For a metal ion, the calculated magnetic moment is 4.90 BM. This metal ion has _______ number of unpaired electrons

We need to find the number of unpaired electrons in a metal ion whose calculated magnetic moment is 4.90 BM.

The spin-only magnetic moment (in Bohr Magnetons, BM) is given by:

where $$n$$ is the number of unpaired electrons. This formula arises from quantum mechanics, where the spin magnetic moment depends on the total spin quantum number $$S = n/2$$, and $$\mu = \sqrt{4S(S+1)} = \sqrt{n(n+2)}$$ BM.

We need a positive integer $$n$$ such that $$n^2 + 2n \approx 24$$. Testing values:

$$n = 3$$: $$3^2 + 2(3) = 9 + 6 = 15$$ (too small)

$$n = 4$$: $$4^2 + 2(4) = 16 + 8 = 24$$ (matches!)

$$n = 5$$: $$5^2 + 2(5) = 25 + 10 = 35$$ (too large)

With $$n = 4$$: $$\mu = \sqrt{4 \times 6} = \sqrt{24} = 4.899 \approx 4.90$$ BM. This matches the given value.

The metal ion has 4 unpaired electrons.