Question 3 of 3

There are two naturally occurring isotopes of europium, [tex]^{151}Eu \ (151.0 \ \text{amu})[/tex] and [tex]^{153}Eu \ (153.0 \ \text{amu})[/tex]. If the atomic mass of Eu is 151.96, what is the approximate natural abundance of [tex]^{151}Eu[/tex]?

A. 20%

B. 40%

C. 50%

D. 60%

E. 80%



Answer :

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%.)