Question

Gallium has two naturally occurring isotopes, 69Ga and 71Ga, with masses of 68.9257amu and 70.9249amu, respectively. Calculate the percent abundances of these isotopes of gallium.

Answer 1

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:

1. The sum of the fractions must equal 1:
\[
x + y = 1
\]

2. 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:
- Percent of \(^{69}\text{Ga}\):
\[
\text{Percent of }^{69}\text{Ga} = x \times 100 \approx 0.601 \times 100 \approx 60.1\%
\]

- 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%**