The half-life of Th(227) is 18.72 days. It decays by alfa emission to Ra(223), an alfa emitter whose half-life is 11.43 days. A particular sample contains 10^6 [10E(+6)] atoms of Th(227) and no Ra(223) at time t=0.
a) How many atoms of each type will be in the sample at t=15 days?
b) At what time will the number of atoms of each type be equal?
To answer these questions, we can use the concept of exponential decay. The number of atoms in a radioactive sample decays exponentially over time, and the rate of decay is governed by the half-life of the element.
a) How many atoms of each type will be in the sample at t=15 days?
To determine the number of atoms of each type at t=15 days, we need to calculate the remaining number of Th(227) atoms and the number of Ra(223) atoms produced during this time.
Let's start with the Th(227) atoms:
The half-life of Th(227) is 18.72 days. This means that after each half-life, the number of remaining Th(227) atoms is halved. We can calculate the fraction of Th(227) remaining at 15 days using the formula:
Remaining fraction = 0.5^(t/half-life)
where t is the time (15 days) and half-life is the half-life of the element (18.72 days).
Remaining fraction of Th(227) = 0.5^(15/18.72) ≈ 0.465
To find the actual number of Th(227) atoms at t=15 days, we multiply the remaining fraction by the initial number of Th(227) atoms:
Number of Th(227) atoms at t=15 days = Remaining fraction * Initial number of Th(227) atoms = 0.465 * 10^6 = 4.65 * 10^5
Next, let's calculate the number of Ra(223) atoms produced during this time:
Since Th(227) decays into Ra(223) by alpha emission, and the half-life of Ra(223) is 11.43 days, we can calculate how much Ra(223) is produced from the decaying Th(227) over the 15-day period.
The number of Ra(223) atoms produced = Initial number of Th(227) atoms * (1 - remaining fraction of Th(227))
Number of Ra(223) atoms produced = 10^6 * (1 - 0.465) = 5.35 * 10^5
Therefore, at t=15 days, there will be approximately 4.65 * 10^5 Th(227) atoms and 5.35 * 10^5 Ra(223) atoms in the sample.
b) At what time will the number of atoms of each type be equal?
To find the time at which the number of Th(227) and Ra(223) atoms becomes equal, we need to solve the following equation:
Number of Th(227) atoms = Number of Ra(223) atoms
Using the decay equations, we can set up the following equation:
Remaining fraction of Th(227) = 1 - Remaining fraction of Ra(223)
0.5^(t/half-life Th) = 1 - 0.5^(t/half-life Ra)
Simplifying the equation, we can raise both sides to the power of -1 and solve for t:
2^(t/half-life Th) = 1 + 2^(t/half-life Ra)
t/half-life Th = log2(1 + 2^(t/half-life Ra))
t = half-life Th * log2(1 + 2^(t/half-life Ra))
Plugging in the specific values for half-life Th and half-life Ra:
t = 18.72 * log2(1 + 2^(t/11.43))
Unfortunately, it is not possible to solve this equation analytically, and we need to use numerical methods or approximation techniques to find the exact time when the number of atoms of each type becomes equal.
To solve this problem, we need to use the concept of radioactive decay and the relationship between half-life and the number of atoms present.
a) To find the number of atoms of each type at t = 15 days, we can use the following steps:
Step 1: Find the number of half-lives for Th(227) in 15 days.
Number of half-lives = t / half-life
Number of half-lives for Th(227) = 15 / 18.72
Step 2: Calculate the number of remaining Th(227) atoms.
Remaining atoms of Th(227) = initial atoms of Th(227) * (1/2)^(number of half-lives)
Remaining Th(227) atoms = 10^6 * (1/2)^(15 / 18.72)
Step 3: Calculate the number of Ra(223) atoms.
Number of Ra(223) atoms = initial atoms of Th(227) - Remaining Th(227) atoms
Number of Ra(223) atoms = 10^6 - Remaining Th(227) atoms
Now you can calculate the values for Remaining Th(227) atoms and Number of Ra(223) atoms.
b) To find the time when the number of atoms of each type will be equal, we can set up an equation based on the decay of both elements.
Let t be the time in days. Since the half-life of Th(227) is longer than Ra(223), we can equate the number of remaining atoms for both elements:
Remaining atoms of Th(227) = Remaining atoms of Ra(223)
Similarly, using the equations established in part (a), we can solve for t.
10^6 * (1/2)^(t / 18.72) = 10^6 - (10^6 * (1/2)^(t / 18.72))
Solving this equation will give us the value for t when the number of atoms of each type is equal.
Please note that you'll need to use numerical methods like trial and error or graphing to solve the equation.