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Chapter 3: Problem 43

The element europium exists in nature as two isotopes: 151 \(\mathrm{Eu}\) has a mass of 150.9196 \(\mathrm{u}\) and \(^{153} \mathrm{Eu}\) has a mass of 152.9209 \(\mathrm{u}\) . The average atomic mass of europium is 151.96 u. Calculate the relative abundance of the two europium isotopes.

Short Answer

Expert verified
The relative abundance of the two isotopes of europium is approximately 46.09% for 151Eu and 53.91% for 153Eu.

Step by step solution

01

Variables representing relative abundances

Let x be the relative abundance of 151Eu (in fractions, so 0 < x < 1) and let y be the relative abundance of 153Eu (again, 0 < y < 1). We know that the sum of the relative abundances should equal 1, since they are the only two isotopes of europium: \( x + y = 1 \)
02

Equation for average atomic mass

Let's create an equation that represents the average atomic mass of europium (151.96 u). The average atomic mass is the weighted sum of the isotopes' masses, with their relative abundances as weights: \(Averag­¬e¬¬_mass=Eur­¬opiu¬¬m=(mass_­¬{151E¬ńu}×abundance_­¬{151E¬ëu})+(mass_­¬{153E=ě¬u}×abundance_­¬{153E¬ęu})\) Then, we can substitute the given values: \(151.96 u = (150.9196 \, u \cdot x) + (152.9209 \, u \cdot y)\)
03

Solve the system of equations

Now, we have a system of two equations with two variables: \(x+y=1\) \(151.96 = 150.9196x + 152.9209y\) We can solve this system of equations using substitution or elimination method. Using the elimination method, we'll first multiply the first equation by -150.9196 to eliminate the x term: \(-150.9196x - 150.9196y = -150.9196\) Next, we'll add this new equation to the second equation to eliminate x: \((-150.9196y - 152.9209y) = 151.96 - 150.9196\) \(-1.9313y = 1.0404\) Now, divide by -1.9313 to solve for y: \(y = \frac{1.0404}{-1.9313} \approx 0.5391\) Finally, substitute the value of y back into the first equation to find x: \(x + 0.5391 = 1\) \(x = 1 - 0.5391\) \(x \approx 0.4609\)
04

Calculate relative abundances as percentages

To express the relative abundances as percentages, multiply the values of x and y by 100: Relative abundance of 151Eu = 0.4609 × 100% ≈ 46.09% Relative abundance of 153Eu = 0.5391 × 100% ≈ 53.91% The relative abundance of the two isotopes of europium is approximately 46.09% for 151Eu and 53.91% for 153Eu.

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