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Chapter 1: Problem 59

Calculate the mass of each of the following: (a) a sphere of gold with a radius of \(10.0 \mathrm{~cm}\) [ the volume of a sphere with a radius \(r\) is \(V=(4 / 3) \pi r^{3} ;\) the density of gold \(\left.=19.3 \mathrm{~g} / \mathrm{cm}^{3}\right],\) (b) a cube of platinum of edge length \(0.040 \mathrm{~mm}\) (the density of platinum \(=\) \(\left.21.4 \mathrm{~g} / \mathrm{cm}^{3}\right),\) (c) \(50.0 \mathrm{~mL}\) of ethanol (the density of cthanol \(=0.798 \mathrm{~g} / \mathrm{mL}\) ).

Short Answer

Expert verified
The mass of the gold sphere is approximately 80385 g. The mass of the platinum cube is approximately 0.014 g. The mass of the 50.0 mL ethanol is approximately 39.9 g.

Step by step solution

01

Calculate the volume of the gold sphere

The volume formula for a sphere is \(V=(4 / 3) \pi r^{3}\). Substitute \(10.0 cm\) for \(r\), and calculate the volume.
02

Calculate the mass of the gold sphere

Density equals mass divided by volume. The mass can therefore be calculated by rearranging the equation to be mass = density * volume. Substitute the volume calculated in Step 1 and the given density of gold, \(19.3 g/cm^{3}\), into the equation to find the mass.
03

Calculate the volume of the platinum cube

The volume of a cube is calculated by the formula \(V = a^{3}\), where \(a\) is the length of an edge of the cube. Substitute \(.040 mm\) for \(a\), noting that you need to convert from \(mm\) to \(cm\) first, as 1 \(mm\) = .1 \(cm\).
04

Calculate the mass of the platinum cube

Again, use the rearranged density formula mass = density * volume to find the mass. Substitute the volume calculated in Step 3 and the given density of platinum, \(21.4 g/cm^{3}\), into the equation to find the mass.
05

Calculate the mass of the 50.0 mL of ethanol

Because the volume of the ethanol is given in \(mL\) and the density in \(g/mL\), you can directly apply the rearranged density formula mass = density * volume to find the mass. Substitute \(50.0 mL\) for the volume and the given density of ethanol, \(0.798 g/mL\), into the equation to find the mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Density
Density is a crucial concept in physics and chemistry, describing how much mass is contained in a given volume. It helps us understand how heavy an object is for its size.
The formula for density is:
  • Density = Mass / Volume
This relationship implies that if you know two of these quantities (e.g., mass and volume), you can calculate the third.
Understanding density is like understanding how tightly packed matter is inside an object. For instance, gold and platinum are known to have high densities, meaning they have a lot of mass in a small volume. This property makes them heavy and compact.
Volume of a Sphere
A sphere is a perfectly symmetrical three-dimensional shape, like a basketball or a perfect soap bubble. To calculate its volume, you can use the formula:
  • \(V = \frac{4}{3} \pi r^3\)
Here, \(r\) is the radius of the sphere. It's the distance from the center of the sphere to any point on its surface.
For example, if you have a gold sphere with a radius of 10 cm, plug that radius into the formula to find its volume. Calculating the volume is the first step to finding out how much space the gold occupies.
Volume of a Cube
A cube is a three-dimensional shape with six equal square faces, like a dice. Its volume can be determined using the formula:
  • \(V = a^3\)
Here, \(a\) represents the side length of the cube.
For instance, if you have a tiny platinum cube with a side length of 0.040 mm, you need to convert that length to centimeters before using the formula. Convert by remembering that 1 mm equals 0.1 cm.
Calculating the volume this way allows us to understand how much space the material occupies in cubic centimeters.
Unit Conversion Basics
Unit conversion is essential when calculating density, volume, or mass. It's the process of converting one unit of measure into another, ensuring consistency and accuracy in your calculations.
For example, when dealing with a cube measured in millimeters, and you need the volume in cubic centimeters, you have to convert using:
  • 1 mm = 0.1 cm
Conversions ensure that your equations make sense and match the units you are working with.
Always check the units presented in the problem and convert them to the desired unit of calculation to avoid any confusion.
Density Formula Application
The density formula, expressed as Density = Mass / Volume, is an essential tool in science for calculating one of these quantities if the other two are known.
  • Rearranged, it becomes Mass = Density \(\times\) Volume
To calculate the mass of an object, multiply its density by its volume.
For example, if you need to find the mass of ethanol given its volume and density, you directly substitute the given values into this equation. This straightforward approach helps solve many real-world problems involving solids, liquids, and gases.
Knowing how to manipulate the density formula enables you to understand and predict the behavior of different substances in diverse conditions.

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Most popular questions from this chapter

One gallon of gasoline in an automobile's engine produces on the average \(9.5 \mathrm{~kg}\) of carbon dioxide, which is a greenhouse gas, that is, it promotes the warming of Earth's atmosphere. Calculate the annual production of carbon dioxide in kilograms if there are 250 million cars in the United States and each car covers a distance of \(5000 \mathrm{mi}\) at a consumption rate of 20 miles per gallon.

The radius of a copper (Cu) atom is roughly \(1.3 \times\) \(10^{-10} \mathrm{~m} .\) How many times can you divide evenly a piece of \(10-\mathrm{cm}\) copper wire until it is reduced to two separate copper atoms? (Assume there are appropriate tools for this procedure and that copper atoms are lined up in a straight line, in contact with each other. Round off your answer to an integer.)

A \(250-\mathrm{mL}\) glass bottle was filled with \(242 \mathrm{~mL}\) of water at \(20^{\circ} \mathrm{C}\) and tightly capped. It was then left outdoors overnight, where the average temperature was \(-5^{\circ} \mathrm{C}\). Predict what would happen. The density of water at \(20^{\circ} \mathrm{C}\) is \(0.998 \mathrm{~g} / \mathrm{cm}^{3}\) and that of ice at \(-5^{\circ} \mathrm{C}\) is $0.916 \mathrm{~g} / \mathrm{cm}^{3}.

The density of methanol, a colorless organic liquid used as solvent, is \(0.7918 \mathrm{~g} / \mathrm{mL}\). Calculate the mass of \(89.9 \mathrm{~mL}\) of the liquid.

Classify each of the following statements as a hypothesis, a law, or a theory. (a) Beethoven's contribution to music would have been much greater if he had married. (b) An autumn leaf gravitates toward the ground because there is an attractive force between the leaf and Earth. (c) All matter is composed of very small particles called atoms.
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