View source for Relative density

{{Short description|Ratio of two densities}}
{{redirect-distinguish-for|Specific gravity|specific weight|the album by Joe McPhee and Joe Giardullo|Specific Gravity (album)}}
{{Infobox physical quantity
| name = Specific gravity
| unit = Unitless
| symbols = ''SG''
| derivations = <math> SG_\mathrm{true} = \frac {\rho_\mathrm{sample}}{\rho_\mathrm{H_2O}}</math>
}}

[[File:US Navy 111005-N-ZN781-031 Aviation Boatswain's Mate (Fuel) 3rd Class Rolando Calilung tests for a specific gravity test on JP-5 fuel.jpg|right|thumb|A [[United States Navy]] [[Aviation boatswain's mate]] tests the specific gravity of [[JP-5]] fuel]]

'''Relative density''', also called '''specific gravity''',<ref name=specificgravity1>{{cite book| last = Dana| first = Edward Salisbury
|title = A text-book of mineralogy: with an extended treatise on crystallography...
|publisher = John Wiley and Sons| date = 1922| location = New York, London(Chapman Hall)
|pages = 195–200, 316
|url = https://books.google.com/books?id=rCwaAAAAYAAJ&pg=PA156}}</ref><ref name=specificgravity2>{{cite book
|last = Schetz| first = Joseph A.|author2=Allen E. Fuhs
 |title = Fundamentals of fluid mechanics| publisher =Wiley, John & Sons, Incorporated
|date =1999-02-05|pages = 111, 142, 144, 147, 109, 155, 157, 160, 175
|url =https://books.google.com/books?id=YCSSolzuu9IC&pg=PP1|isbn =0-471-34856-2}}</ref> is a [[dimensionless quantity]] defined as the [[ratio]] of the [[density]] (mass of a unit volume) of a substance to the density of a given reference material. Specific gravity for solids and liquids is nearly always measured with respect to [[water]] at its densest (at {{cvt|4|°C|°F|1|disp=or}}); for gases, the reference is air at [[room temperature]] ({{cvt|20|°C|°F|disp=or}}). The term "relative density" (abbreviated '''r.d.''' or '''RD''') is preferred in [[SI]], whereas the term "specific gravity" is gradually being abandoned.<ref>{{cite book |author1=United States Bureau of Reclamation |title=Metric Manual |date=1978 |publisher=U.S. Department of the Interior, Bureau of Reclamation |url=https://books.google.com/books?id=E6BLn01jRz8C&pg=PA37|page=37 |language=en}}</ref>

If a substance's relative density is less than 1 then it is less dense than the reference; if greater than 1 then it is denser than the reference. If the relative density is exactly 1 then the densities are equal; that is, equal volumes of the two substances have the same mass.  If the reference material is water, then a substance with a relative density (or specific gravity) less than 1 will float in water. For example, an ice cube, with a relative density of about 0.91, will float. A substance with a relative density greater than 1 will sink.

Temperature and pressure must be specified for both the sample and the reference. Pressure is nearly always 1 [[Atmosphere (unit)|atm]] (101.325 [[Pascal (unit)|kPa]]). Where it is not, it is more usual to specify the density directly. Temperatures for both sample and reference vary from industry to industry. In British brewing practice, the specific gravity, as specified above, is multiplied by 1000.<ref name=briggs /> Specific gravity is commonly used in industry as a simple means of obtaining information about the concentration of solutions of various materials such as [[brine]]s, [[must weight]] ([[syrup]]s, juices, honeys, brewers [[wort]], [[must]], etc.) and acids.

==Basic calculation==
Relative density (<math>RD</math>) or specific gravity (<math>SG</math>) is a [[dimensionless quantity]], as it is the ratio of either densities or weights
<math display="block">\mathit{RD} = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}},</math>
where <math>RD</math> is relative density, <math>\rho_\mathrm{substance}</math> is the density of the substance being measured, and <math>\rho_\mathrm{reference}</math> is the density of the reference. (By convention <math>\rho</math>, the Greek letter [[rho]], denotes density.)

