Most of us have long understood that "oil is lighter than water", or that iron is "heavier" than sugar. But in making such statements, we are implicitly comparing equal volumes of these substances: after all, we know that a cup of sugar will weigh more than a single ordinary steel nail.
Mass and volume, as we learned in the previous unit, are measures of the quantity of a substance, and as such are defined as extensive properties of matter. You will recall that the ratio of two extensive properties is always an intensive property — one that characterizes a particular kind of matter, independently of its size or mass. It is this ratio, (mass ÷ volume), that we are concerned with in this lesson.
The only difference between these plots is their slopes. Denoting mass and volume by m and V respectively, we can write the equation of each line as m = ρV, where the slope ρ (Greek lower-case rho) is the proportionality constant that relates mass to volume. This quantity ρ is known as the density, which is usually defined as the mass per unit volume: ρ = m/V.
The volume units milliliter (mL) and cubic centimeter (cm3) are almost identical and are commonly used interchangeably.
The general meaning of density is the amount of anything per unit volume. What we conventionally call the "density" is more precisely known as the "mass density".
Density can be expressed in any combination of mass and volume units; the most commonly seen units are grams per mL (g mL–1, g cm–3), or kilograms per litre.
1 kg m–3 = 1 g L–1 = .0624 lb ft–3
It is sometimes more convenient to express the volume occupied by a unit mass of a substance. This is just the inverse of the density and is known as the specific volume.
A quantity that is very closely related to density, and which is frequently used in its place, is specific gravity.
Specific gravity is the ratio of the mass of a material to that of an equal volume of water. Because the density of water is about 1.00 g mL–1, the specific gravity is numerically very close to that of the density, but being a ratio, it is dimensionless.
The presence of "volume" in this definition introduces a slight complication: volumes are temperature-dependent owing to thermal expansion. At 4°C, water has its maximum density of almost exactly 1.000 g mL–1, so if the equivalent volume of water is assumed to be at this temperature, then the density and specific gravity can be considered numerically identical. In making actual comparisons, however, the temperatures of both the material being measured and of the equivalent volume of water are frequently different, so in order to specify a specific gravity value unambiguously, it is necessary to state the temperatures of both the substance in question and of the water.
Thus if we find that a given volume of a substance at 20°C weighs 1.11 times as much as the same volume of water measured at 4°C, we would express its specific gravity as
Although most chemists find density to be more convenient to work with and consider "specific gravity" to be rather old-fashioned, the latter quantity is still widely used in many industrial and technical fields ranging from winemaking to urinalysis.
The range of densities encountered in the world spans a remarkably wide range, from essentially zero in outer space to the unimaginably huge values found in stellar bodies. These very high densities represent the ultimate limits of how much mass can be packed into a given volume.
The following chart will give you some feeling for the values of density found in nature generally (top), in common solids (middle), and in gases and liquids (bottom). Please note that in order to depict reasonably wide ranges of values in limited space, the density scales are logarithmic; thus zero on these scales corresponds to the density of water (100 = 1 g cm–3). Densities listed for ordinary substances (including gases) are mostly those at around 20° C.
All substances tend to expand as they are heated, causing the same mass to occupy a greater volume, and thus lowering the density. For most solids, this expansion is relatively small, but it is far from negligible; for liquids, it is greater. The volumes of gases, as you may already know (see here for details), are highly temperature-sensitive, and so, of course, are their densities.
What is the cause of thermal expansion? As molecules acquire thermal energy, they move about more vigorously. In condensed phases (liquids and solids), this motion has the character of an irregular kind of bumping or jostling that causes the average distances between the molecules to increase, thus leading to increased volume and smaller density.
One might expect the densities of the chemical elements to increase uniformly with atomic weight, but this is not what happens; density depends on the volume as well as the mass, and the volume occupied by a given mass of an element, and these volumes can vary in a non-uniform way for two reasons:
The sizes (atomic radii) follow the zig-zag progression that characterizes the other periodic properties of the elements, with atomic volumes diminishing with increasing nuclear charge across each period (more here).
The plot is taken from the popular WebElements site.
