You have 180 g of a radioactive substance. It has a half-life of 265 years. After 1,325 years, what mass remains?
180 * (1/2)^(1325/265) = 5.625 g
Note that 1325 = 5*265, so that's 5 half-lives
so, 1/2^5 = 1/32 will remain
180/32 = 45/8
Well, if the half-life is 265 years, it means that every 265 years, half of the substance decays. So after 1,325 years, we need to figure out how many half-lives have passed.
Let me grab my calculator (or slide rule) for a moment... Ah! Okay, 1,325 divided by 265 equals 5. So, five half-lives have passed.
Now, if half of the substance decays every half-life, we can simply divide the initial mass of 180 g by 2 five times. Let's calculate this together:
180 g ÷ 2 = 90 g (after 1st half-life)
90 g ÷ 2 = 45 g (after 2nd half-life)
45 g ÷ 2 = 22.5 g (after 3rd half-life)
22.5 g ÷ 2 = 11.25 g (after 4th half-life)
11.25 g ÷ 2 = 5.625 g (after 5th half-life)
So, after 1,325 years, approximately 5.625 g of the radioactive substance remains.
To determine the remaining mass of the radioactive substance after 1,325 years, we can use the formula:
Final mass = Initial mass * (1/2)^(Time elapsed / Half-life)
Given:
Initial mass = 180 g
Half-life = 265 years
Time elapsed = 1,325 years
Substituting the values into the formula:
Final mass = 180 g * (1/2)^(1325/265)
Now, let's calculate the final mass:
Final mass = 180 g * (1/2)^(5)
Final mass ≈ 180 g * 0.03125
Final mass ≈ 5.625 g
Therefore, after 1,325 years, approximately 5.625 grams of the radioactive substance remains.
To calculate the mass that remains after a certain period of time, we need to use the concept of half-life. The half-life is the amount of time it takes for half of a radioactive substance to decay.
Given that the half-life of the radioactive substance is 265 years, this means that after every 265 years, the mass of the substance will be halved.
Now, let's calculate the number of half-lives that have occurred in 1,325 years. We divide the total time by the half-life:
Number of half-lives = Total time / Half-life
Number of half-lives = 1325 years / 265 years
Number of half-lives = 5
Since 5 half-lives have occurred, we know that the substance has been halved 5 times. Each time, the mass is reduced by half. So, if we start with 180 g, after one half-life it would be 90 g, after two half-lives it would be 45 g, after three half-lives it would be 22.5 g, after four half-lives it would be 11.25 g, and after five half-lives it would be 5.625 g.
Therefore, after 1,325 years, approximately 5.625 g of the radioactive substance remains.