The reference material can be indicated using subscripts: <math>RD_\mathrm{substance/reference}</math> 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&nbsp;°[[Celsius|C]] (or, more precisely, 3.98&nbsp;°C, which is the temperature at which water reaches its maximum density). In [[International System of Units|SI]] units, the density of water is (approximately) 1000&nbsp;[[Kilogram|kg]]/[[Cubic metre|m<sup>3</sup>]] or 1&nbsp;[[Gram|g]]/[[Cubic centimetre|cm<sup>3</sup>]], 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&nbsp;°C and a pressure of 101.325 kPa absolute, which has a density of 1.205&nbsp;kg/m<sup>3</sup>. Relative density with respect to air can be obtained by
<math display="block">\mathit{RD} = \frac{\rho_\mathrm{gas}}{\rho_\mathrm{air}} \approx \frac{M_\mathrm{gas}}{M_\mathrm{air}},</math>
where <math>M</math> is the [[molar mass]] and the approximately equal sign is used because equality pertains only if 1 [[Mole (unit)|mol]] of the gas and 1&nbsp;mol of air occupy the same volume at a given temperature and pressure, i.e., they are both [[ideal gas]]es. Ideal behaviour is usually only seen at very low pressure. For example, one mol of an ideal gas occupies 22.414 L at 0&nbsp;°C and 1 atmosphere whereas [[carbon dioxide]] has a molar volume of 22.259 L under those same conditions.

Those with SG greater than 1 are denser than water and will, disregarding [[surface tension]] effects, sink in it. Those with an SG less than 1 are less dense than water and will float on it. In scientific work, the relationship of mass to volume is usually expressed directly in terms of the density (mass per unit volume) of the substance under study. It is in industry where specific gravity finds wide application, often for historical reasons.

True specific gravity of a liquid can be expressed mathematically as:
<math display="block"> SG_\mathrm{true} = \frac {\rho_\mathrm{sample}}{\rho_\mathrm{H_2O}},</math>
where <math>\rho_\mathrm{sample}</math> is the density of the sample and <math>\rho_\mathrm{H_2O}</math> is the density of water.

The apparent specific gravity is simply the ratio of the weights of equal volumes of sample and water in air:
<math display="block"> SG_\mathrm{apparent} = \frac {W_{\mathrm{A},\text{sample}}}{W_{\mathrm{A},\mathrm{H_2O}}},</math>
where <math>W_{A,\text{sample}}</math> represents the weight of the sample measured in air and <math>{W_{\mathrm{A},\mathrm{H_2O}}}</math> the weight of an equal volume of water measured in air.

It can be shown that true specific gravity can be computed from different properties:
<math display="block"> SG_\mathrm{true} = \frac {\rho_\mathrm{sample}}{\rho_\mathrm{H_2O}} = \frac {\frac{m_\mathrm{sample}}{V}}{\frac{m_\mathrm{H_2O}}{V}} = \frac {m_\mathrm{sample}}{m_\mathrm{H_2O}} \frac{g}{g} = \frac {W_{\mathrm{V},\text{sample}}}{W_{\mathrm{V},\mathrm{H_2O}}},</math>

where ''g'' is the local acceleration due to gravity, ''V'' is the volume of the sample and of water (the same for both), ''ρ''<sub>sample</sub> is the density of the sample, ''ρ''<sub>H<sub>2</sub>O</sub> is the density of water, ''W''<sub>V</sub> represents a weight obtained in vacuum, <math>\mathit{m}_\mathrm{sample}</math> is the mass of the sample and <math>\mathit{m}_\mathrm{H_2 O}</math> is the mass of an equal volume of water.