Note that the volumes used to compute densities of the solid elements depend, in part, on the geometrical structure of these solids, which vary from element to element. Also, non-metallic solids are often composed of molecules (rather than individual atoms) that are more spread out in space, and which have shapes that cannot be arranged as compactly. so they tend to form more open crystal lattices than do the metals, and therefore have lower densities.
|Element||Solid density||Liquid density||m.p. (°C)|
For example, five solid elements, all semimetals, have lower densities in the solid state than in the molten state, and therefore will float in their own melts, just as ice does.
The 4° density maximum is one of many "anomalous" behaviors of water. As you may know, the H2O molecules in liquid and solid water are loosely joined together through a phenomenon known as hydrogen bonding. Any single water molecule can link up to four other H2O molecules, but this occurs only when the molecules are locked into place within an ice crystal. This is what leads to a relatively open lattice arrangement, and thus to the relatively low density of ice.
When ice melts, thermal energy begins to overcome the hydrogen-bonding forces so that each H2O molecule, instead of being permanently connected to four neighbors, is now only linked to an average of three other molecules through hydrogen bonds that continually break and re-form. With fewer hydrogen bonds, the geometrical requirements that formerly mandated a more open structural arrangement now diminish, so the entire network tends to collapse, rendering the water more dense. As the temperature rises, the fraction of H2O molecules that occupy ice-like clusters diminishes, contributing to the rise in density that is seen between 0° and 4°.
Whenever a continuously varying quantity such as density passes through a maximum or a minimum value as the temperature or some other variable is changing, you know that two opposing effects are at work.
The 4° density maximum of water corresponds to the temperature at which the breakup of ice-like clusters (leading to higher density) and thermal expansion (leading to lower density) achieve a balance.
What do an ice cube and a block of wood have in common? Throw either material into water, and it will float. Well, mostly; each object will have its bottom part immersed, but the upper part will ride high and dry. People often say that wood and ice float because they are "lighter than water", but this is true only if we compare the masses of equal volumes of the substances. In other words, we need to compare the masses-per-unit-volume, meaning the densities, of each material with that of water. So we would more properly say that objects capable of floating in water must have densities smaller than that of water.
Air is of course a fluid, and buoyancy can be a problem when weighing a large object such as an empty flask. The following problem illustrates a more extreme case:
When an object floats in a liquid, the portion of it that is immersed has a volume that depends on the mass of the same volume of displaced liquid.
An object (such as an iceberg) whose density is smaller than that of the water in which it is immersed, will float on the surface. This occurs because the volume V of the more dense fluid required to support the object is smaller than the volume of the object itself. Thus only that portion of the object having this same volume V need be submerged in order to counter the object's weight. This condition is known as positive buoyancy.
It is interesting to note that it takes much less force to lift a sunken object such as a ship up to just under the water's surface, than it does to lift itcompletely out of the water, at which point the buoyancy force that reduces its apparent weight is no longer present.
If the density of the object exceeds that of the fluid, the weight of fluid it displaces (the buoyant force) will be smaller than the object’s true weight, and thus unable to support it. In this condition of negative buoyancy, the object experiences a net downward force at any depth, so it will sink until it strikes the bottom.
If the density of an object is the same as that of the fluid, the upward and downward forces balance out to zero. In this condition of neutral buoyancy, the object "floats" within the fluid.
Most fish possess an organ known as a swim bladder whose degree of inflation enables them to switch buoyancy modes. These bladders are not connected to the mouth; they are inflated or deflated by oxygen gas that is extracted from (or returned to) the blood.
The most obvious way of finding the density of a material is to measure its mass and its volume. This is the only option we have for gases, but observing the mass of a fixed volume of a liquid is time-consuming and awkward, and measuring the volumes of solids whose shapes are irregular or which are finely divided is usually impractical.
The traditional hydrometer is a glass tube having a weighted bulb near the bottom. The hydrometer is lowered into a container of the liquid to be measured, and comes to a rest with the upper part protruding above the liquid surface at a height (read from a calibrated scale) that depends on the density of the liquid. This will only work, of course, if the overall density of the hydrometer itself is smaller than the density of the liquid to be measured. For this reason, hydrometers intended for general use come in sets. Because liquid densities are temperature dependent, hydrometers intended for precise measurements also contain an internal thermometer so that this information can be collected in the event that temperature corrections will be made.
Owing to the ease with which they can be observed, densities are widely employed to estimate the composition or quality of liquid mixtures or solutions, and in some cases determine their commercial value. This has given rise to many kinds of hydrometers that are specialized for specific uses:
Battery hydrometer - How it works
Aquarium salinity hydrometer
A boat with depth markings on its body can be thought of as a gigantic hydrometer!
