If you were to double the volume of a substance, what would happen to the mass?
Learning Objectives
By the end of this section, y'all will be able to:
- Define density.
- Calculate the mass of a reservoir from its density.
- Compare and dissimilarity the densities of various substances.
Which weighs more, a ton of feathers or a ton of bricks? This onetime riddle plays with the distinction between mass and density. A ton is a ton, of course; but bricks take much greater density than feathers, and and then we are tempted to think of them as heavier. (See Figure 1.)
Figure 1. A ton of feathers and a ton of bricks have the same mass, but the feathers brand a much bigger pile because they take a much lower density.
Density, as you will see, is an of import characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid. Density is the mass per unit volume of a substance or object. In equation grade, density is defined as
[latex]\rho =\frac{m}{V}\\[/latex],
where the Greek letter ρ (rho) is the symbol for density, thousand is the mass, and V is the volume occupied past the substance.
Density
Density is mass per unit volume.
[latex]\rho =\frac{m}{5}\\[/latex],
where ρ is the symbol for density, 1000 is the mass, and V is the volume occupied by the substance.
In the riddle regarding the feathers and bricks, the masses are the same, but the book occupied by the feathers is much greater, since their density is much lower. The SI unit of density is kg/thou3, representative values are given in Table i. The metric organization was originally devised so that water would have a density of 1 g/cm3, equivalent to 103 kg/miii. Thus the basic mass unit, the kilogram, was first devised to be the mass of thou mL of water, which has a volume of 1000 cmthree.
| Substance | [latex]\rho \left({\text{10}}^{iii}{\text{kg/m}}^{3}\text{or}\text{g/mL}\right)\\[/latex] | Substance | [latex]\rho \left({\text{ten}}^{3}{\text{kg/m}}^{three}\text{or}\text{g/mL}\right)\\[/latex] | Substance | [latex]\rho \left({\text{10}}^{3}{\text{kg/thou}}^{iii}\text{or}\text{g/mL}\right)\\[/latex] |
|---|---|---|---|---|---|
| Solids | Liquids | Gases | |||
| Aluminum | 2.seven | Water (4ºC) | 1.000 | Air | i.29 × 10−3 |
| Brass | 8.44 | Blood | 1.05 | Carbon dioxide | one.98 × 10−3 |
| Copper (average) | 8.8 | Sea h2o | i.025 | Carbon monoxide | 1.25 × 10−3 |
| Gold | 19.32 | Mercury | thirteen.vi | Hydrogen | 0.090 × ten−iii |
| Atomic number 26 or steel | 7.8 | Ethyl booze | 0.79 | Helium | 0.xviii × 10−3 |
| Lead | 11.3 | Petrol | 0.68 | Methane | 0.72 × 10−three |
| Polystyrene | 0.10 | Glycerin | 1.26 | Nitrogen | 1.25 × x−iii |
| Tungsten | 19.thirty | Olive oil | 0.92 | Nitrous oxide | 1.98 × 10−3 |
| Uranium | 18.lxx | Oxygen | ane.43 × ten−three | ||
| Concrete | two.30–3.0 | Steam (100º C) | 0.60 × 10−3 | ||
| Cork | 0.24 | ||||
| Glass, common (average) | 2.vi | ||||
| Granite | 2.seven | ||||
| Earth's chaff | 3.3 | ||||
| Wood | 0.3–0.9 | ||||
| Water ice (0°C) | 0.917 | ||||
| Bone | one.7–2.0 |
As you tin run across by examining Table one, the density of an object may help identify its composition. The density of gilt, for instance, is near 2.5 times the density of fe, which is well-nigh 2.5 times the density of aluminum. Density as well reveals something about the stage of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty infinite.
Take-Home Experiment Saccharide and Salt
A pile of carbohydrate and a pile of salt look pretty similar, just which weighs more? If the volumes of both piles are the same, whatsoever difference in mass is due to their different densities (including the air space between crystals). Which practice you recollect has the greater density? What values did you notice? What method did you use to determine these values?
Example one. Calculating the Mass of a Reservoir From Its Book
A reservoir has a surface surface area of fifty.0 km2 and an average depth of 40.0 1000. What mass of water is held behind the dam? (See Effigy 2 for a view of a large reservoir—the Three Gorges Dam site on the Yangtze River in central China.)
Strategy
Nosotros tin calculate the volume 5 of the reservoir from its dimensions, and notice the density of water ρ in Tabular array 1. Then the mass m tin be found from the definition of density
[latex]\rho =\frac{chiliad}{5}\\[/latex].