The density of water and of the sample varies with temperature and pressure, so it is necessary to specify the temperatures and pressures at which the densities or weights were determined. Measurements are nearly always made at 1 nominal atmosphere (101.325&nbsp;kPa ± variations from changing weather patterns), but as specific gravity usually refers to highly incompressible aqueous solutions or other incompressible substances (such as petroleum products), variations in density caused by pressure are usually neglected at least where apparent specific gravity is being measured. For true (''in vacuo'') specific gravity calculations, air pressure must be considered (see below). Temperatures are specified by the notation (''T''<sub>s</sub>/''T''<sub>r</sub>), with ''T''<sub>s</sub> representing the temperature at which the sample's density was determined and ''T''<sub>r</sub> the temperature at which the reference (water) density is specified. For example, SG (20&nbsp;°C/4&nbsp;°C) would be understood to mean that the density of the sample was determined at 20&nbsp;°C and of the water at 4&nbsp;°C. Taking into account different sample and reference temperatures, while ''SG''<sub>H<sub>2</sub>O</sub> = {{val|1.000000}} (20&nbsp;°C/20&nbsp;°C), it is also the case that ''SG''<sub>H<sub>2</sub>O</sub> = {{frac|{{val|0.9982008}}|{{val|0.9999720}}}} = {{val|0.9982288}} (20&nbsp;°C/4&nbsp;°C). Here, temperature is being specified using the current [[ITS-90]] scale and the densities <ref name="PTB-Mitteilungen" /> used here and in the rest of this article are based on that scale. On the previous IPTS-68 scale, the densities at 20&nbsp;°C and 4&nbsp;°C are {{val|0.9982041}} and {{val|0.9999720}} respectively,<ref name="„kell">{{cite journal |last1=Kell |first1=George S. |title=Density, Thermal Expansivity, and Compressibility of Liquid Water from 0 to 150°C: Correlations and Tables for Atmospheric Pressure and Saturation Reviewed and Expressed on 1968 Temperature Scale |journal=Journal of Chemical and Engineering Data |volume=20 |pages=97–105 |doi=10.1021/je60064a005 |url=https://doi.org/10.1021/je60064a005}}</ref> resulting in an SG (20&nbsp;°C/4&nbsp;°C) value for water of {{val|0.998232}}.

As the principal use of specific gravity measurements in industry is determination of the concentrations of substances in aqueous solutions and as these are found in tables of SG versus concentration, it is extremely important that the analyst enter the table with the correct form of specific gravity. For example, in the brewing industry, the [[Plato scale|Plato table]] lists sucrose concentration by weight against true SG, and was originally (20&nbsp;°C/4&nbsp;°C)<ref name="ReferenceA">ASBC Methods of Analysis Preface to Table 1: Extract in Wort and Beer, American Society of Brewing Chemists, St Paul, 2009</ref> i.e. based on measurements of the density of sucrose solutions made at laboratory temperature (20&nbsp;°C) but referenced to the density of water at 4&nbsp;°C which is very close to the temperature at which water has its maximum density, ''ρ''<sub>H<sub>2</sub>O</sub> equal to 999.972&nbsp;kg/m<sup>3</sup> in SI units ({{val|0.999972|u=g/cm<sup>3</sup>}} in [[CGS system|cgs units]] or 62.43&nbsp;lb/cu&nbsp;ft in [[United States customary units]]). The [[American Society of Brewing Chemists|ASBC]] table<ref name="ReferenceB">ASBC Methods of Analysis ''op. cit.'' Table 1: Extract in Wort and Beer</ref> in use today in North America for apparent specific gravity measurements at (20&nbsp;°C/20&nbsp;°C) is derived from the original Plato table using Plato et al.‘s value for SG(20&nbsp;°C/4&nbsp;°C) = {{val|0.9982343}}. In the sugar, soft drink, honey, fruit juice and related industries, sucrose concentration by weight is taken from a table prepared by [[Brix|A. Brix]], which uses SG (17.5&nbsp;°C/17.5&nbsp;°C). As a final example, the British SG units are based on reference and sample temperatures of 60&nbsp;°F and are thus (15.56&nbsp;°C/15.56&nbsp;°C).

Given the specific gravity of a substance, its actual density can be calculated by rearranging the above formula:
<math display="block">\rho_\mathrm{substance} = SG \times \rho_\mathrm{H_2O}.</math>

Occasionally a reference substance other than water is specified (for example, air), in which case specific gravity means density relative to that reference.

==Temperature dependence==
:''See [[Density]] for a table of the measured densities of water at various temperatures.''