Sugar and syrup hydrometer
Hydrometers for general purpose use are normally calibrated in units of specific gravity, but often defined at temperatures other than 25°C. A very common type of calibration is in "degrees" on various arbitrary scales, of which the best known are the Baumé scales. Special-purpose hydrometer scales can get quite esoteric; thus alcohol hydrometers may directly measure percentage alcohol by weight on a 0–100% scale, or "proof" (twice the volume-percent of alcohol) on a 0-200 scale.
Measuring the density of a solid that is large enough to weigh accurately is largely a matter of determining its volume. For an irregular solid such as a rock, this is most easily done by observing the amount of water it displaces.
A small containerl having a precisely determined volume can be used to determine the density of powdered or granular samples. The vessel (known as a pycnometer) is weighed while empty, and again when filled; the density is found from the weight difference and the calibrated volume of the pycnometer. This method is also applicable to liquids and gases.
In forensic work it is often necessary to determine the density of very small particles such as fibres, flakes of paint or metal, or grains of sand. Neither the weight nor volumes of such samples can be determined directly, so the simplest solution is to place the sample in a series of liquids of different densities, and see if it floats, sinks, or remains suspended within the liquid. A more sophisticated method is to layer two liquids in a vertical glass tube and allow them to slowly mix, creating a density gradient. When a particle is dropped into the tube, it sinks to a depth that matches its density.
The most famous application of buoyancy is due to Archimedes of Syracuse around 250 BCE. He was asked to determine whether the new crown that King Hiero II had commissioned contained all the gold that he had provided to the goldsmith for that purpose; apparently he suspected that the smith might have set aside some of the gold for himself and substituted less-valuable silver instead.
According to legend, Archimedes devised the principle of the “hydrostatic balance” after he noticed his own apparent loss in weight while sitting in his bath. The story goes that he was so enthused with his discovery that he jumped out of his bath and ran through the town, shouting "eureka" to the bemused populace.
One of the delights of chemical science is to find a way of using the macroscopic properties of bulk matter to uncover information about the microscopic world at the atomic level. The following problem example is a good illustration of this.
When the weather turns cold in the fall, the surface waters lose heat and cool to 4°C . This more dense layer of water sinks to the bottom, displacing the water below, which rises to the surface and restores nutrients that were removed when dead algae sank to the bottom. This “fall turnover” renews the lake for the next season.
The density of water containing dissolved salts is greater than that of pure water, and the density increases with the salt concentration (salinity). This seemingly trivial fact exerts a profound effect on our planet, driving the circulation in the oceans and affecting the climate and oceanic productivity.
The world's oceans consist of two huge reservoirs, one on top of the other. The upper ocean comprises about 5% of the total volume; owing to turbulence produced by wind and waves, its vertical profile is fairly uniform in composition and temperature. Its currents are driven mainly by surface winds which exert little effect below about 100 meters or so.
The remaining 95% of the ocean's volume resides in a deeper and colder layer. The temperature and density of this bottom ocean increase with depth, the coldest (4° C, and therefore the most dense) water being at the sea floor. So unlike the well-mixed upper ocean, the bottom one is stratified, similarly to that of lakes in cold winters.
Owing to this permanent stratification, mixing between the upper and lower oceans is very slow; only about 0.01% moves between the two layers per year. The mean residence time of a water molecule in the deep layer is about 1600 years. If it were not for cold polar winters, there would be very little movement of water in the bottom ocean.
But when seasonal ice begins to form in the polar regions,"holes" open up that allow water to drain from the upper ocean to the lower one. Because the salts dissolved in seawater cannot be accommodated within the ice structure, they are largely excluded from the new ice and remain in solution. This increases the salinity, and thus the density, of the surrounding unfrozen water, causing it to sink into the bottom ocean.
There are two major locations at which surface waters enter the deep ocean. The northern entry point is in the Norwegian Sea off Greenland; this water forms a mass known as the North Atlantic Deep Water (NADW) which flows southward across the equator.
Most of the transport into the deep ocean takes place in the Weddell Sea off the coast of Antarctica. The highly saline water flows down the submerged Antarctic Slope to begin a 5000-year trip to the north across the bottom of the ocean. This is the major route by which dissolved CO2 and O2 (which are more soluble in this cold water) are transported into the deep ocean where it forms a water mass known as the Antarctic Bottom Water (AABW) which can be traced into all three oceans.
