Question

the element europium exists in nature as two isotopes: ^151eu has a mass of 150.9196 u and ^153eu has a mass of 152.9209 u. the average atomic mass of europium is 151.96 u. calculate the relative abundance of the two europium isotopes.

Explanation:

Step1: Let the relative abundance of \(^{151}\text{Eu}\) be \(x\), then the relative abundance of \(^{153}\text{Eu}\) is \(1 - x\).

The formula for the average - atomic mass \(A\) of an element with two isotopes is \(A=m_1x + m_2(1 - x)\), where \(m_1\) and \(m_2\) are the masses of the isotopes. Here, \(m_1 = 150.9196\ u\), \(m_2=152.9209\ u\), and \(A = 151.96\ u\).
So, \(151.96=150.9196x+152.9209(1 - x)\).

Step2: Expand the right - hand side of the equation.

\[

$$\begin{align*} 151.96&=150.9196x + 152.9209-152.9209x\\ 151.96&=152.9209+(150.9196x - 152.9209x)\\ 151.96&=152.9209-2.0013x \end{align*}$$

\]

Step3: Solve for \(x\).

First, move the terms around: \(2.0013x=152.9209 - 151.96\).
\(2.0013x = 0.9609\).
Then, \(x=\frac{0.9609}{2.0013}\approx0.48\).
The relative abundance of \(^{151}\text{Eu}\) is approximately \(48\%\), and the relative abundance of \(^{153}\text{Eu}\) is \(1 - 0.48 = 0.52\) or \(52\%\).

Answer:

The relative abundance of \(^{151}\text{Eu}\) is approximately \(48\%\) and the relative abundance of \(^{153}\text{Eu}\) is approximately \(52\%\).