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

The densities for each scenario are: a. \(106.36 \frac{g}{cm^3}\) b. \(0.0010636 \frac{g}{cm^3}\) c. \(0.0091465 \frac{g}{cm^3}\) d. \(0.0939766 \frac{g}{cm^3}\)

Step-by-step solution

Problem a - Calculating density (grams per cubic centimeter)

In this part, mass and volume are already in appropriate units. We just need to apply the density formula. Density = Mass/Volume Density = \( \frac{234g}{2.2 cm^3} \) Density = \( \frac{234}{2.2} = 106.36 \frac{g}{cm^3}\)

Problem b - Converting units and calculating density

First, we need to convert the mass from kilograms to grams and the volume from cubic meters to cubic centimeters. 1 kg = 1000 g 1 m³ = 1000000 cm³ Mass = 2.34 kg * 1000 = 2340 g Volume = 2.2 m³ * 1000000 = 2200000 cm³ Now we can apply the density formula: Density = Mass/Volume Density = \( \frac{2340g}{2200000 cm^3} \) Density = \( \frac{2340}{2200000} = 0.0010636 \frac{g}{cm^3}\)

Problem c - Converting units and calculating density

First, we need to convert mass from pounds to grams and volume from cubic feet to cubic centimeters. 1 lb = 453.592 g 1 ft³ = 28316.8 cm³ Mass = 1.2 lb * 453.592 = 544.3104 g Volume = 2.1 ft³ * 28316.8 = 59495.28 cm³ Now we can apply the density formula: Density = Mass/Volume Density = \( \frac{544.3104g}{59495.28 cm^3} \) Density = \( \frac{544.3104}{59495.28} = 0.0091465 \frac{g}{cm^3}\)

Problem d - Converting units and calculating density

First, we need to convert the mass from tons to grams and the volume from cubic yards to cubic centimeters. 1 ton = 907185 g 1 yd³ = 764554.9 cm³ Mass = 4.3 tons * 907185 = 3895849.5 g Volume = 54.2 yd³ * 764554.9 = 41458470.78 cm³ Now we can apply the density formula: Density = Mass/Volume Density = \( \frac{3895849.5g}{41458470.78 cm^3} \) Density = \( \frac{3895849.5}{41458470.78} = 0.0939766 \frac{g}{cm^3}\) Now we have calculated the densities for all four scenarios in grams per cubic centimeter.

Key concepts

Unit Conversion

Unit conversion is essential when calculating density, especially when the initial units of mass or volume are not in the desired units of grams or cubic centimeters. When converting units, it's helpful to use conversion factors, which are ratios that express how many of one unit equals another unit. Doing so allows you to express quantities in the same measurement system, ensuring calculations are accurate.
For example:

• To convert kilograms to grams, multiply by 1000 (since 1 kg = 1000 g).
• To convert cubic meters to cubic centimeters, multiply by 1,000,000 (since 1 m³ = 1,000,000 cm³).
• For pounds to grams, use the factor 453.592 (as 1 lb = 453.592 g).
• For cubic feet to cubic centimeters, multiply by 28,316.8 (since 1 ft³ = 28,316.8 cm³).
• Finally, to convert tons to grams and cubic yards to cubic centimeters, use 907,185 g and 764,554.9 cm³, respectively.

Measurement Units

Measurement units are crucial in all scientific calculations as they provide a standard way to quantify and communicate quantities like mass and volume. There are various measurement systems in use worldwide, such as the Metric system (meters, grams) and the Imperial system (pounds, feet).
When working with these units, it's important to choose consistent measurement systems to ease calculations and avoid errors. The Metric system is often preferred in scientific calculations for its decimal base, which simplifies unit conversion.

Grams per Cubic Centimeter

Grams per cubic centimeter

• is a common unit for density, describing how much mass is present per unit volume.
• This unit tells us how tightly matter is packed in a given space.
• The more grams in a cubic centimeter, the denser the object.

In density calculations, converting all measurements to grams per cubic centimeter ensures a standard form that allows easy comparison across different materials. This standardization is key to understanding an object's density in a clear and uniform manner.

Mass and Volume

Mass and volume are two fundamental properties that describe the amount of matter in an object and the space it occupies, respectively. In density calculations:

• Mass is typically measured in grams or kilograms and indicates how much matter an object contains.
• Volume, on the other hand, measures the space occupied by an object and is generally expressed in cubic centimeters, cubic meters, etc.

Understanding the mass and volume of an object is crucial as they serve as the foundation for calculating density. Together, they help us understand how concentrated material is within a specific space.

Density Formula

The density formula is a straightforward way to calculate how dense an object is. The formula is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This formula illustrates the ratio of an object's mass to its volume, providing insight into its compactness. Using the density formula, we can derive values in units of grams per cubic centimeter, allowing us to understand and compare how different materials distribute their mass within a given volume. By consistently applying this formula, one can accurately ascertain the density of various substances with ease.