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 one hour. Rounded to three decimal
after t minutes, the two isotopes have remaining amounts of
2^(-t/22) and 2^(-t/37)
So, after one hour (60 minutes), the ratio of 162Ho/164Ho is just
2^(-60/22) / 2^(-60/37)
= 2^(-60/22 + 60/37)
= 2^(-450/407)
= 2^-1.10565
= 0.465
Well, we could do some math to calculate the exact ratio, but where's the fun in that? Let's have a little laugh instead!
Why did the scientist break up with his radioactive girlfriend?
Because she had way too much decay for him to handle!
Now, let's get back to business. After one hour, we need to consider how many half-lives each isotope goes through.
For 162Ho, with a half-life of 22 minutes, after one hour we have 60 minutes / 22 minutes = 2.7272... half-lives.
For 164Ho, with a half-life of 37 minutes, after one hour we have 60 minutes / 37 minutes = 1.6216... half-lives.
Now, to find the ratio, we simply divide the number of half-lives of 162Ho by the number of half-lives of 164Ho:
2.7272... / 1.6216... = 1.681 (rounded to three decimal places)
So, the ratio of the amounts of 162Ho to 164Ho after one hour is approximately 1.681.
Remember, my friend, laughter is the best solution to any problem!
To find the ratio of the amounts of 162Ho to 164Ho after one hour, we need to determine how many half-lives have passed for each isotope during that time.
First, let's calculate the number of half-lives of 162Ho in one hour:
Since the half-life of 162Ho is 22 minutes, we divide 60 minutes (one hour) by 22 minutes:
1 hour / 22 minutes = 2.7272 half-lives (rounded to four decimal places)
Next, let's calculate the number of half-lives of 164Ho in one hour:
Since the half-life of 164Ho is 37 minutes, we divide 60 minutes (one hour) by 37 minutes:
1 hour / 37 minutes = 1.6216 half-lives (rounded to four decimal places)
Now, let's calculate the ratio of the amounts of 162Ho to 164Ho after one hour:
Since the amount of each isotope is equal at the start, the ratio is determined by the number of half-lives.
First, we calculate the amount of each isotope after the corresponding number of half-lives:
For 162Ho: (1/2)^(2.7272) ≈ 0.2036
For 164Ho: (1/2)^(1.6216) ≈ 0.3939
Finally, we divide the amount of 162Ho by the amount of 164Ho:
0.2036 / 0.3939 ≈ 0.517 (rounded to three decimal places)
Therefore, the ratio of the amounts of 162Ho to 164Ho after one hour is approximately 0.517.
To find the ratio of the amounts of 162Ho to 164Ho after one hour, we need to determine the number of half-lives that have passed for each isotope.
The general formula for calculating the number of half-lives elapsed is:
(number of elapsed time) / (half-life of the isotope)
For 162Ho, the half-life is 22 minutes, so the number of half-lives elapsed after one hour (60 minutes) would be:
60 minutes / 22 minutes = 2.727
Rounded to the nearest whole number, this gives us 3 half-lives for 162Ho.
Similarly, for 164Ho, the half-life is 37 minutes, so the number of half-lives elapsed after one hour would be:
60 minutes / 37 minutes = 1.622
Rounded to the nearest whole number, this gives us 2 half-lives for 164Ho.
Now, let's calculate the remaining amount of each isotope:
For 162Ho: Since three half-lives have passed, the remaining fraction of the original sample would be (1/2)^(3) = 0.125.
For 164Ho: Since two half-lives have passed, the remaining fraction of the original sample would be (1/2)^(2) = 0.25.
Finally, let's calculate the ratio of the amounts of 162Ho to 164Ho:
Ratio = (remaining amount of 162Ho) / (remaining amount of 164Ho)
= 0.125 / 0.25
= 0.5
Therefore, after one hour, the ratio of the amounts of 162Ho to 164Ho is 0.5 (or 1:2 if expressed as a ratio).