Thallium-208 has a half life of 3.053 min. How long will it take for 120.0 g to decay to 7.50 g?
To find the time it takes for a given amount of a substance to decay, we can use the half-life formula. The formula is as follows:
N = N₀ * (1/2)^(t / T)
where:
N: the final amount of the substance
N₀: the initial amount of the substance
t: the time elapsed
T: the half-life of the substance
In this case, we need to solve for the time t. The initial amount N₀ is 120.0 g and the final amount N is 7.50 g. The half-life T is given as 3.053 min. We can now substitute these values into the formula:
7.50 g = 120.0 g * (1/2)^(t / 3.053)
Next, we can rearrange the formula to isolate the variable t:
(1/2)^(t / 3.053) = 7.50 g / 120.0 g
Now, we take the logarithm of both sides of the equation to solve for t:
log[(1/2)^(t / 3.053)] = log(7.50 / 120.0)
(t / 3.053) * log(1/2) = log(7.50 / 120.0)
t = (log(7.50 / 120.0)) / log(1/2)
Using a calculator, we can evaluate the right side of the equation:
t ≈ (log(7.50 / 120.0)) / log(1/2)
t ≈ (−2.077) / (−0.301)
t ≈ 6.905
Therefore, it will take approximately 6.905 minutes for 120.0 g of thallium-208 to decay to 7.50 g.