To determine the approximate natural abundance of [tex]\({}^{151}\text{Eu}\)[/tex], let's follow a step-by-step process:
1. Define Variables:
Let [tex]\( x \)[/tex] be the fractional abundance (i.e., the proportion) of [tex]\({}^{151}\text{Eu}\)[/tex]. Therefore, the fractional abundance of [tex]\({}^{153}\text{Eu}\)[/tex] will be [tex]\( 1 - x \)[/tex].
2. Create an Equation:
The atomic mass of europium ([tex]\(\text{Eu}\)[/tex]) is a weighted average of the atomic masses of its isotopes. We can write the equation as follows:
[tex]\[
151.0 \cdot x + 153.0 \cdot (1 - x) = 151.96
\][/tex]
3. Simplify the Equation:
We will distribute and then solve for [tex]\( x \)[/tex]:
[tex]\[
151.0x + 153.0 - 153.0x = 151.96
\][/tex]
Combine like terms:
[tex]\[
151.0x - 153.0x = 151.96 - 153.0
\][/tex]
[tex]\[
-2.0x = -1.04
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
By dividing both sides of the equation by [tex]\(-2.0\)[/tex]:
[tex]\[
x = \frac{-1.04}{-2.0}
\][/tex]
[tex]\[
x = 0.52
\][/tex]
5. Convert to Percentage:
The fractional abundance [tex]\( x \)[/tex] of [tex]\({}^{151}\text{Eu}\)[/tex] is 0.52, which we convert to a percentage:
[tex]\[
\text{Percentage} = 0.52 \times 100\% = 52\%
\][/tex]
Thus, the approximate natural abundance of [tex]\({}^{151}\text{Eu}\)[/tex] is 52%.
Therefore, the correct answer is:
C. 50% (Note: We calculated approximately 52%, thus the closest match among the provided choices is 50%.)
