Question

What is the density (g/mL) of each of the following samples? a. A 20.0-mL sample of a salt solution that has a mass of \(24.0 \mathrm{~g}\). b. A solid object with a mass of \(1.65 \mathrm{lb}\) and a volume of \(170 \mathrm{~mL} .\) c. A gem has a mass of \(45.0 \mathrm{~g}\). When the gem is placed in a graduated cylinder containing \(20.0 \mathrm{~mL}\) of water, the water level rises to \(34.5 \mathrm{~mL}\). d. A lightweight head on the driver of a golf club is made of titanium. If the volume of a sample of titanium is \(114 \mathrm{~cm}^{3}\) and the mass is \(514.1 \mathrm{~g}\), what is the density of titanium?

Short answer

a. 1.2 g/mL b. 4.4 g/mL c. 3.1 g/mL d. 4.51 g/cm³

Step-by-step solution

Define the Formula for Density

Density is defined as mass divided by volume. The formula is: \[ \text{Density} = \frac{\text{mass}}{\text{volume}} \]

Calculate Density for Sample A

For sample a, the mass is 24.0 grams and the volume is 20.0 milliliters. Use the formula: \[ \text{Density} = \frac{24.0 \mathrm{~g}}{20.0 \mathrm{~mL}} = 1.2 \mathrm{~g / mL} \]

Convert Mass to grams for Sample B

Sample b has a mass in pounds, so convert it to grams first. 1 pound is equal to 453.592 grams, so \[ 1.65 \mathrm{~lb} \times 453.592 \mathrm{~g/lb} = 748.43 \mathrm{~g} \]

Calculate Density for Sample B

Use the converted mass and given volume to find density of sample b: \[ \text{Density} = \frac{748.43 \mathrm{~g}}{170 \mathrm{~mL}} = 4.4 \mathrm{~g / mL} \]

Determine Volume for Sample C

For the gem in sample c, the volume is equal to the change in water level in the graduated cylinder: \[ \text{Volume} = 34.5 \mathrm{~mL} - 20.0 \mathrm{~mL} = 14.5 \mathrm{~mL} \]

Calculate Density for Sample C

Use the mass and the determined volume to find the density of the gem: \[ \text{Density} = \frac{45.0 \mathrm{~g}}{14.5 \mathrm{~mL}} = 3.1 \mathrm{~g / mL} \]

Calculate Density for Sample D

For titanium in sample d, use the given mass and volume to determine the density: \[ \text{Density} = \frac{514.1 \mathrm{~g}}{114 \mathrm{~cm}^3} = 4.51 \mathrm{~g / cm}^3 \]

Key concepts

mass

Mass is a measure of the amount of matter in an object. In the context of density calculations, it's essential to use the correct units for mass. Typically, mass is measured in grams (g) or kilograms (kg). For example, if you have a 24.0-gram salt solution, the mass is represented as 24.0 g. Make sure to convert any given mass into the required units before performing density calculations. An example of this is when you need to convert pounds (lb) to grams (g), knowing that 1 pound equals 453.592 grams.

volume

Volume is the amount of space an object occupies. The common units for volume in density calculations are milliliters (mL) or cubic centimeters (cm³). It's crucial to measure the exact volume of the sample. For instance, if a gem's presence raises the water level in a graduated cylinder from 20.0 mL to 34.5 mL, the volume of the gem is the difference: 34.5 mL - 20.0 mL = 14.5 mL. Always ensure the volume is accurately measured for precise density calculations.

density formula

The density formula is a simple yet powerful tool in science. It is given by: \ \( \text{Density} = \frac{\text{mass}}{\text{volume}} \) \ It tells us how much mass is contained in a unit volume. For example, if you have 24.0 grams of a salt solution in a 20.0 mL container, the density is calculated as follows: \ \( \text{Density} = \frac{24.0 \mathrm{~g}}{20.0 \mathrm{~mL}} = 1.2 \mathrm{~g/mL} \). \ Understanding and using this formula is essential for solving various real-world problems requiring density calculations.

unit conversion

Unit conversion is an integral part of solving density problems because it ensures consistency in measurements. For instance, mass should be in grams and volume in milliliters or cubic centimeters for the density formula to work correctly. If mass is given in pounds, convert it to grams using: \ \(1 \text{ lb} = 453.592 \text{ grams} \). \ Suppose you have a mass of 1.65 lb. Converting to grams: \ \( 1.65 \text{ lb} \times 453.592 \text{ g/lb} = 748.43 \text{ g} \). \ Correct unit conversion is critical for accurate density calculations.

problem-solving steps

To solve any density calculation problem, follow these key steps: \ 1. Determine the mass and volume of the sample. Ensure the units are consistent. \ 2. Convert any given units to the required units if necessary (e.g., pounds to grams). \ 3. Use the formula: \( \text{Density} = \frac{\text{mass}}{\text{volume}} \). Plug in the values for mass and volume. \ 4. Calculate the density. \ For example, for a sample with a mass of 45.0 g and a volume of 14.5 mL: \ \( \text{Density} = \frac{45.0 \mathrm{~g}}{14.5 \mathrm{~mL}} = 3.1 \mathrm{~g/mL} \). \ By following these steps, you ensure accurate and thorough problem-solving.