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Chapter 2: Problem 20

An atom of rhodium (Rh) has a diameter of about \(2.7 \times 10^{-8} \mathrm{cm} .\) (a) What is the radius of a rhodium atom in angstroms (A) and in meters \((\mathrm{m}) ?\) (b) How many Rh atoms would have to be placed side by side to span a distance of 6.0\(\mu \mathrm{m}\) ? (c) If you assume that the Rh atom is a sphere, what is the volume in \(\mathrm{m}^{3}\) of a single atom?

Short Answer

Expert verified
The radius of a rhodium atom is \(1.35 \times 10^{-8}\) cm, which is equivalent to 1.35 Å and \(1.35 \times 10^{-10}\) m. To span a distance of 6.0 µm, approximately 22,222 Rh atoms are needed side by side. The volume of a single Rh atom, assuming it is a sphere, is approximately \(1.03 \times 10^{-29}\) m³.

Step by step solution

01

Convert the diameter to radius in angstroms and meters

First, we convert the diameter of the rhodium atom to radius by dividing it by 2. Radius (in cm) = Diameter / 2 = \((2.7\times10^{-8}\mathrm{cm})\) / 2 Next, convert the radius from centimeters to angstroms and meters. 1 cm = \(10^{8}\) Å (angstroms) and 1 cm = 0.01 m (meters).
02

Calculate the number of Rhodium atoms needed to span 6.0 µm

We are given the distance, d, to be spanned as 6.0 μm (micrometers). First, convert the distance from μm to cm. 1 μm = \(10^{-4}\) cm d (in cm) = 6.0 μm * \(10^{-4}\) Now, divide the distance by the diameter of a single rhodium atom to find the number of atoms needed to span the distance: Number of atoms = Distance / Diameter
03

Calculate the volume of a single Rhodium atom

We are given the assumption that rhodium atom is a sphere. The volume formula for a sphere is: Volume = \(\frac{4}{3}\pi r^{3}\) Using the radius in meters, calculate the volume of a single rhodium atom in m³.

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