Question

The compound sodium chloride has a solid-state structure in which there is a repeating cubic arrangement of sodium ions and chloride ions. The volume of the cube is \(1.81 \times 10^{-22} \mathrm{~cm}^{3} .\) Calculate the length of an edge of the cube in \(\mathrm{pm}\left(1 \mathrm{pm}=1 \times 10^{-12} \mathrm{~m}\right)\).

Short answer

The edge of the cube is approximately 564 pm.

Step-by-step solution

Understand the Problem

The problem asks us to find the length of an edge of a cube given its volume. We are given the volume of the cube as \(1.81 \times 10^{-22} \mathrm{~cm}^3\).

Recall the Formula for Volume of a Cube

The formula for the volume of a cube is given by \( V = a^3 \), where \( V \) is the volume and \( a \) is the length of an edge of the cube.

Express the Equation for Edge Length

Since \( V = a^3 \), the length of an edge \( a \) can be calculated by taking the cube root of the volume: \( a = \sqrt[3]{V} \).

Insert Given Volume and Calculate Edge Length in cm

We insert the given volume into the equation to find the edge length: \[ a = \sqrt[3]{1.81 \times 10^{-22}} \mathrm{~cm}^3 \]Use a calculator to find the cube root.

Calculate the Edge Length

After calculating, \( a \approx 5.64 \times 10^{-8} \mathrm{~cm}\).

Convert the Edge Length to pm

To convert cm to pm, remember that \(1 \mathrm{~cm} = 10^{10} \mathrm{~pm}\). Therefore, multiply the edge length by \(10^{10}\) to convert:\[ a \approx 5.64 \times 10^{-8} \times 10^{10} \mathrm{~pm} \approx 564 \mathrm{~pm} \]

Conclusion

The edge length of the cube is calculated to be approximately \(564 \mathrm{~pm}\).

Key concepts

Solid State Structure

The sodium chloride structure is fascinating in the realm of solid-state chemistry. It forms a cubic crystalline structure where sodium ions (\( \text{Na}^+ \)) and chloride ions (\( \text{Cl}^- \)) are arranged in an alternating pattern.
This pattern yields a highly symmetrical and stable arrangement.

Here's what you need to know about this structure:

• The cube is a repeating unit called a "unit cell," which is fundamental in understanding the entire solid.
• Each corner and the center of each face of the cube hosts a chloride ion, while sodium ions occupy the other available spaces.
• This arrangement maximizes ionic interactions, giving sodium chloride its characteristic properties, such as high melting point and solubility in water.

Solid-state structures like that of sodium chloride are vital for predicting material behaviors and properties.

Cube Edge Calculation

When given the volume of a cube, calculating the edge length requires a simple formula: \( V = a^3 \) , where \( a \) is the edge length.
To find \( a \), you simply take the cube root of the volume: \( a = \sqrt[3]{V} \).

For example, with the volume of \(1.81 \times 10^{-22} \text{ cm}^3\):

• Use a calculator to determine \( a \approx \sqrt[3]{1.81 \times 10^{-22}} \text{ cm} \).
• This results in \( a \approx 5.64 \times 10^{-8} \text{ cm} \).

Understanding this calculation allows you to grasp how the dimensions of microscopic structures are quantified in physical space.

Unit Conversion in Chemistry

Converting units is a staple in chemistry, especially when working with measurements at different scales. In this exercise, we converted centimeters to picometers.
Remember, \(1 \text{ cm} = 10^{10} \text{ pm}\).

Here's how you perform the conversion:

• Once you have the edge length in centimeters (\(5.64 \times 10^{-8} \text{ cm} \)), multiply by \(10^{10}\)
• This yields \( a \approx 564 \text{ pm} \).

Unit conversions like this enable students to toggle between varying scales, essential for understanding measurements in chemistry and physics.