The density of substances varies with temperature and pressure so that it is necessary to specify the temperatures and pressures at which the densities or masses were determined. It is nearly always the case that measurements are made at nominally 1 atmosphere (101.325&nbsp;kPa ignoring the variations caused by changing weather patterns) but as relative density usually refers to highly incompressible aqueous solutions or other incompressible substances (such as petroleum products) variations in density caused by pressure are usually neglected at least where apparent relative density is being measured. For true (''in vacuo'') relative density calculations air pressure must be considered (see below). Temperatures are specified by the notation (''T<sub>s</sub>''/''T<sub>r</sub>'') with ''T<sub>s</sub>'' representing the temperature at which the sample's density was determined and ''T<sub>r</sub>'' the temperature at which the reference (water) density is specified. For example, SG (20&nbsp;°C/4&nbsp;°C) would be understood to mean that the density of the sample was determined at 20&nbsp;°C and of the water at 4&nbsp;°C. Taking into account different sample and reference temperatures, while SG<sub>H<sub>2</sub>O</sub> = 1.000000 (20&nbsp;°C/20&nbsp;°C) it is also the case that RD<sub>H<sub>2</sub>O</sub> = {{sfrac|0.9982008|0.9999720}} = 0.9982288 (20&nbsp;°C/4&nbsp;°C). Here temperature is being specified using the current [[ITS-90]] scale and the densities<ref name="PTB-Mitteilungen">{{cite journal |last1=Bettin |first1=H. |last2=Spieweck |first2=F. |title=Die Dichte des Wassers als Funktion der Temperatur nach Einführung des Internationalen Temperaturskala von 1990 |journal=[[PTB-Mitteilungen]] |volume=100 |year=1990 |pages=195–196 |language=de}}</ref> used here and in the rest of this article are based on that scale. On the previous IPTS-68 scale the densities<ref name="„kell" /> at 20&nbsp;°C and 4&nbsp;°C are, respectively, 0.9982041 and 0.9999720 resulting in an RD (20&nbsp;°C/4&nbsp;°C) value for water of 0.99823205.

The temperatures of the two materials may be explicitly stated in the density symbols; for example:
:relative density: 8.15{{su|b=4&nbsp;°C|p=20&nbsp;°C}}; or specific gravity: 2.432{{su|b=0|p=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 also help to quantify the [[buoyancy]] of a substance in a [[fluid]] or gas, or determine the density of an unknown substance from the known density of another. Relative density is often used by [[geologist]]s and [[Mineralogy|mineralogists]] to help determine the [[mineral]] content of a rock or other sample. [[Gemology|Gemologists]] use it as an aid in the identification of [[gemstone]]s. Water is preferred as the reference because measurements are then easy to carry out in the field (see below for examples of measurement methods).

As the principal use of relative density measurements in industry is determination of the concentrations of substances in aqueous solutions and these are found in tables of RD vs concentration it is extremely important that the analyst enter the table with the correct form of relative density. For example, in the brewing industry, the [[Plato scale|Plato table]], which lists sucrose concentration by mass against true RD, were originally (20&nbsp;°C/4&nbsp;°C)<ref name="ReferenceA"/> that is based on measurements of the density of sucrose solutions made at laboratory temperature (20&nbsp;°C) but referenced to the density of water at 4&nbsp;°C which is very close to the temperature at which water has its maximum density of ''ρ''({{chem|H|2|O}}) equal to 0.999972&nbsp;g/cm<sup>3</sup> (or 62.43&nbsp;lb·ft<sup>−3</sup>). The [[American Society of Brewing Chemists|ASBC]] table<ref name="ReferenceB"/> in use today in North America, while it is derived from the original Plato table is for apparent relative density measurements at (20&nbsp;°C/20&nbsp;°C) on the IPTS-68 scale where the density of water is 0.9982071&nbsp;g/cm<sup>3</sup>. In the sugar, soft drink, honey, fruit juice and related industries sucrose concentration by mass is taken from this work<ref name=briggs>Hough, J.S., Briggs, D.E., Stevens, R and Young, T.W. Malting and Brewing Science, Vol. II Hopped Wort and Beer, Chapman and Hall, London, 1997, p. 881 {{ISBN|0-412-16590-2}}</ref> which uses SG (17.5&nbsp;°C/17.5&nbsp;°C). As a final example, the British RD units are based on reference and sample temperatures of 60&nbsp;°F and are thus (15.56&nbsp;°C/15.56&nbsp;°C).<ref name=briggs/>

Relative density is use in medicine particularly in pharmaceutical field. It is used in [[compounding|automated compounders]] in preparation of multicomponent mixtures for [[parenteral nutrition]], while it is an important factor in [[urinalysis]], relative density is an indicator of both the concentration of particles in the urine and a patient's degree of hydration.<ref>{{cite book |last1=Stockton |first1=Shelly J. |title=Stoklosa and Ansel's Pharmaceutical Calculations |date=2022 |publisher=[[Wolters Kluwer]] |isbn=978-1-9752-0680-2 |page=81 |edition=16|language=en}}</ref>

==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.

