Question

a fictitious element is composed of isotopes a and b with masses of 61.9887 and 64.9846 amu, respectively. the atomic mass of the element is 64.52. what can you conclude about the natural abundances of the two isotopes?
the natural abundance of isotope b must be greater than the natural abundance of isotope a.
the natural abundance of isotope a must be greater than the natural abundance of isotope b.
nothing can be concluded about the natural abundances of the two isotopes from the given information.
the natural abundances of both isotopes must be about equal.

Explanation:

Step1: Recall the formula for atomic mass

The atomic mass of an element composed of two isotopes \(A\) and \(B\) is given by \(M = M_A\times x+M_B\times(1 - x)\), where \(M\) is the atomic mass of the element, \(M_A\) and \(M_B\) are the masses of the isotopes, and \(x\) is the natural abundance of isotope \(A\) (and \(1 - x\) is the natural abundance of isotope \(B\)).

Step2: Analyze the relationship between mass and abundance

We know \(M_A=61.9887\) amu, \(M_B = 64.9846\) amu, and \(M = 64.52\) amu. The atomic mass \(M\) is closer to \(M_B\) than to \(M_A\).

If we assume \(x\) (abundance of \(A\)) and \(1 - x\) (abundance of \(B\)), we can rewrite the formula as \(64.52=61.9887x + 64.9846(1 - x)\).

Expanding gives \(64.52=61.9887x+64.9846 - 64.9846x\).

Combining like - terms: \(64.52-64.9846=(61.9887 - 64.9846)x\).

\(- 0.4646=-2.9959x\), then \(x=\frac{0.4646}{2.9959}\approx0.155\) and \(1 - x\approx0.845\)

Since the atomic mass is closer to the mass of isotope \(B\), the isotope with the higher mass (\(B\)) has a higher natural abundance.

Answer:

The natural abundance of isotope B must be greater than the natural abundance of isotope A.