How old is a sample that contains 125 atoms of U-235 and 375 atoms of Pb-207?
...Given that the half life values on my handout are 713 million years.
*I do not even know where to start so could you please explain in steps too? Thanks.
Well, let's start with some clown math, shall we? So, you have 125 atoms of U-235 and 375 atoms of Pb-207.
Now, since U-235 is radioactive and it decays over time, we need to figure out how many U-235 atoms have decayed to form Pb-207.
The half-life of U-235 is 713 million years, which means that every 713 million years, half of the U-235 atoms will decay.
So, let's calculate how many half-lives have passed by dividing the total number of U-235 atoms (125) by the amount of U-235 atoms that decay in one half-life (which is 125 / 2 = 62.5).
Since we can't have half a half-life, let's round it down to the nearest whole number. This means that approximately 2 half-lives have passed.
Now, let's calculate the remaining U-235 atoms by subtracting the number of decayed U-235 atoms (125 - 62.5 = 62.5). Again, let's round it down to the nearest whole number, which gives us 62 remaining U-235 atoms after 2 half-lives.
Since U-235 atoms decay into Pb-207 atoms, the number of Pb-207 atoms formed from the decayed U-235 atoms should be the same as the number of remaining U-235 atoms, which is 62.
But wait a second, you already started with 375 Pb-207 atoms! So, to get the age of the sample, we need to sum up the number of Pb-207 atoms formed from the decayed U-235 atoms (62) with the initial number of Pb-207 atoms (375).
Therefore, the total number of Pb-207 atoms in the sample is 62 + 375 = 437.
Now, since we know that the half-life of U-235 is 713 million years, and two half-lives have passed, we can multiply the number of half-lives (2) by the half-life value to get the age of the sample.
So, the age of the sample is approximately 2 * 713 million years = 1.426 billion years.
Ta-da! There you have it, the age of the sample.
To determine the age of a sample, you can use the concept of radioactive decay. Here are the steps:
Step 1: Understand the concept
The idea behind radioactive decay is that certain atoms, known as parent isotopes, decay into stable daughter isotopes over time. This decay process follows an exponential decay model, where the number of parent isotopes decreases while the number of daughter isotopes increases.
Step 2: Determine the decay equation
In this case, uranium-235 (U-235) is the parent isotope, and lead-207 (Pb-207) is the daughter isotope. The decay equation for U-235 to Pb-207 is as follows:
U-235 → Pb-207 + α
Step 3: Calculate the decay constant
The decay constant (λ) can be calculated using the half-life (t½) value given in your handout. The decay constant is related to the half-life by the equation:
λ = ln(2) / t½
Substituting the given half-life value (t½ = 713 million years) into the equation, you can find the decay constant (λ).
Step 4: Calculate the age of the sample
To find the age of the sample, you need to determine the ratio of U-235 to Pb-207 atoms in the sample, and then use this ratio to calculate the age. Here's how:
- Calculate the ratio of U-235 to Pb-207 atoms:
U-235 atoms / Pb-207 atoms = 125 / 375 = 1/3
- Use the exponential decay equation:
ln(N0/N) = λt
Where:
N0 = initial number of U-235 atoms
N = current number of U-235 atoms
λ = decay constant
t = time (in years)
- Convert the ratio 1/3 to a fraction of U-235 atoms remaining (N/N0):
N/N0 = 1 - 1/3 = 2/3
- Substitute the values into the decay equation:
ln(2/3) = λt
- Solve for t (the age of the sample), using the calculated decay constant.
By following these steps, you should be able to determine the age of the sample.
To determine the age of the sample, we can use the concept of radioactive decay, specifically the decay of uranium-235 (U-235) into lead-207 (Pb-207) over time.
Step 1: Understand the concept of radioactive decay and half-life:
Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation. A half-life is the time required for half of the atoms in a radioactive substance to undergo decay.
Step 2: Calculate the ratio of U-235 to Pb-207 in the sample:
We are given that the sample contains 125 atoms of U-235 and 375 atoms of Pb-207. The ratio of U-235 to Pb-207 is calculated as follows:
U-235/Pb-207 = (number of U-235 atoms) / (number of Pb-207 atoms)
U-235/Pb-207 = 125 / 375
U-235/Pb-207 = 1 / 3
Step 3: Determine the number of half-lives that have occurred:
Since we know the half-life of U-235 is 713 million years, we can find the number of half-lives that have occurred by dividing the age of the sample by the half-life. Let X represent the number of half-lives:
X * Half-Life = Age of the Sample
Step 4: Use the ratio of U-235 to Pb-207 in radioactive decay:
During each half-life, half of the U-235 atoms decay into Pb-207. Therefore, for every 1 atom of U-235, half an atom of U-235 and half an atom of Pb-207 are created.
Step 5: Calculate the number of U-235 atoms that decayed:
Since the initial ratio of U-235 to Pb-207 is 1/3, after X half-lives, the number of U-235 atoms remaining can be calculated as:
(Number of U-235 atoms) = (initial number of U-235 atoms) * (1/2)^(Number of half-lives)
Step 6: Calculate the number of U-235 atoms decayed into Pb-207:
The difference between the initial number of U-235 atoms and the number of U-235 atoms remaining gives the number of U-235 atoms that have decayed into Pb-207:
(Number of U-235 atoms decayed) = (initial number of U-235 atoms) - (number of U-235 atoms remaining)
Step 7: Calculate the age of the sample in years:
Using the number of U-235 atoms that have decayed and the ratio of U-235 to Pb-207 atoms produced during decay, we can determine the age of the sample. The age can be calculated as:
Age = (Number of U-235 atoms decayed) * (Half-Life)
Now, let's plug in the given values and solve for the age of the sample:
(Number of U-235 atoms) = 125 * (1/2)^(X)
(Number of U-235 atoms decayed) = 125 - (125 * (1/2)^(X))
Age = (Number of U-235 atoms decayed) * (Half-Life)
To solve for the age, you would need to iteratively try different values of X until you achieve a close approximation. Start with X = 0, then incrementally increase X until the number of U-235 atoms remaining is close to 0. The corresponding Age value at that point will be the approximate age of the sample.