One isotope of holmium, 162Ho, has a half life of 22 minutes. The half-life of a second isotope, 164Ho, is 37 minutes. Starting with a sample containing equal amounts, find the ratio of the amounts of 162Ho to 164Ho after two hour. Rounded to three decimal
To solve this problem, we need to use the exponential decay formula:
A = A0 * (1/2)^(t/h)
Where A is the amount of isotope remaining after time t, A0 is the initial amount of the isotope, t is the time elapsed, and h is the half-life of the isotope.
Let's start by calculating the number of half-lives that have passed after two hours for each isotope.
For 162Ho:
Hours to minutes: 2 * 60 = 120 minutes
Number of half-lives: 120 / 22 = 5.454
For 164Ho:
Hours to minutes: 2 * 60 = 120 minutes
Number of half-lives: 120 / 37 = 3.243
Now, let's calculate the remaining amount of each isotope.
Remaining 162Ho = (1/2)^(5.454) = 0.0425
Remaining 164Ho = (1/2)^(3.243) = 0.1569
Finally, let's find the ratio of the amounts of 162Ho to 164Ho after two hours.
Ratio = Remaining 162Ho / Remaining 164Ho
Ratio = 0.0425 / 0.1569
Ratio ≈ 0.270
To find the ratio of the amounts of 162Ho to 164Ho after two hours, we need to calculate how many half-lives have passed for each isotope during that time period.
Given that the half-life of 162Ho is 22 minutes, we can calculate the number of half-lives by dividing the total time (2 hours) by the half-life:
Number of half-lives for 162Ho = (2 hours * 60 minutes per hour) / 22 minutes per half-life
Number of half-lives for 162Ho = 5.455
Rounded to the nearest whole number, the number of half-lives for 162Ho is 5.
Similarly, for the half-life of 164Ho, which is 37 minutes:
Number of half-lives for 164Ho = (2 hours * 60 minutes per hour) / 37 minutes per half-life
Number of half-lives for 164Ho = 3.243
Rounded to the nearest whole number, the number of half-lives for 164Ho is 3.
Now, we can calculate the ratio of the amounts of 162Ho to 164Ho using the formula:
Ratio = (amount of 162Ho after 2 hours) / (amount of 164Ho after 2 hours)
Since the initial amounts of both isotopes are equal, the ratio simplifies to:
Ratio = (1/2)^(number of half-lives for 162Ho) / (1/2)^(number of half-lives for 164Ho)
Ratio = (1/2)^5 / (1/2)^3
Ratio = (1/32) / (1/8)
Ratio = 1/4
Therefore, the ratio of the amounts of 162Ho to 164Ho after two hours is 1:4.