Question

One mole of magnesium (6 × 1023 atoms) has a mass of 24 grams, as shown in the periodic table on the inside front cover of the textbook. The density of magnesium is 1.7 grams/cm3. What is the approximate diameter of a magnesium atom (length of a bond) in a solid block of the material? Make the simplifying assumption that the atoms are arranged in a "cubic" array, as shown in the figure. Remember to convert to SI units.

Answer 1

This question involves the concepts of density, volume, and mass.

The approximate diameter of a magnesium atom is "3.55 x 10⁻¹⁰ m".

STEP 1 (FINDING MASS OF INDIVIDUAL ATOM)

It is given that:

Mass of one mole = 24 grams

Mass of 6 x 10²³ atoms = 24 grams

Mass of 1 atom = $\frac{24\ grams}{6\ x\ 10^{23}\ atoms}$ = 4 x 10⁻²³ grams

STEP 2 (FINDING VOLUME OF A SINGLE ATOM)

$\rho = \frac{m}{V}\\\\V=\frac{m}{\rho}$

where,

• $\rho$ = density = 1.7 grams/cm³
• m = mass of single atom = 4 x 10⁻²³ grams
• V = volume of single atom = ?

Therefore,

$V=\frac{4\ x\ 10^{-23}\ grams}{1.7\ grams/cm^3}$

V = 2.35 x 10⁻²³ cm³

STEP 3 (FINDING DIAMETER OF ATOM)

The atom is in a spherical shape. Hence, its Volume can be given as follows:

$V =\frac{\pi d^3}{6}\\\\d=\sqrt[3]{ \frac{6V}{\pi}}\\\\d=\sqrt[3]{ \frac{6(2.35\ x\ 10^{-23}\ cm^3)}{\pi}}$

d = 0.355 x 10⁻⁷ cm = 3.55 x 10⁻¹⁰ m

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