Solution
Solving equation ρ=1000/5for m gives yard = ρV. The volume V of the reservoir is its surface expanse A times its boilerplate depth h:
[latex]\begin{array}{lll}V& =& {Ah}=\left(\text{l.0}{\text{km}}^{two}\right)\left(\text{twoscore.0}\text{m}\right)\\ & =& \left[\left(\text{50.0 thou}{\text{m}}^{2}\right){\left(\frac{{\text{ten}}^{3}\text{m}}{ane\text{km}}\right)}^{two}\correct]\left(\text{40.0 m}\correct)=2\text{.}\text{00}\times {\text{10}}^{9}{\text{m}}^{3}\end{array}\\[/latex]
The density of water ρ from Tabular array 1 is one.000 × 103. Substituting V and ρ into the expression for mass gives
[latex]\begin{array}{lll}m& =& \left(1\text{.}\text{00}\times {\text{10}}^{iii}{\text{ kg/thousand}}^{3}\right)\left(2\text{.}\text{00}\times {\text{10}}^{9}{\text{chiliad}}^{3}\right)\\ & =& 2.00\times {\text{ten}}^{\text{12}}\text{ kg}\terminate{array}\\[/latex].
Word
A large reservoir contains a very large mass of h2o. In this example, the weight of the water in the reservoir is mg = i.96 × 1013 North, where g is the acceleration due to the Globe's gravity (nigh nine.fourscore thousand/s2). It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As nosotros shall see in the following sections, the force the dam must supply can be much smaller than the weight of the h2o it holds back.
Figure 2. Three Gorges Dam in central China. When completed in 2008, this became the world's largest hydroelectric establish, generating ability equivalent to that generated by 22 average-sized nuclear power plants. The concrete dam is 181 m loftier and 2.iii km across. The reservoir made past this dam is 660 km long. Over ane million people were displaced past the creation of the reservoir. (credit: Le G Portage)
Section Summary
- Density is the mass per unit book of a substance or object. In equation form, density is defined equally
[latex]\rho =\frac{m}{V}\\[/latex].
- The SI unit of density is kg/grand3.
Conceptual Questions
1. Approximately how does the density of air vary with altitude?
2. Give an instance in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than than ane material?
three. Effigy 3 shows a glass of ice h2o filled to the brim. Will the water overflow when the ice melts? Explain your respond.
Figure iii.
Problems & Exercises
1. Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gilt?
ii. Mercury is commonly supplied in flasks containing 34.five kg (about 76 lb). What is the volume in liters of this much mercury?
3. (a) What is the mass of a deep breath of air having a volume of 2.00 Fifty? (b) Discuss the effect taking such a jiff has on your body'south volume and density.
4, A straightforward method of finding the density of an object is to measure out its mass and then measure out its volume by submerging it in a graduated cylinder. What is the density of a 240-m rock that displaces 89.0 cmiii of water? (Annotation that the accurateness and practical applications of this technique are more than limited than a diversity of others that are based on Archimedes' principle.)
5. Suppose you lot take a coffee mug with a circular cross department and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density every bit water.
6. (a) A rectangular gasoline tank tin can agree l.0 kg of gasoline when total. What is the depth of the tank if information technology is 0.500-one thousand wide past 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.
7. A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, past what factor is the density of the rubbish increased?
8. A two.50-kg steel gasoline tin holds 20.0 50 of gasoline when full. What is the average density of the full gas can, taking into account the book occupied by steel every bit well as past gasoline?
ix. What is the density of 18.0-karat gold that is a mixture of xviii parts gilded, 5 parts silverish, and 1 part copper? (These values are parts by mass, non volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.
x. In that location is relatively trivial empty infinite between atoms in solids and liquids, and then that the boilerplate density of an cantlet is about the same as matter on a macroscopic calibration—approximately teniii kg/one thousandiii. The nucleus of an atom has a radius about 10-5 that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can take the density of a nucleus. What would exist the radius of a neutron star with a mass ten times that of our Lord's day (the radius of the Sunday is 7 × 108)?
Glossary
- density:
- the mass per unit volume of a substance or object
Selected Solutions to Problems & Exercises
1. 1.610 cmiii
3. (a) two.58 g (b) The volume of your trunk increases by the volume of air you inhale. The boilerplate density of your torso decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body earlier you took the deep breath.
iv. 2.70 thousand/cm3
6. (a) 0.163 k (b) Equivalent to 19.iv gallons, which is reasonable
8. 7.9 × 102 kg/m3
9. 15.6 thou/cmiii
10. (a) 1018 kg/gthree (b) ii × 10four m
Source: https://courses.lumenlearning.com/physics/chapter/11-2-density/
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