Use a calculator to help solve the problem.
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 hours. (Round your answer to three decimal places.)
Assume we start with 1 unit of each
after t = 2 hrs = 120 minutes
amount of 162ho = 1(1/2)^(120/22)
amount of 164ho = 1(1/2)^(120/37)
ratio of the two = 1(1/2)^(120/22) / 1(1/2)^(120/37)
= (1/2)^(120/22 - 120/37)
= (1/2)^(900/407)
= .2159
= .216 to 3 decimals
= 216 : 1000
=27 : 125
amount162/amount164= 100/100 * e^(-.692*60/22)/e^(-.692*60/37)
e^-.692*60(1/22-1/37)=.465
check my work.
thank you Reiny and Bob
To solve this problem using a calculator, we need to understand the concept of half-life and how it affects the decay of isotopes.
The half-life of an isotope is the time it takes for half of the initial amount of the isotope to decay. In this case, the half-life of 162Ho is 22 minutes, and the half-life of 164Ho is 37 minutes.
Let's break down the problem step by step:
1. Determine the number of half-lives in two hours:
Since one hour consists of 60 minutes, two hours would be 2 x 60 = 120 minutes.
For 162Ho: 120 minutes / 22 minutes (half-life of 162Ho) = 5.4545 half-lives
For 164Ho: 120 minutes / 37 minutes (half-life of 164Ho) = 3.2432 half-lives
2. Calculate the remaining amounts of each isotope after the given number of half-lives:
For 162Ho: Remaining amount = Initial amount × (1/2)^(number of half-lives) = 1 × (1/2)^5.4545
For 164Ho: Remaining amount = Initial amount × (1/2)^(number of half-lives) = 1 × (1/2)^3.2432
3. Calculate the ratio of the amounts of 162Ho to 164Ho:
Ratio = Amount of 162Ho / Amount of 164Ho
Now, you can use a calculator to solve each part separately.
First, calculate the remaining amount of 162Ho:
Remaining amount of 162Ho = 1 × (1/2)^5.4545
Next, calculate the remaining amount of 164Ho:
Remaining amount of 164Ho = 1 × (1/2)^3.2432
Finally, calculate the ratio of the amounts:
Ratio = Remaining amount of 162Ho / Remaining amount of 164Ho
Using a calculator, perform the calculations and round your answer to three decimal places.