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Relative density

Relative density, sometimes called specific density, is the ratio of the density of a substance to the density of a given reference material. If a substance's relative density is less than one then it is less dense than the reference; if greater than one then it is denser than the reference. If the relative density is exactly one then the densities are equal; that is, equal volumes of the two substances have the same mass.

Relative density is a generalisation of, or in some usages synonymous with, specific gravity (which specifically means relative density with respect to water), with the former term often preferred in modern scientific usage.

In symbols,

RD = frac{rho_mathrm{substance}}{rho_mathrm{reference}},

where RD is relative density, rho_mathrm{substance}, is the density of the substance being measured, and rho_mathrm{reference}, is the density of the reference. (By convention ρ, the Greek letter rho, denotes density.)

The reference material can be indicated using subscripts:

RD_mathrm{substance/reference} , which means "the relative density of substance with respect to reference". If the reference is not explicitly stated then it is normally assumed to be water at 4 °C (or, more precisely, 3.98 °C, which is the temperature at which water reaches its maximum density). In SI units, the density of water is (approximately) 1000 kg/ or 1 g/cm³, which makes relative density calculations particularly convenient: the density of the object only needs to be divided by 1000 or 1, depending on the units.

The relative density of gases is often measured with respect to dry air at a temperature of 20 degrees Celsius and a pressure of 101.325 kPa absolute, which has a density of 1.205 kg/m3.

Relative density with respect to air can be obtained by

mbox{SG} = frac{rho_mathrm{gas}}{rho_{mathrm{air}}} = frac{M_mathrm{gas}}{M_{mathrm{air}}}

Where {M} is the molar mass.

Examples

Taking the relative density with respect to ethanol, the relative densities of ethanol, water and iron are as follows:

  • Ethanol: 1.0 (by definition)
  • Water: 1.2 (i.e. water is 1.2 times as dense as ethanol)
  • Iron: 10.0 (i.e. iron is 10.0 times as dense as ethanol)

Taking the relative density relative to water, the numbers are 0.78, 1.0, and 7.9 respectively. With respect to iron, the numbers are 0.1, 0.12, and 1.0 respectively

Dimensions

Since it is a ratio of two quantities of the same type, relative density is dimensionless (it has no units). Although relative density does not depend on the unit system being used, the two densities must as necessary be converted to the same units (e.g., kg/ or g/cm³) before calculating the numerical value of the ratio.

For example, suppose an object has a density of 4 g/cm3. To calculate its relative density with respect to water, which has a density of 1 g/cm3, we divide the former by the latter:

RD = frac{4 mathrm{g/cm^3}}{1 mathrm{g/cm^3}} = 4

If the densities are instead measured in kg/m3 then the calculation becomes

RD = frac{4000 mathrm{kg/m^3}}{1000 mathrm{kg/m^3}} = 4

which gives exactly the same answer.

Relative density can also be calculated as the ratio of gravitational densities rather than "ordinary" mass-based densities. Defining gravitational density as ρg, where ρ is ordinary density and g is the local gravitational constant, it is seen that the gravitational constant simply cancels out when the ratio is computed.

Temperature dependence

See Density for a table of the measured densities of water at various temperatures.

Changes in temperature affect the densities of different materials differently, and thus alter their relative densities. For precision work the temperatures of the two materials may be explicitly stated; for example:

relative density: 8.15_{4^circ mathrm{C}}^{20^circ mathrm{C}} , or specific gravity: 2.432_0^{15}
where the superscript indicates the temperature at which the density of the material is measured, and the subscript indicates the temperature of the reference substance to which it is compared.

Uses

Relative density can help quantify the buoyancy of a substance in a fluid, or determine the density of an unknown substance from the known density of another.

Relative density is often used by geologists and mineralogists to help determine the mineral content of a rock or other sample. Gemologists use it as an aid in the identification of gemstones. Water is preferred as the reference because measurements are then easy to carry out in the field (see below for examples of measurement methods).

The relative density of liquids has numerous practical uses, as described under hydrometer.

Relative density may be a more intelligible quantity than density itself, especially to the layman. For example, the density of iridium may be stated as 22650 kg/m³, but this is not a number that can easily be grasped unless one is already familiar with the numerical densities of various materials. However, if iridium it is said to be nearly twice as dense as lead, or over 22 times as dense as water, then it is easy to understand how heavy iridium really is.

Measurement

Relative density can be calculated directly by measuring the density of a sample and dividing it by the (known) density of the reference substance. The density of the sample is simply its mass divided by its volume. Although mass is easy to measure, the volume of an irregularly shaped sample can be more difficult to ascertain. One method is to put the sample in a water-filled graduated cylinder and read off how much water it displaces. Alternatively the container can be filled to the brim, the sample immersed, and the volume of overflow measured. The surface tension of the water may keep a significant amount of water from overflowing, which is especially problematic for small samples. For this reason it is desirable to use a water container with as small a mouth as possible.

The fact that relative density is a unitless ratio often simplifies calculations. For example, suppose a certain rock sample deflects a spring by 3 inches, and a sample of the reference substance deflects the spring by 5 inches. Furthermore, the rock sample causes the water in a certain graduated cylinder rise by 20 mm, and the reference substance causes it to rise by 34 mm. The relative density between these two objects can easily be determined without having to figure out several constants that would be needed to determine the density directly (such as the spring constant or the cross sectional area of the cylinder).