For each substance, the density, ''ρ'', is given by
<math display="block">\rho = \frac{\text{Mass}}{\text{Volume}}
 = \frac{\text{Deflection} \times \frac{\text{Spring Constant}}{\text{Gravity}}}{\text{Displacement}_\mathrm{Water Line} \times \text{Area}_\mathrm{Cylinder}}.</math>

When these densities are divided, references to the spring constant, gravity and cross-sectional area simply cancel, leaving
<math display="block">
RD = \frac{\rho_\mathrm{object}}{\rho_\mathrm{ref}}
= \frac{\frac{\text{Deflection}_\mathrm{Obj.}}{\text{Displacement}_\mathrm{Obj.}}}{\frac{\text{Deflection}_\mathrm{Ref.}}{\text{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.
</math>

===Hydrostatic weighing===
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:
<math display="block">RD = \frac{W_\mathrm{air}}{W_\mathrm{air} - W_\mathrm{water}},</math>
where
*''W''<sub>air</sub> is the weight of the sample in air (measured in [[newton (unit)|newton]]s, [[pound-force|pounds-force]] or some other unit of force)
*''W''<sub>water</sub> 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. ''W''<sub>water</sub> 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 allows the use of scales which cannot handle a suspended sample. A sample less dense than water can also be handled, but it has to be held down, and the error introduced by the fixing material must be considered.

===Hydrometer===
{{main article|hydrometer}}
[[File:Hydro.PNG|right|300px]]
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 adjacent diagram.

First the hydrometer is floated in the reference liquid (shown in light blue), and the [[displacement (fluid)|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, Δ''x'', 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 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,
*''ρ''<sub>ref</sub> is the known density ([[mass]] per unit [[volume]]) of the reference liquid (typically water).
*''ρ''<sub>new</sub> is the unknown density of the new (green) liquid.
*''RD''<sub>new/ref</sub> 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 [[Earth's gravity|local gravitational constant]].
*Δ''x'' is the change in displacement. In accordance with the way in which hydrometers are usually graduated, Δ''x'' 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, Δ''x'' 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 ''ρ''<sub>ref</sub>''Vg''. Setting these equal, we have
<math display="block">mg = \rho_\mathrm{ref}Vg</math>

or just
{{NumBlk||<math display="block">m = \rho_\mathrm{ref} V.</math>|{{EquationRef|1}}}}

Exactly the same equation applies when the hydrometer is floating in the liquid being measured, except that the new volume is {{math|''V'' − ''A''Δ''x''}} (see note above about the sign of Δ''x''). Thus,
{{NumBlk||<math display="block">m = \rho_\mathrm{new} (V - A \Delta x).</math>|{{EquationRef|2}}}}

Combining ({{EquationNote|1}}) and ({{EquationNote|2}}) yields
{{NumBlk||<math display="block">RD_\mathrm{new/ref} = \frac{\rho_\mathrm{new}}{\rho_\mathrm{ref}} = \frac{V}{V - A \Delta x}.</math>|
{{EquationRef|3}}}}

But from ({{EquationNote|1}}) we have {{math|1=''V'' = ''m''/''ρ''<sub>ref</sub>}}. Substituting into ({{EquationNote|3}}) gives
{{NumBlk||<math display="block">RD_\mathrm{new/ref} = \frac{1}{1 - \frac{A \Delta x}{m} \rho_\mathrm{ref}}.</math>|{{EquationRef|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 Δ''x'' is small then, as a [[Orders of approximation|first-order approximation]] of the [[geometric series]] equation ({{EquationNote|4}}) can be written as:
<math display="block">RD_\mathrm{new/ref} \approx 1 + \frac{A \Delta x}{m} \rho_\mathrm{ref}.</math>

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

===Pycnometer===
[[File:PycnometerEmpty.jpg|thumb|upright|An empty glass pycnometer and stopper]]
[[File:Pycnometer full.jpg|thumb|upright|A filled pycnometer]]
{{see also|Gas pycnometer}}
A '''pycnometer''' (from {{langx|grc|πυκνός|puknos|dense}}), also called '''pyknometer''' or '''specific gravity bottle''', is a device used to determine the [[density]] of a liquid. A pycnometer is usually made of [[glass]], with a close-fitting [[ground glass joint|ground glass]] [[stopper (plug)|stopper]] with a [[capillary tube]] through it, so that air bubbles may escape from the apparatus.  This device enables a liquid's density to be measured accurately by reference to an appropriate working fluid, such as [[water]] or [[Mercury (element)|mercury]], using an [[analytical balance]].{{Citation needed|date=May 2012}}

If the flask is weighed empty, full of water, and full of a liquid whose relative density is desired, the relative density of the liquid can easily be calculated. The [[particle density (packed density)|particle density]] of a powder, to which the usual method of weighing cannot be applied, can also be determined with a pycnometer. The powder is added to the pycnometer, which is then weighed, giving the weight of the powder sample.  The pycnometer is then filled with a liquid of known density, in which the powder is completely insoluble.  The weight of the displaced liquid can then be determined, and hence the relative density of the powder.

