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 \(151.96 \mathrm{u} .\) Calculate the relative abundance of the two europium isotopes.

Let x be the relative abundance of \({ }^{151} \mathrm{Eu}\) and y be the relative abundance of \({ }^{153} \mathrm{Eu}\). Since these are the only two isotopes, their relative abundances sum up to 100% or 1, so we can write: x + y = 1

Using the given data and the relationship between the average atomic mass and the individual atomic masses with their relative abundance, the equation is: 151.96 = (150.9196 × x) + (152.9209 × y)

Now we have a system of two linear equations with two variables, x and y: 1. x + y = 1 2. 151.96 = (150.9196 × x) + (152.9209 × y) From equation (1), we can obtain x as follows: x = 1 - y Now, substitute x in equation (2): 151.96 = (150.9196 × (1 - y)) + (152.9209 × y) Expand and rearrange to solve for y: 151.96 = 150.9196 - 150.9196y + 152.9209y 151.96 = 150.9196 + 2.0013y Now, solve for y: 2.0013y = 151.96 - 150.9196 y = \(\frac{1.0404}{2.0013}\) y ≈ 0.5203 Now, substitute the value of y back into the equation for x: x = 1 - 0.5203 x ≈ 0.4797 Thus, the relative abundance of \({ }^{151} \mathrm{Eu}\) (x) is approximately 47.97% and the relative abundance of \({ }^{153} \mathrm{Eu}\) (y) is approximately 52.03%.