Question

Tungsten crystallizes in a body-centered cubic unit cell with an edge length of \(317 \mathrm{pm}\). What is the length in picometers of a unitcell diagonal that passes through the center atom?

Short answer

The unit cell diagonal length is approximately 549 pm.

Step-by-step solution

Understand Unit Cell Geometry

A body-centered cubic (BCC) unit cell has atoms at each corner and one atom at the center of the cube. The diagonal that passes through the center atom stretches across the body of the cube.

Determine the Diagonal Formula

In a cube, the body diagonal can be calculated using the Pythagorean theorem in three dimensions. For a cube with edge length \(a\), the formula for the body diagonal \(d\) is: \[d = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3}\].

Apply Edge Length to Diagonal Formula

Substitute the given edge length \(a = 317 \text{ pm}\) into the formula for the body diagonal: \[d = 317 \sqrt{3} \text{ pm}\].

Calculate the Diagonal Length

Use the value \(\sqrt{3} \approx 1.732\) to find the length of the diagonal: \[d = 317 \times 1.732 \approx 548.684 \text{ pm}\].

Round to Appropriate Significant Figures

Since the edge length has three significant figures (317 pm), round the diagonal length to three significant figures: \(d \approx 549 \text{ pm}\).

Key concepts

Unit Cell Geometry

Understanding the structure of a body-centered cubic (BCC) unit cell is essential to solving problems related to it. A BCC unit cell is a type of crystal structure where each cube has atoms located at every one of its eight corners and a single atom at the center of the cube. This central atom helps distinguish the BCC structure from other cubic arrangements, such as face-centered cubic (FCC) structures, where additional atoms are found on the faces of the cube.

In a BCC unit cell, the atoms at the corners and the center result in one complete atom per unit cell. This differs from a simple cubic structure, which only has corner atoms. Keep in mind that these geometric distinctions result in unique calculations when determining factors like volume, density, and diagonal lengths of unit cells.

To fully appreciate BCC structures, we need not only visualize these points within the cube but also be familiar with how atoms interact across the diagonal plane, extending through the center atom—this is crucial for understanding further calculations.

Pythagorean Theorem in 3D

The Pythagorean Theorem, typically encountered in two-dimensional problems, extends into three dimensions, particularly in the context of geometry involving cubes. In a 3D space, the theorem is used to calculate the length of the diagonal along the body of a cube, an essential component of determining various physical properties of crystal lattices.

For a cube with edge length \(a\), the formula for determining the body diagonal (which passes through the central atom in BCC structures) is derived from the Pythagorean Theorem:

• The traditional 2D formula \(a^2 + b^2 = c^2\) adjusts to a 3D formula.
• This 3D extension becomes \(d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}\).

This formula highlights how the cube's three-dimensional nature expands our typical understanding of geometry.

By understanding how to apply the Pythagorean Theorem in three dimensions, students can calculate and comprehend physical properties such as internal atomic distances in crystals.

Significant Figures in Calculation

Significant figures play a crucial role in maintaining precision and accuracy in scientific calculations. When calculating the body diagonal in a BCC unit cell, it is vital to consider significant figures correctly to ensure that the accuracy of measurements and calculations reflects the precision of the initial data.

In our specific problem, the edge length of the unit cell is given as 317 pm, which consists of three significant figures. Because the length itself determines the precision of the calculation, all subsequent results should be rounded to match this level of significance. Thus, the diagonal length, initially calculated as approximately 548.684 pm, must be rounded to three significant figures, resulting in 549 pm.

• Always match the number of significant figures to that of the least precise measurement.
• Significant figures ensure that reported results are neither artificially precise nor vague.

Adhering to proper usage of significant figures helps produce results in scientific exercises that are both meaningful and contextually accurate.