Question

Calculate the density in grams per milliliter for each of the following. (a) \(25.5-\mathrm{g}\) solid whose volume is found by displacement to be \(4.5 \mathrm{~mL}\) (b) 95.5-g rectangular solid measuring \(3.55 \mathrm{~cm} \times 2.50 \mathrm{~cm} \times 1.75 \mathrm{~cm}\)

Short answer

(a) 5.67 g/mL; (b) 6.15 g/mL.

Step-by-step solution

Understand the formula for density

Density (2) is calculated using the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Here, the mass is measured in grams (g) and the volume in milliliters (mL), leading to a density in grams per milliliter (g/mL).

Calculate density for part (a)

For part (a), we have a mass of \( 25.5 \, \text{g} \) and a volume of \( 4.5 \, \text{mL} \). Using the density formula: \( \text{Density} = \frac{25.5}{4.5} \approx 5.67 \, \text{g/mL} \).

Determine volume for part (b)

First, we need to find the volume of the rectangular solid in part (b). The volume \( V \) is calculated using the formula for volume of a rectangle: \( V = \text{length} \times \text{width} \times \text{height} \). Substituting the values: \( V = 3.55 \, \text{cm} \times 2.50 \, \text{cm} \times 1.75 \, \text{cm} \approx 15.53125 \, \text{cm}^3 \).

Calculate density for part (b)

For part (b), we have a mass of \( 95.5 \, \text{g} \) and a calculated volume of \( 15.53125 \, \text{cm}^3 \) (which is the same numerically as mL for water-based conversions). Using the density formula: \( \text{Density} = \frac{95.5}{15.53125} \approx 6.15 \, \text{g/mL} \).

Key concepts

Mass and Volume

To understand how density is calculated, it's important to first grasp the concepts of mass and volume. Mass refers to the amount of matter in an object and is measured in grams (g) when working with chemistry problems. It is essentially how much stuff is packed into the object.

On the other hand, volume is a measure of the space that an object occupies. In this exercise, volume is given in milliliters (mL), which is a common unit for measuring liquid volumes. When dealing with solid objects, volume can be determined via methods like displacement or straightforward calculations for shaped objects like boxes.

• Mass: Amount of matter, measured in grams.
• Volume: Space occupied, often measured in milliliters or cubic centimeters.

Knowing these basics is crucial because density is derived from both these properties.

Density Formula

The density of a substance is a measure of how much mass is contained in a given volume. It's expressed using the formula: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]This means that the density will tell you how tightly packed the mass is, within the volume. Given in units of grams per milliliter (g/mL), density can tell us something fundamental about the nature of the substance. A high density means a large amount of mass is packed into a small volume, while a low density would mean the mass is more spread out.

When applying this formula, ensure that both mass and volume are in compatible units. For instance, mass should be in grams and volume in milliliters (or cubic centimeters, which are numerically the same as mL for solid objects). This will ensure that your density calculations are accurate and meaningful.

Rectangular Solid Volume

For calculating the density of objects with a clear geometric shape, like a block or cube, understanding how to find volume is key. For a rectangular solid, the volume is calculated by multiplying its length, width, and height. In mathematical terms, this is shown as:\[V = \text{length} \times \text{width} \times \text{height}\]In the exercise, the dimensions of the rectangular solid must first be multiplied to determine the volume, which in this case yields cubic centimeters (cm³) or milliliters (mL) since they represent the same volume units for water-related measurements.

• Step 1: Measure each dimension: length, width, and height.
• Step 2: Plug these values into the formula and solve for volume.
• Step 3: Use this volume in the density formula.

Using these calculated volumes, you can easily find the density of the solid object by following the subsequent steps in the density calculation process.