The radius of an atom of gold (Au) is about . (a) Express this distance in nanometers (nm) and in picometers (pm). (b) How many gold atoms would have to be lined up to span 1.0 mm

(a) 0.135 nm; 135 pm

(b) 3.70 × 10⁶ Au atoms

Explanation:

The radius of an atom of gold (Au) is about 1.35 A. (a) Express this distance in nanometers (nm) and in picometers (pm). (b) How many gold atoms would have to be lined up to span 1.0 mm.

(a)

Step 1: Convert the radius to meters

We will use the conversion factor 1 m = 10¹⁰ A.

1.35 A × 1 m/10¹⁰ A = 1.35 × 10⁻¹⁰ m

Step 2: Convert the radius from m to nm

We will use the conversion factor 1 m = 10⁹ nm.

1.35 × 10⁻¹⁰ m × 10⁹ nm/1 m = 0.135 nm

Step 3: Convert the radius from m to pm

We will use the conversion factor 1 m = 10¹² pm.

1.35 × 10⁻¹⁰ m × 10¹² pm/1 m = 135 pm

(b)

Step 1: Convert "1.0 mm" to meters

We will use the conversion factor 1 m = 10³ mm.

1.0 mm × 1 m/10³ mm = 1.0 × 10⁻³ m

Step 2: Calculate how many gold atoms would have to be lined up to span 1.0 × 10⁻³ m

We will divide 1.0 × 10⁻³ m by 2 times the radius (diameter) of an atom of gold.

1.0 × 10⁻³ m / (2 × 1.35 × 10⁻¹⁰ m) = 3.70 × 10⁶