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Chapter 2: Problem 17

The radius of an atom of gold \((\mathrm{Au})\) is about \(1.35 \AA\). (a) \(\mathrm{Ex}\) press this distance in nanometers (nm) and in picometers (pm). (b) How many gold atoms would have to be lined up to span \(1.0 \mathrm{~mm} ?\) (c) If the atom is assumed to be a sphere, what is the volume in \(\mathrm{cm}^{3}\) of a single Au atom?

Short Answer

Expert verified
(a) 1.35 Å = 0.135 nm = 135 pm. (b) Approximately 3.7 * 10^6 gold atoms would line up to span 1.0 mm. (c) The volume of a single gold atom is 1.29 * 10^-28 cm^3.

Step by step solution

01

Part (a) Î Angstrom to Nanometer and Picometer Conversion

To convert from Angstroms \(\mathrm{(Å)}\) to nanometers \(\mathrm{(nm)}\) and picometers \(\mathrm{(pm)}\), the student must know the following conversion factors: 1 Angstrom (\(\mathrm{Å}\)) = 0.1 nanometers (\(\mathrm{nm}\)) 1 Angstrom (\(\mathrm{Å}\)) = 100 picometers (\(\mathrm{pm}\)) Given that the radius of a gold atom is 1.35 Å, the student can use these conversion factors to convert between the units: 1.35 Å * 0.1 nm/Å = 0.135 nm 1.35 Å * 100 pm/Å = 135 pm
02

Part (b) - Number of Gold Atoms to Span 1.0 mm

To calculate how many gold atoms would line up to span 1.0 mm, the student needs to divide the given distance (1.0 mm) by the diameter of a single gold atom (which is double the radius). First, convert 1.0 mm to picometers: 1.0 mm * 10^6 nm/mm * 10^3 pm/nm = 10^9 pm Next, calculate the diameter of a gold atom in picometers: Diameter = 2 * Radius = 2 * 135 pm = 270 pm Finally, divide the given distance by the diameter of the gold atom: Number of Gold Atoms = 10^9 pm / 270 pm ≈ 3.7 * 10^6 gold atoms
03

Part (c) - Volume of a Single Gold Atom

To calculate the volume of a single gold atom, assuming it's a sphere, the student can use the formula for the volume of a sphere: Volume = \(\frac{4}{3}\pi r^{3}\) Given the radius of a gold atom in picometers, the student must first convert it to cm before calculating the volume: 1.35 Å * 10^-10 cm/Å = 1.35 * 10^-10 cm Now, the student can plug the radius in cm into the formula for the volume of a sphere: Volume = \(\frac{4}{3}\pi (1.35 * 10^{-10})^{3}\) = 1.29 * 10^-28 cm^3

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