Final answer:
To calculate the time required for 50.0 g of a substance with a half-life of 4.50 years to decrease to 0.0500 g, one must find the number of half-lives (approximately 9.966) and then multiply by the half-life duration. It will take roughly 44.85 years for the substance to be depleted to 0.0500 g.
Step-by-step explanation:
To determine how many years it will take for 50.0 g of a substance with a half-life of 4.50 years to be depleted to 0.0500 g, one must calculate the number of half-lives that occur in the process of decay from 50.0 g to 0.0500 g.
Let's denote the number of half-lives as 'n'. After each half-life, the amount of substance is reduced by half. Therefore, after 'n' half-lives, the amount remaining (R) can be calculated using the formula:
R = Initial amount * (1/2)^n
Plugging in the values we have:
0.0500 g = 50.0 g * (1/2)^n
To find 'n', we divide both sides of the equation by 50.0 g:
0.0500 g / 50.0 g = (1/2)^n
1/1000 = (1/2)^n
Using a logarithmic operation, we can solve for 'n':
n = log(1/1000) / log(1/2)
n ≈ 9.966
Since the half-life of the substance is 4.50 years, we multiply this value by 'n' to find the total time ('t'):
t = n * 4.50 years
t ≈ 9.966 * 4.50 years
t ≈ 44.85 years
So it takes approximately 44.85 years for 50.0 g of the substance to be depleted to 0.0500 g.