Calculate the fraction of empty space in cubic closest packing to five significant figures.

검증된 단계별 안내

Understand that cubic closest packing (CCP) is also known as face-centered cubic (FCC) packing.

In FCC, each unit cell contains 4 atoms. The atoms are located at each corner and the centers of each face of the cube.

Calculate the volume of the unit cell. If 'a' is the edge length of the cube, the volume is given by \( V_{cell} = a^3 \).

Determine the volume occupied by the atoms in the unit cell. The effective radius of each atom is \( r \), and the relationship between 'a' and 'r' in FCC is \( a = 2\sqrt{2}r \). The volume occupied by the atoms is \( 4 \times \frac{4}{3}\pi r^3 \).

Calculate the fraction of empty space by subtracting the volume occupied by the atoms from the total volume of the unit cell, and then divide by the total volume of the unit cell: \( \text{Fraction of empty space} = 1 - \frac{\text{Volume occupied by atoms}}{V_{cell}} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cubic Closest Packing (CCP)

Cubic Closest Packing (CCP) is a method of packing spheres in three-dimensional space to maximize density. In this arrangement, spheres are stacked in layers, with each layer fitting into the gaps of the layer below. This structure achieves a packing efficiency of approximately 74%, meaning that 74% of the volume is occupied by spheres, while the remaining 26% is empty space.

Packing Efficiency

Packing efficiency is a measure of how effectively a given volume is filled with particles, expressed as a percentage. In the case of cubic closest packing, the packing efficiency is calculated by dividing the volume occupied by the spheres by the total volume of the packing arrangement. Understanding packing efficiency is crucial for determining the fraction of empty space in a given packing structure.

Fraction of Empty Space

The fraction of empty space in a packing arrangement is calculated by subtracting the packing efficiency from 1 (or 100% if expressed as a percentage). For cubic closest packing, with a packing efficiency of 74%, the fraction of empty space is 1 - 0.74 = 0.26. This concept is essential for understanding how much of the volume in a packed structure is not occupied by the particles.