Question

Vitamin \(\mathrm{B}_{12}\) is a coordination compound with cobalt as its central atom. It contains \(4.4 \%\) cobalt by mass and has a molar mass of \(1.3 \times 10^{3} \mathrm{~g} / \mathrm{mol}\). How many cobalt atoms are in a molecule of vitamin \(\mathrm{B}_{12} ?\)

Short answer

Answer: There is approximately 1 cobalt atom present in a molecule of vitamin B12.

Step-by-step solution

Find the mass of cobalt in one mole of vitamin B12

To find the mass of cobalt in one mole of vitamin B12, we will multiply the molar mass of vitamin B12 by the percentage of cobalt in it (given as \(4.4 \%\)). Mass of cobalt in one mole of vitamin B12 = $$\frac{4.4}{100} \times 1.3 \times 10^{3} \mathrm{~g} / \mathrm{mol}$$

Calculate the number of moles of cobalt in one mole of vitamin B12

Now, we'll divide the mass of cobalt in one mole of vitamin B12 by the molar mass of cobalt (which is approximately \(58.93 \mathrm{~g} / \mathrm{mol}\) for cobalt) to find the number of moles of cobalt atoms in one mole of vitamin B12. Number of moles of cobalt in one mole of vitamin B12 = $$\frac{\frac{4.4}{100} \times 1.3 \times 10^{3} \mathrm{~g} / \mathrm{mol}}{58.93 \mathrm{~g} / \mathrm{mol}} = \frac{4.4 \times 1.3 \times 10^{3}}{58.93 \times 100} \ \mathrm{mol} $$

Calculate the number of cobalt atoms in a molecule of vitamin B12

The number of cobalt atoms in a molecule of vitamin B12 is equal to the number of moles of cobalt in one mole of vitamin B12 multiplied by Avogadro's number (\(6.022 \times 10^{23}\)). Then, we'll round the result to the nearest whole number to get the number of cobalt atoms in a molecule of vitamin B12. Number of cobalt atoms in a molecule of vitamin B12 = $$\frac{4.4 \times 1.3 \times 10^{3}}{58.93 \times 100} \ \mathrm{mol} \times 6.022 \times 10^{23} \ \mathrm{~atoms/mol}$$ Calculating the above expression, we have: Number of cobalt atoms in a molecule of vitamin B12 ≈ $$1 \ \mathrm{Co \ atom}$$ So, there is approximately 1 cobalt atom present in a molecule of vitamin B12.