ is the initial amount of the substance
t is the time that has passed
half-life is the time it takes for half of the substance to decay
In this case, we can assume that the initial amount of iodine-131 (N₀) is 130 g and we want to find the time it takes for it to decay to 74 g.
We can set up the equation as follows:
74 g = 130 g * (1/2)^(t / 2.2 × 10^-2)
Now we can solve for t by taking the logarithm of both sides:
log(74 g / 130 g) = log((1/2)^(t / 2.2 × 10^-2))
Using logarithmic properties, we can bring down the exponent from the right side:
log(74 g / 130 g) = (t / 2.2 × 10^-2) * log(1/2)
Dividing both sides by log(1/2):
(t / 2.2 × 10^-2) = log(74 g / 130 g) / log(1/2)
Now we can solve for t by multiplying both sides by 2.2 × 10^-2:
t = (2.2 × 10^-2) * (log(74 g / 130 g) / log(1/2))
Now we can calculate t using a calculator or software that can evaluate logarithmic functions.
Substituting the values:
t ≈ (2.2 × 10^-2) * (-0.2469 / -0.3010)
Calculating t:
t ≈ (2.2 × 10^-2) * 0.8206
t ≈ 0.01805 years
Therefore, it will take approximately 0.01805 years for a 130 g sample of iodine-131 to reduce to 74 g of iodine-131.
k = 0.693/t1/2
ln(No/N) = kt
No = 130 g
N = 74 g
k = from above
Solve for t in years.