A rock containing 238U and 206Pb was examined to determine its appropriate age. The molar ratio of 206Pb to 238U was 0.1462. Assuming no lead was originally present and that all lead formed remained in the rock. What is the age of the rock in years? The half life of 238U is 4.500*10^9yr
Let's start with 100 atoms of U238. Suppose x atoms of U238 decompose to Pb. Then there must be 100-x atoms U238 not yet decomposed. We know the ratio is 0.1462; therefore,
(x/100-x) = 0.1462 and solve for x which is the amount of Pb formed and 100-x is the amount of U left.
So No = 100 and N = 100-x, whatever that value is. Substitute into
ln(No/N) = kt
You know No and N. You want to find t. k = 0.693/t1/2 and you have the half life. Solve for k and substitute into the equation above.
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To determine the age of the rock, we can use the concept of half-life. The half-life of 238U is given as 4.500 * 10^9 years.
Let's denote the initial amount of 238U as N0. Since all the Pb-206 found in the rock is the result of decay from 238U, we can denote the amount of Pb-206 as N.
Given that the molar ratio of 206Pb to 238U is 0.1462, we have:
N / N0 = 0.1462
To find the age of the rock, we need to determine how many half-lives have passed. Since each half-life reduces the amount of 238U by half, we can express the relationship as:
N / N0 = (1/2)^(number of half-lives)
Now we can rewrite the equation as:
0.1462 = (1/2)^(number of half-lives)
To find the number of half-lives, we can take the logarithm of both sides:
log(0.1462) = log[(1/2)^(number of half-lives)]
Using logarithm rules, we can bring the exponent down:
log(0.1462) = number of half-lives * log(1/2)
Now we can solve for the number of half-lives:
number of half-lives = log(0.1462) / log(1/2)
Calculating this using a calculator:
number of half-lives = 0.8621 / (-0.3010) = -2.866
Since the number of half-lives cannot be negative, we must take the absolute value:
number of half-lives ≈ 2.866
Now we can find the age of the rock by multiplying the number of half-lives by the half-life of 238U:
age of the rock = number of half-lives * half-life of 238U
age of the rock ≈ 2.866 * 4.500 * 10^9 years
Therefore, the age of the rock is approximately 12.897 * 10^9 years or 1.2897 * 10^10 years.
To determine the age of the rock, we can use the concept of radioactive decay. Uranium-238 (238U) undergoes radioactive decay and eventually decays into lead-206 (206Pb) over time. We can use the molar ratio of 206Pb to 238U to calculate the age of the rock.
The molar ratio of 206Pb to 238U is given as 0.1462. This ratio represents the amount of lead-206 relative to the amount of uranium-238 present in the rock.
We can use the equation for radioactive decay to relate the molar ratio to the age of the rock:
Age = (ln(ratio) / ln(0.5)) * half-life
Where:
- Age refers to the age of the rock
- Ratio is the molar ratio of 206Pb to 238U (0.1462 in this case)
- Half-life is the half-life of uranium-238 (4.500 * 10^9 years)
Let's plug the values into the formula and calculate the age of the rock:
Age = (ln(0.1462) / ln(0.5)) * (4.500 * 10^9 years)
Using a scientific calculator, calculate ln(0.1462) and ln(0.5), and then divide the natural logarithm of the ratio by the natural logarithm of 0.5:
Age = (-1.921 / -0.693) * (4.500 * 10^9 years)
Simplify the equation further:
Age = 2.773 * (4.500 * 10^9 years)
Finally, calculate the value to find the age of the rock:
Age ≈ 12.477 * 10^9 years
Therefore, the age of the rock is approximately 12.477 billion years.