(b) Because atoms are spherical, they cannot occupy all of the space of the cube. The silver atoms pack in the solid in such a way that 74% of the volume of the solid is actually filled with the silver atoms. Calculate the volume of a single silver atom.

Verified step by step guidance

First, understand that the problem involves calculating the volume of a single silver atom given that 74% of the volume of the solid is filled with silver atoms. This percentage is known as the packing efficiency.

Next, recognize that the packing efficiency is the ratio of the volume occupied by the atoms to the total volume of the solid. In this case, the packing efficiency is 0.74 (or 74%).

Assume the solid is a cube with a side length 'a'. The total volume of the cube is then given by the formula: \( V_{cube} = a^3 \).

The volume occupied by the silver atoms is 74% of the total volume of the cube. Therefore, the volume occupied by the atoms is: \( V_{atoms} = 0.74 \times a^3 \).

To find the volume of a single silver atom, you need to know the number of atoms in the cube. If the crystal structure is known (e.g., face-centered cubic), you can calculate the number of atoms per unit cell and then divide the total volume occupied by the atoms by this number to find the volume of a single atom.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Atomic Packing Efficiency

Atomic packing efficiency refers to the fraction of volume in a crystal structure that is occupied by atoms. In the case of silver, which has a face-centered cubic (FCC) structure, the packing efficiency is approximately 74%. This means that while the atoms are spherical, they do not fill the entire volume of the unit cell, leaving some space unoccupied.

Volume of a Unit Cell

The volume of a unit cell is the basic repeating unit in a crystal lattice. For a face-centered cubic structure, the volume can be calculated using the edge length of the cube. Understanding the relationship between the unit cell volume and the number of atoms it contains is crucial for determining the volume occupied by individual atoms within the solid.

Calculating Volume of an Atom

To calculate the volume of a single atom, one can use the formula for the volume of a sphere, V = (4/3)πr³, where r is the atomic radius. By knowing the packing efficiency and the volume of the unit cell, one can derive the volume of a single atom by dividing the total volume occupied by the atoms in the unit cell by the number of atoms present.

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