Calculation of the Diameter of a Mercury Atom

To calculate the diameter of a mercury atom, we need to determine the volume occupied by a single atom and then find the side length of the cube that would contain that volume.

Step 1: Finding the Volume of a Mercury Atom

1. The density of mercury is given as 13.6 g/mL. This means that 1 mL (or 1 cm³) of mercury has a mass of 13.6 grams.
2. The atomic mass of mercury is given as 200 atomic mass units (amu).
3. We know that 1 mole of any substance contains Avogadro's number (6.022 × 10^23) of atoms.
4. Therefore, the mass of 1 mole of mercury is 200 grams.
5. Using the molar mass and density, we can calculate the volume occupied by 1 mole of mercury atoms.

Volume of 1 mole of mercury atoms = (Mass of 1 mole of mercury) / (Density of mercury)
= 200 g / 13.6 g/mL
= 14.71 mL

6. Since 1 mL is equal to 1 cm³, the volume of 1 mole of mercury atoms is 14.71 cm³.

Step 2: Finding the Volume Occupied by a Single Atom

1. To find the volume occupied by a single atom, we divide the volume of 1 mole of mercury atoms by Avogadro's number.

Volume occupied by a single mercury atom = (Volume of 1 mole of mercury atoms) / (Avogadro's number)
= 14.71 cm³ / (6.022 × 10^23)
= 2.44 × 10^(-23) cm³

Step 3: Calculating the Side Length of the Cube

1. Since we assume that the atom occupies a cube with edge length equal to the diameter of the mercury atom, we can find the side length of the cube by taking the cube root of the volume occupied by a single atom.

Side length of the cube = (Volume occupied by a single atom)^(1/3)
= (2.44 × 10^(-23) cm³)^(1/3)
= 2.9 × 10^(-8) cm

Step 4: Finding the Diameter of the Mercury Atom

1. The diameter of the mercury atom is twice the side length of the cube, as it passes through the center of the cube.

Diameter of the mercury atom = 2 × (Side length of the cube)
= 2 × (2.9 × 10^(-8) cm)
= 5.8 × 10^(-8) cm

Therefore, the diameter of a mercury atom is approximately 5.8 × 10^(-8) cm, which can be rounded to 2.9 × 10^(-8) cm.