Element X has three istotopes, X⁹⁵, X⁹⁶, and X⁹⁸. Element X has an atomic mass of 96.344u. If X-98 has a mass of 98.0001u and an abundance of 19.8%, what are the abundances of X-96 (mass of 95.9988u) and X-95 (mass of 95.0011u)?
We know X-98 is 98.0001 and 0.198; therefore, 1-0.198 = 0.802 must be the fraction for 96 and 95.
Let X = fractional abundance of X-96
Then 0.8020-x = fractional abundance of 95.
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98.0001(0.198) + 95.9988(x) + 95.0011(0.802-x) = 96.344
Solve for x.
To find the abundances of isotopes X-96 and X-95, we can use the information given and set up a system of equations.
Let's assume the abundances of X-96 and X-95 are A and B, respectively. We know that the sum of the abundances of all isotopes should be equal to 100%, so we have the equation:
A + B + 19.8 = 100 (equation 1)
Additionally, we know that the sum of the masses of each isotope multiplied by their abundances should be equal to the atomic mass of element X, which is 96.344u. Using this information, we can set up another equation:
(95.9988 * A) + (95.0011 * B) + (98.0001 * 19.8) = 96.344 (equation 2)
Now, we can solve this system of equations to find the values of A and B.
First, simplify equation 2:
95.9988A + 95.0011B + 1938.0198 = 96.344
Rearrange this equation:
95.9988A + 95.0011B = 96.344 - 1938.0198
95.9988A + 95.0011B = -1841.6758 (equation 3)
Now, we can solve equations 1 and 3 simultaneously. We'll use the substitution method.
From equation 1, isolate A:
A = 100 - B - 19.8
Substitute this value of A into equation 3:
95.9988(100 - B - 19.8) + 95.0011B = -1841.6758
Now, simplify this equation:
9599.88 - 95.9988B - 1918.028 + 95.0011B = -1841.6758
9599.88 - 1918.028 = 95.9988B - 95.0011B + 1841.6758
Combine like terms:
7681.852 = 0.9977B + 1841.6758
Rearrange this equation:
0.9977B = 7681.852 - 1841.6758
0.9977B = 5840.1762
Divide both sides by 0.9977:
B = 5840.1762 / 0.9977
B ≈ 5850.16
To find the value of A, substitute this value of B into equation 1:
A + 5850.16 + 19.8 = 100
A ≈ 100 - 5850.16 - 19.8
A ≈ 95.04
Therefore, the abundance of X-96 (mass of 95.9988u) is approximately 95.04% and the abundance of X-95 (mass of 95.0011u) is approximately 5850.16%.