Problem 74 - Chemistry and Chemical Reactivity

Question

A ln an experiment, you need 0.125 mol of sodium metal. Sodium can be cut easily with a knife (Figure 2.12 ), so if you cut out a block of sodium, what should the volume of the block be in cubic centimeters? If you cut a perfect cube, what is the length of the edge of the cube? (The density of sodium is \(0.97 \mathrm{g} / \mathrm{cm}^{3} .\) )

Step-by-Step Solution

Answer

Volume is approximately 2.963 cm³; edge length is about 1.43 cm.

Step 1: Determine Mass from Moles

To find the mass of sodium needed, we use the formula:\[ \text{mass} = \text{moles} \times \text{molar mass} \]Given that the molar mass of sodium (Na) is approximately 22.99 g/mol, we calculate the mass required for 0.125 mol:\[ \text{mass} = 0.125 \times 22.99 = 2.87375 \text{ g} \]

Step 2: Calculate the Volume

Using the density formula \( \text{density} = \frac{\text{mass}}{\text{volume}} \), we rearrange to find the volume:\[ \text{volume} = \frac{\text{mass}}{\text{density}} \]Substitute the known values (2.87375 g for mass and 0.97 g/cm³ for density):\[ \text{volume} = \frac{2.87375}{0.97} \approx 2.963 \text{ cm}^3 \]

Step 3: Determine Edge Length of the Cube

To find the edge length of a cube with a given volume, use the formula for the volume of a cube \( V = s^3 \) where \(s\) is the edge length. Solve for \(s\):\[ s = \sqrt[3]{V} = \sqrt[3]{2.963} \approx 1.43 \text{ cm} \]

Key Concepts

Molar Mass

Molar mass plays a crucial role in chemical calculations, helping you convert between grams and moles. It is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol).

• To find the molar mass, you often refer to the periodic table. Each element's atomic mass is approximately equal to its molar mass.
• For sodium (\[ ext{Na}\]), this value is about 22.99 g/mol. This means that one mole of sodium atoms weighs 22.99 grams.

By knowing the molar mass, you can easily calculate the mass required for a specific mole quantity, as shown in the exercise. When you need 0.125 mol of sodium, you multiply the molar mass by the number of moles:\[ ext{mass} = 0.125 imes 22.99 = 2.87375 ext{ g}\] This result tells you that 0.125 moles of sodium weighs approximately 2.874 grams, which is essential for experiments requiring precise amounts.

Volume Calculation

Volume calculation is essential when dealing with solid materials, especially when converting from mass using density. Density is defined as mass per unit volume and can be rearranged to find the volume.

• To calculate volume, use the formula: \[ \text{volume} = \frac{\text{mass}}{\text{density}} \]
• This formula allows you to determine how much space an object will occupy based on its mass and density.

In the context of the exercise, once you calculate the mass of sodium as 2.874 grams, you can use the given density of sodium (\[0.97\text{ g/cm}^3\]) to find the volume: \[ \text{volume} = \frac{2.87375}{0.97} \approx 2.963 \text{ cm}^3 \] This calculation tells you that this mass of sodium occupies around 2.963 cubic centimeters.

Edge Length of a Cube

When you want to cut a material into a perfect cube, knowing the cube's volume allows you to determine the length of each edge.

• The formula for the volume of a cube is \[ V = s^3 \], where \(s\) is the edge length.
• To find the edge length (\(s\)), simply take the cube root of the volume.

Using the calculated volume from the previous section, we find the edge length this way:\[ s = \sqrt[3]{2.963} \approx 1.43 \text{ cm} \] This calculation reveals that to achieve the volume of a perfect cube, each side of the sodium block must be about 1.43 cm long. Knowing this helps in evenly cutting the sodium for an accurate experimental setup.