For each substance, the density, ρ, is given by

rho = frac{Mass}{Volume}
= frac{Deflection times Spring Constant/Gravity}{Displacement_mathrm{Water Line} times Area_mathrm{Cylinder}},

When these densities are divided, references to the spring constant, gravity and cross-sectional area simply cancel, leaving

RD=frac{rho_mathrm{object}}{rho_mathrm{ref}} = frac{frac{Deflection_mathrm{Obj.}}{Displacement_mathrm{Obj.}}}{frac{Deflection_mathrm{Ref.}}{Displacement_mathrm{Ref.}}} = frac{frac{3 mathrm{in}}{20 mathrm{mm}}}{frac{5 mathrm{in}}{34 mathrm{mm}}}=frac{3 mathrm{in} times 34 mathrm{mm}}{5 mathrm{in} times 20 mathrm{mm}} = 1.02,

Relative density is more easily and perhaps more accurately measured without measuring volume. Using a spring scale, the sample is weighed first in air and then in water. Relative density (with respect to water) can then be calculated using the following formula:

RD = frac{W_mathrm{air}}{W_mathrm{air} - W_mathrm{water}},

where

Wair is the weight of the sample in air (measured in pounds-force, newtons, or some other unit of force)
Wwater is the weight of the sample in water (measured in the same units).

This technique cannot easily be used to measure relative densities less than one, because the sample will then float. Wwater becomes a negative quantity, representing the force needed to keep the sample underwater.

Another practical method uses three measurements. The sample is weighed dry. Then a container filled to the brim with water is weighed, and weighed again with the sample immersed, after the displaced water has overflowed and been removed. Subtracting the last reading from the sum of the first two readings gives the weight of the displaced water. The relative density result is the dry sample weight divided by that of the displaced water. This method works with scales that can't easily accommodate a suspended sample, and also allows for measurement of samples that are less dense than water.

Relative density and hydrometers

The relative density of a liquid can be measured using a hydrometer. This consists of a bulb attached to a stalk of constant cross-sectional area, as shown in the diagram to the right.

First the hydrometer is floated in the reference liquid (shown in light blue), and the displacement (the level of the liquid on the stalk) is marked (blue line). The reference could be any liquid, but in practice it is usually water.

The hydrometer is then floated in a liquid of unknown density (shown in green). The change in displacement, Δx, is noted. In the example depicted, the hydrometer has dropped slightly in the green liquid; hence its density is lower than that of the reference liquid. It is, of course, necessary that the hydrometer floats in both liquids.

The application of simple physical principles allows the relative density of the unknown liquid to be calculated from the change in displacement. (In practice the stalk of the hydrometer is pre-marked with graduations to facilitate this measurement.)

In the explanation that follows,

ρref is the known density (mass per unit volume) of the reference liquid (typically water).
ρnew is the unknown density of the new (green) liquid.
RDnew/ref is the relative density of the new liquid with respect to the reference.
V is the volume of reference liquid displaced, i.e. the red volume in the diagram.
m is the mass of the entire hydrometer.
g is the local gravitational constant.
Δx is the change in displacement. In accordance with the way in which hydrometers are usually graduated, Δx is here taken to be negative if the displacement line rises on the stalk of the hydrometer, and positive if it falls. In the example depicted, Δx is negative.
A is the cross sectional area of the shaft.

Since the floating hydrometer is in static equilibrium, the downward gravitational force acting upon it must exactly balance the upward buoyancy force. The gravitational force acting on the hydrometer is simply its weight, mg. From the Archimedes buoyancy principle, the buoyancy force acting on the hydrometer is equal to the weight of liquid displaced. This weight is equal to the mass of liquid displaced multiplied by g, which in the case of the reference liquid is ρrefVg. Setting these equal, we have

mg = rho_mathrm{ref}Vg,

or just

m = rho_mathrm{ref} V, (1)

Exactly the same equation applies when the hydrometer is floating in the liquid being measured, except that the new volume is V − AΔx (see note above about the sign of Δx). Thus,

m = rho_mathrm{new} (V - A Delta x), (2)

Combining (1) and (2) yields

RD_{mathrm{new/ref}} = frac{rho_mathrm{new}}{rho_mathrm{ref}} = frac{V}{V - A Delta x} (3)

But from (1) we have V = m/ρref. Substituting into (3) gives

RD_{mathrm{new/ref}} = frac{1}{1 - frac{A Delta x}{m} rho_mathrm{ref}} (4)

This equation allows the relative density to be calculated from the change in displacement, the known density of the reference liquid, and the known properties of the hydrometer. If & Delta;x is small then, as a first-order approximation of the geometric series equation (4) can be written as:

RD_mathrm{new/ref} approx 1 + frac{A Delta x}{m} rho_mathrm{ref}

This shows that, for small Δx, changes in displacement are approximately proportional to changes in relative density.

See also

References

  • Munson, B.R.; D.F. Young, T.H. Okishi (2001). Fundamentals of Fluid Mechanics. 4th Edition, Wiley.
  • Fox, R.W.; McDonald, A.T. (2003). Introduction to Fluid Mechanics. 4th Edition, Wiley.

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