Use the density values in Table 2.8 to solve each of the...

Use the density values in Table 2.8 to solve each of the following problems: a. What is the mass, in grams, of a cube of copper that has a volume of $74.1 \mathrm{~cm}^{3} ?$ b. How many kilograms of gasoline fill a 12.0 -gal gas tank? c. What is the volume, in cubic centimeters, of an ice cube that has a mass of $27 \mathrm{~g}$ ?

Use the density values in Table 2.8 to solve each of the following problems:
a. What is the mass, in grams, of a cube of copper that has a volume of $74.1 \mathrm{~cm}^{3} ?$
b. How many kilograms of gasoline fill a 12.0 -gal gas tank?
c. What is the volume, in cubic centimeters, of an ice cube that has a mass of $27 \mathrm{~g}$ ?

Key Concepts

Density

Density is a fundamental property of matter defined as mass per unit volume. It describes how much mass is contained in a given volume and is key in linking the concepts of mass and volume through the formula density = mass/volume.

Mass-Volume Relationship

The mass-volume relationship is derived from the concept of density, allowing one to calculate mass when volume and density are known, or volume when mass and density are provided. This relationship is essential in solving problems that require interconversion between mass and volume.

Unit Conversion

Unit conversion is the process of changing a quantity expressed in one set of units to another, using conversion factors. This is crucial in density-related problems where measurements in different unit systems must be reconciled, such as converting between gallons and cubic centimeters or grams and kilograms.

Dimensional Analysis

Dimensional analysis is a method used to check the consistency of units in calculations. By ensuring that the units appropriately cancel out or combine, it verifies that the mathematical manipulations using density, mass, and volume yield results in the desired units.

Transcript

00:01 Okay, for this question, we are using the table 2 .8 to solve the problem.

00:06 So the first one is, asks us for the mass.

00:11 If we're given a cube of copper that has a volume of 74 .1 cubic centimeter, just a reminder our density formula is mass over volume.

00:23 So for a, we have a volume of 74 .1 cubic centimeter, which is, equal to 74 .1 milliliter.

00:38 Cubic centimeters and milliliters are one to one.

00:44 If we look at table 2 .8, the density of copper is 8 .92 grams per milliliter.

01:00 Please be reminded, when you look at table 2 .8, there are two different units on that table.

01:06 So whenever you're selecting yours, make sure that you use the correct units.

01:11 So we want to know what the mass is.

01:15 So if we take our formula, density, equals mass over volume, we can solve for that, and we get mass is equal to volume times density.

01:27 We can just plug in our numbers here, and we get a volume of 74 .1 millimeters times our density, 8 .92 grams per millimeter.

01:46 Those milliliters are going to cancel.

01:49 And so when you solve for that, you'll take that 74 .1, and you'll multiply it by 8 .92.

01:56 And you will get a mass of 660.

02:02 Actually, 661.

02:08 We're going to keep to that three significant figures.

02:14 So now if we look at b, we're given a gasoline, 12 -gallon gas tank.

02:22 And we want to know how many kilograms of gasoline we can put in it.

02:28 So we are given a volume of 12 .0 gallons.

02:36 Now we're going to use kilograms here, so we're going to need to move that to not move.

02:52 We will need to convert that to liters.

02:55 So in this case, we'll take 12 .0 gallons times liters, gallons.

03:06 If you look at that conversion, you have a conversion of 3 .79 liters per gallon.

03:21 And again, you're just going to multiply, which will give us 45 .48 liters.

03:38 Now, we are also given our density of gasoline, which can be found in that table.

03:51 The density of gasoline is 0 .74 grams per milliliter.

04:01 Grams per milliliter.

04:03 Now, we're looking for kilograms per liter.

04:08 Now, we know that there are 1 ,000 grams and a kilogram, and there are 1 ,000 grams...

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