As the cold water drains into the deep ocean. it spills over ridges and valleys, creating huge undersea cascades which rival the greatest terrestrial waterfalls in height and the largest rivers in volume. These deep water currents are much slower than the surface currents; estimated rates are of the order of kilometers per month, in contrast to the few kilometers per hour of surface waters.
As the cold water begins to fill up the bottom ocean, it displaces the less dense bottom water, pushing it back up into the upper ocean. This recirculation occurs to a small extent in many regions, but is especially pronounced along continental coasts where prevailing winds tend to roll surface waters over.
The deep ocean contains few organisms to deplete the water of the nutrients it receives from the remains of the dead organisms raining down from above; this upwelled water is therefor exceptionally rich in nutrients, and strongly encourages the growth of new organisms that extend up the food chain to fish. For example, the upwelling that occurs off the west coast of South America is responsible for the Peruvian fishing and guano fertilizer industry.
About every seven years the prevailing winds that help drive upwelling disappear for a while, allowing warm equatorial waters to move in. This phenomenon is known as El Niño and it results in massive kills of plankton and fish. Decomposition of the dead organisms reduces the oxygen content of the water, causing the death of still more fish, and allowing reduced compounds such as hydrogen sulfide to accumulate.
The grandest scale in which density and buoyancy reveal themselves lies in the structure of the earth and its continents.
The sun formed by gravitational attraction of the the massive amounts of interstellar atoms (mostly hydrogen and helium) that happened to be more concentrated in our part of the galaxy.
The remaining material probably formed a disk that rotated around the sun. As the temperature dropped to around 2000K, some of the more stable combinations of the elements began to condense out. These substances might have been calcium aluminum silicates, followed by the more volatile iron-nickel system, and then magnesium silicates. The further aggregation of these materials, together with the other constituents of the cooling disk, is now believed to be the origin of the planets. Density estimates indicate that the planets closest to the sun are predominantly rocky in nature, and probably condensed first. The outer planets (Uranus, Neptune and Pluto) appear to consist largely of water ice, methane, and ammonia, with a smaller rocky core.
We usually think of buoyancy as something that happens in liquids, but under sufficiently high pressures, the solid crust and mantle can deform and flow similarly to liquids. (Only the outer core and portions of the upper mantle are liquids.)
Half-dome peak, in California's Yosemite National Park, is one of the most well known and widely visited batholiths. [Wikimedia]
Pop-up mountains. In the earth's early stages the stronger granitic rocks had not yet appeared, and the crust was mechanically weak. Upwelling flows of lava would cause the crust to subside. In some places, magma would solidify underground, forming low-density rock (batholiths) that would eventually rise by buoyancy and push up overlying crust. These mountain-building periods probably occurred in 6-8 major episodes, each lasting about 800 million years.
Continental and oceanic crusts float and slide on the lower lithosphere.
The earth's crust together with the upper part of the mantle (the asthenosphere) are collectively known as the lithosphere. It is in this region that the dynamics of geological change take place: continental drift, mountain building, earthquakes and volcanoes.
The crust itself comes in two varieties having different compositions and densities. The continental crust is thicker and less dense (2.7 g cm–3) than the oceanic crust (3.0 g cm–3) and therefore floats higher on the plastic material of the upper mantle. Both kinds of crust can slide along on the mantle in response to convection currents. When these motions cause the two kinds of crust to collide, the less-dense oceanic crust invariably loses the battle and undergoes subduction and melting.
Known more generically as "fluid motion lamps", these devices became popular in the 1970's and provide a nice, if somewhat mesmerizing illustration of density and buoyancy in action as the blobs of oozing goo move up and down in ever-changing shapes.
These lamps consist of a container of water in which is placed a colored organic oily liquid that does not mix with water, thus constituting a second phase. The composition of the oil phase is such that its density is slightly greater than that of water at room temperature, so it normally resides at the bottom of the container. When the lamp is turned on, a heat source (usually an incandescent light bulb) concealed in the base of the container heats the oil phase. This reduces its density to a value below that of the water, causing blobs of oil to rise to the top of the container. Being now far removed from the heat source, the blobs cool down and sink back to the bottom, where they repeat the cycle.