Question

For the masses and volumes indicated, calculate the density in grams per cubic centimeter. a. mass \(=234 \mathrm{~g} ;\) volume \(=2.2 \mathrm{~cm}^{3}\) b. mass \(=2.34 \mathrm{~kg} ;\) volume \(=2.2 \mathrm{~m}^{3}\) c. mass \(=1.2\) lb ; volume \(=2.1 \mathrm{ft}^{3}\) d. mass \(=4.3\) ton ; volume \(=54.2 \mathrm{yd}^{3}\)

Short answer

a. Density ≈ \(106.36 \mathrm{~g/cm}^3\) b. Density ≈ \(0.001064 \mathrm{~g/cm}^3\) c. Density ≈ \(0.009153 \mathrm{~g/cm}^3\) d. Density ≈ \(0.094116 \mathrm{~g/cm}^3\)

Step-by-step solution

1. Convert mass and volume to g and cm³.

In this case, the mass and volume are already given in grams and cubic centimeters. No conversion necessary.

2. Calculate the density:

Now we can plug in the mass and volume into the density formula: Density = \(\frac{234 \mathrm{~g}}{2.2 \mathrm{~cm}^3}\) Density ≈ \(106.36 \mathrm{~g/cm}^3\) Object b: mass=2.34 kg and volume=2.2 m³

1. Convert mass and volume to g and cm³.

To convert kilograms to grams, multiply by 1000: \(2.34 \mathrm{~kg} \times 1000 = 2340 \mathrm{~g}\) To convert cubic meters to cubic centimeters, multiply by \(10^6\): \(2.2 \mathrm{~m}^3 \times 10^6 = 2.2 \times 10^6 \mathrm{~cm}^3\)

2. Calculate the density:

Now we will plug the mass and volume into the density formula: Density = \(\frac{2340 \mathrm{~g}}{2.2 \times 10^6 \mathrm{~cm}^3}\) Density ≈ \(0.001064 \mathrm{~g/cm}^3\) Object c: mass=1.2 lb and volume=2.1 ft³

1. Convert mass and volume to g and cm³.

To convert pounds to grams, multiply by 453.592: \(1.2 \mathrm{~lb} \times 453.592 = 544.3104 \mathrm{~g}\) To convert cubic feet to cubic centimeters, multiply by 28316.8466: \(2.1 \mathrm{~ft}^3 \times 28316.8466 = 59463.37846 \mathrm{~cm}^3\)

2. Calculate the density:

Now we will plug the mass and volume into the density formula: Density = \(\frac{544.3104 \mathrm{~g}}{59463.37846 \mathrm{~cm}^3}\) Density ≈ \(0.009153 \mathrm{~g/cm}^3\) Object d: mass=4.3 ton and volume=54.2 yd³

1. Convert mass and volume to g and cm³.

To convert tons to grams, multiply by 907184.74: \(4.3 \mathrm{~ton} \times 907184.74 = 3898890.378 \mathrm{~g}\) To convert cubic yards to cubic centimeters, multiply by 764554.857: \(54.2 \mathrm{~yd}^3 \times 764554.857 = 41432681.466 \mathrm{~cm}^3\)

2. Calculate the density:

Now we will plug the mass and volume into the density formula: Density = \(\frac{3898890.378 \mathrm{~g}}{41432681.466 \mathrm{~cm}^3}\) Density ≈ \(0.094116 \mathrm{~g/cm}^3\) Final answers: a. Density ≈ \(106.36 \mathrm{~g/cm}^3\) b. Density ≈ \(0.001064 \mathrm{~g/cm}^3\) c. Density ≈ \(0.009153 \mathrm{~g/cm}^3\) d. Density ≈ \(0.094116 \mathrm{~g/cm}^3\)

Key concepts

Unit Conversion

When working with density calculations, it's crucial to convert all measurements to a consistent unit system, typically grams (g) for mass and cubic centimeters (cm³) for volume. This is because density is most commonly expressed in units of grams per cubic centimeter (g/cm³). Understanding how to convert between different units is an essential skill in chemistry. Here are some common conversions needed:

• For mass, if you start with kilograms (kg), you multiply by 1,000 to convert to grams, because 1 kg equals 1,000 g.
• For volume, converting from cubic meters (m³) to cubic centimeters (cm³) involves multiplying by 1,000,000, as 1 m³ equals 1,000,000 cm³.
• Pounds (lb) must be converted into grams by multiplying the number of pounds by 453.592.
• Cubic feet (ft³) need to be converted to cubic centimeters by multiplying by 28,316.8466.
• Tons can be converted to grams by multiplying by 907,184.74, and cubic yards (yd³) to cubic centimeters by multiplying by 764,554.857.

By ensuring all measurements are in the correct units, density calculations become straightforward and accurate.

Density Formula

Density is an essential property in physics and chemistry that relates the mass of a substance to how much space it occupies. The formula to calculate density is simple: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]This equation states that density is equal to the mass divided by the volume. It's crucial to note that mass should be measured in grams and volume in cubic centimeters to ensure that the resulting density is in the standard unit of g/cm³.Using this fundamental formula, one can determine how compact or spread out the matter within an object or substance is. Higher density means the particles of the material are closely packed, while lower density indicates they are more spread out.By substituting the correctly converted mass and volume into the formula, students and professionals alike can solve density problems with ease. Remembering to always check units first ensures that the density calculation is both meaningful and accurate.

Mass and Volume

Mass and volume are the two critical components needed to calculate density. Mass is the amount of matter in an object and is measured in grams or kilograms, while volume is the space an object occupies, measured in cubic centimeters or meters. To successfully calculate density:

• First, accurately measure or determine the mass of the object using a scale.
• Next, measure the volume, which can be done by calculation or by displacement method if the object is irregularly shaped.

Understanding these basic measurements and their conversions allows for a precise calculation of an object's density. When performing these measurements:

• Always ensure that tools are calibrated correctly for precise readings.
• Consider the effects of any environmental factors, such as temperature, that might influence measurements.

Having accurate data for mass and volume is the foundation for any reliable density calculation, allowing for deeper insight into the physical properties of materials.