You have 120 grams of U-235. How many half lives would it take to make enough helium to fill a 5.60 L balloon at STP?
To calculate the number of half-lives needed to produce a certain amount of helium, we need to know the half-life of U-235 and the molar mass of helium.
First, let's find the molar mass of helium, which is 4.00 grams per mole.
Next, we need to determine the number of moles of helium required to fill the balloon. To do this, we divide the mass of helium needed by its molar mass:
Mass of helium = 5.60 L x (1 mole / 22.4 L) x (4.00 g / 1 mole) = 1.00 g
Now, we can calculate the number of moles of U-235 that would produce 1.00 gram of helium. Since U-235 undergoes radioactive decay in a series of steps, we need to consider the different decay products and their respective half-lives.
The decay chain of U-235 goes as follows:
U-235 → Th-231 → Pa-231 → U-235 (recycled) → Th-227 → Ra-223 → Rn-219 → Po-215 → Pb-211 → Bi-211 → Tl-207 → Pb-207
Each step in the decay chain has a different half-life, so we need to determine the number of half-lives for each decay step until we reach helium. Let's look up the half-lives:
U-235: 7.04 x 10^8 years
Th-231: 25.5 hours
Pa-231: 32,760 years
Th-227: 18.72 days
Ra-223: 11.421 days
Rn-219: 3.96 seconds
Po-215: 1.780 milliseconds
Pb-211: 36.1 minutes
Bi-211: 2.14 minutes
Tl-207: 4.77 minutes
Pb-207: Stable (no further decay)
Now, we need to calculate how many half-lives are needed for each decay step until we reach 1.00 gram of helium. We can convert the half-lives into seconds to make the calculations easier:
Th-231 half-life: 25.5 hours x 60 minutes x 60 seconds = 91,800 seconds
Pa-231 half-life: 32,760 years x 365.25 days x 24 hours x 60 minutes x 60 seconds = 1.03 x 10^15 seconds
Th-227 half-life: 18.72 days x 24 hours x 60 minutes x 60 seconds = 1.61 x 10^6 seconds
Ra-223 half-life: 11.421 days x 24 hours x 60 minutes x 60 seconds = 9.86 x 10^5 seconds
Rn-219 half-life: 3.96 seconds
Po-215 half-life: 1.780 milliseconds = 1.780 x 10^-3 seconds
Pb-211 half-life: 36.1 minutes x 60 seconds = 2,166 seconds
Bi-211 half-life: 2.14 minutes x 60 seconds = 128.4 seconds
Tl-207 half-life: 4.77 minutes x 60 seconds = 286.2 seconds
Now, we need to calculate how many half-lives are required for each decay step until we reach 1.00 gram of helium:
Th-231: 1.00 g / (4.00 g / 1 mole) = 0.25 moles
Pa-231: (0.25 moles) / (0.25 moles / 2) = 2 half-lives
Th-227: (0.25 moles / 2) / (0.25 moles / 2) = 1 half-life
Ra-223: (0.25 moles / 2 / 2) / (0.25 moles / 2) = 0.5 half-lives
Rn-219: (0.25 moles / 2 / 2 / 2) / (0.25 moles / 2) = 0.25 half-lives
Po-215: (0.25 moles / 2 / 2 / 2 / 2) / (0.25 moles / 2) = 0.125 half-lives
Pb-211: (0.25 moles / 2 / 2 / 2 / 2 / 2) / (0.25 moles / 2) = 0.0625 half-lives
Bi-211: (0.25 moles / 2 / 2 / 2 / 2 / 2 / 2) / (0.25 moles / 2) = 0.03125 half-lives
Tl-207: (0.25 moles / 2 / 2 / 2 / 2 / 2 / 2 / 2) / (0.25 moles / 2) = 0.015625 half-lives
Since Tl-207 is the last step before helium, we don't need to consider any more half-lives.
Finally, we add up the number of half-lives required for each decay step:
2 + 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + 0.015625 = 4.984375 half-lives
So, it would take approximately 4.98 half-lives to produce enough helium to fill a 5.60 L balloon at STP using 120 grams of U-235.