To find the percent abundances of the isotopes of gallium (\(^{69}\text{Ga}\) and \(^{71}\text{Ga}\)), we will use the average atomic mass of gallium, which is approximately 69.723 amu. We can set up a system of equations to solve for the abundances.
Let:
- \(x\) = the fraction of \(^{69}\text{Ga}\)
- \(y\) = the fraction of \(^{71}\text{Ga}\)
The relations we can use are:
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The sum of the fractions must equal 1: \[ x + y = 1 \]
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The weighted average of the masses gives us the average atomic mass: \[ 68.9257x + 70.9249y = 69.723 \]
Now, we can solve these equations. From the first equation, we have: \[ y = 1 - x \]
Substituting \(y\) into the second equation: \[ 68.9257x + 70.9249(1 - x) = 69.723 \]
Expanding and simplifying: \[ 68.9257x + 70.9249 - 70.9249x = 69.723 \] \[ (68.9257 - 70.9249)x + 70.9249 = 69.723 \] \[ -1.9992x + 70.9249 = 69.723 \]
Now, isolate \(x\): \[ -1.9992x = 69.723 - 70.9249 \] \[ -1.9992x = -1.2019 \] \[ x = \frac{-1.2019}{-1.9992} \approx 0.601 \]
Now, substitute back to find \(y\): \[ y = 1 - x = 1 - 0.601 \approx 0.399 \]
Finally, we can express these abundances as percentages:
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Percent of \(^{69}\text{Ga}\): \[ \text{Percent of }^{69}\text{Ga} = x \times 100 \approx 0.601 \times 100 \approx 60.1% \]
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Percent of \(^{71}\text{Ga}\): \[ \text{Percent of }^{71}\text{Ga} = y \times 100 \approx 0.399 \times 100 \approx 39.9% \]
Thus, the percent abundances of the isotopes of gallium are:
- \(^{69}\text{Ga}\): 60.1%
- \(^{71}\text{Ga}\): 39.9%
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