If the half life of Iodine-131 is 8.10 days, how long will it take for 120.0 g to decay to 7.50g?
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To find out how long it will take for 120.0 g of iodine-131 to decay to 7.50 g, we can use the concept of the half-life.
Step 1: Calculate the number of half-lives that occur
Each time the half-life elapses, the amount of iodine-131 will reduce by half. We need to find out how many half-lives it takes for 120.0 g to decay to 7.50 g.
Initial mass = 120.0 g
Final mass = 7.50 g
Let's calculate the fraction of the initial mass that remains after each half-life:
Fraction remaining after one half-life = (1/2) * Initial mass
Fraction remaining after two half-lives = (1/2) * (1/2) * Initial mass
Fraction remaining after three half-lives = (1/2) * (1/2) * (1/2) * Initial mass
Now, let's set up an equation to solve for the number of half-lives (n) that occur:
Final mass = (1/2)^n * Initial mass
Substituting the given values:
7.50 g = (1/2)^n * 120.0 g
Step 2: Solve for n
To solve for n, we need to take the logarithm of both sides of the equation. The base of the logarithm can be any base, but we'll use the natural logarithm (ln) for this explanation.
ln(7.50 g) = ln((1/2)^n) + ln(120.0 g)
Using the properties of logarithms, we can simplify the equation:
ln(7.50 g) = n * ln(1/2) + ln(120.0 g)
Step 3: Calculate the value of n
Now that we have the equation in the form ln(7.50 g) = n * ln(1/2) + ln(120.0 g), we can substitute the known values and solve for n:
n * ln(1/2) = ln(7.50 g) - ln(120.0 g)
n = (ln(7.50 g) - ln(120.0 g)) / ln(1/2)
Using a calculator, calculate the natural logarithms and substitute the values:
n = (-0.8109 - 4.7875) / ln(1/2)
n = -5.5984 / -0.6931
n ≈ 8.0706
Step 4: Calculate the time it takes
The number of half-lives, n, represents the number of times the half-life duration (8.10 days) elapses. So, to calculate the time it takes for the given mass to decay, multiply the number of half-lives by the half-life duration.
Time = n * Half-life duration
Time = 8.0706 * 8.10 days
Time ≈ 65.54 days
Therefore, it will take approximately 65.54 days for 120.0 g of iodine-131 to decay to 7.50 g.