Question

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

Short answer

Thetwo europium isotopes' relative abundance is:

$$\begin{gathered} {}^{151}\mathrm{Eu}:48.014\% \\ {}^{153}\mathrm{Eu}:51.986\% \end{gathered}$$

Step-by-step solution

Definition of europium isotopes relative abundance

To determine the relative abundance of the two europium isotopes${}^{\mathbf{151}}\mathbf{Eu}$and${}^{\mathbf{153}}\mathbf{Eu}$, consider the percent abundance of ${}^{\mathbf{151}}\mathbf{Eu}$ as ‘X’. Then the percent abundance of ${}^{\mathbf{153}}\mathbf{Eu}$ will be ‘100-X’. The average atomic mass of europium is the sum of the contributions to the average atomic mass for each isotope.

Calculation of relative abundance of isotopes

Theaverage atomic mass of europium is 151.96 amu. The mass of the isotope ${}^{151}\mathrm{Eu}$ is 150.9196 amu, and that of ${}^{153}\mathrm{Eu}$ is 152.9209 amu.

With the known value solve for X as:

$$\begin{gathered} \left(\frac{\mathrm{X}}{100}\right)\left(150.9196amu\right)+\left(\frac{100-X}{100}\right)\left(152.9209amu\right)=151.96\mathrm{amu} \\ \mathrm{X}\left(152.9196\mathrm{amu}-150.9196\right)=100\left(152.9209\mathrm{amu}-151.96\mathrm{amu}\right) \\ \mathrm{X}=\frac{96.09}{2.0013} \\ \mathrm{X}=48.014 \end{gathered}$$

Thus, the relative abundance of two europium isotopes is

$$\begin{gathered} {}^{151}\mathrm{Eu}:48.014\% \\ {}^{153}\mathrm{Eu}:51.986\% \end{gathered}$$