A [[gas pycnometer]], the gas-based manifestation of a pycnometer, compares the change in pressure caused by a measured change in a closed volume containing a reference (usually a steel sphere of known volume) with the change in pressure caused by the sample under the same conditions.  The difference in change of pressure represents the volume of the sample as compared to the reference sphere, and is usually used for solid particulates that may dissolve in the liquid medium of the pycnometer design described above, or for porous materials into which the liquid would not fully penetrate.

When a pycnometer is filled to a specific, but not necessarily accurately known volume, ''V'' and is placed upon a balance, it will exert a force
<math display="block"> F_\mathrm{b} = g\left(m_\mathrm{b} - \rho_\mathrm{a}\frac{m_\mathrm{b}}{\rho_\mathrm{b}}\right),</math>
where ''m''<sub>b</sub> is the mass of the bottle and ''g'' the [[gravitational acceleration]] at the location at which the measurements are being made. ''ρ''<sub>a</sub> is the density of the air at the ambient pressure and ''ρ''<sub>b</sub> is the density of the material of which the bottle is made (usually glass) so that the second term is the mass of air displaced by the glass of the bottle whose weight, by [[Archimedes' principle#Archimedes' principle|Archimedes Principle]] must be subtracted. The bottle is filled with air but as that air displaces an equal amount of air the weight of that air is canceled by the weight of the air displaced. Now we fill the bottle with the reference fluid e.g. pure water. The force exerted on the pan of the balance becomes:
<math display="block"> F_\mathrm{w} = g\left(m_\mathrm{b} - \rho_\mathrm{a} \frac{m_\mathrm{b}}{\rho_\mathrm{b}} + V\rho_\mathrm{w} - V\rho_\mathrm{a}\right). </math>

If we subtract the force measured on the empty bottle from this (or tare the balance before making the water measurement) we obtain.
<math display="block">F_\mathrm{w,n} = gV( \rho_\mathrm{w} - \rho_\mathrm{a}),</math>
where the subscript ''n'' indicated that this force is net of the force of the empty bottle. The bottle is now emptied, thoroughly dried and refilled with the sample. The force, net of the empty bottle, is now:
<math display="block">F_\mathrm{s,n} = gV(\rho_\mathrm{s} - \rho_\mathrm{a}),</math>
where ''ρ''<sub>s</sub> is the density of the sample. The ratio of the sample and water forces is:
<math display="block">SG_\mathrm{A} = \frac{gV(\rho_\mathrm{s} - \rho_\mathrm{a})}{gV( \rho_\mathrm{w} - \rho_\mathrm{a})} = \frac{\rho_\mathrm{s} - \rho_\mathrm{a}}{\rho_\mathrm{w} - \rho_\mathrm{a}}. </math>

This is called the ''apparent relative density'', denoted by subscript A, because it is what we would obtain if we took the ratio of net weighings in air from an analytical balance or used a [[hydrometer]] (the stem displaces air). Note that the result does not depend on the calibration of the balance. The only requirement on it is that it read linearly with force. Nor does ''RD''<sub>A</sub> depend on the actual volume of the pycnometer.

Further manipulation and finally substitution of ''RD''<sub>V</sub>, the true relative density (the subscript V is used because this is often referred to as the relative density {{lang|la|in vacuo}}), for ''ρ''<sub>s</sub>/''ρ''<sub>w</sub> gives the relationship between apparent and true relative density:

<math display="block">RD_\mathrm{A}= {{\rho_\mathrm{s} \over \rho_\mathrm{w}}-{\rho_\mathrm{a} \over \rho_\mathrm{w}} \over 1 - {\rho_\mathrm{a} \over \rho_\mathrm{w}}} ={RD_\mathrm{V}-{\rho_\mathrm{a} \over \rho_\mathrm{w}} \over 1 - {\rho_\mathrm{a} \over \rho_\mathrm{w}}}.</math>

In the usual case we will have measured weights and want the true relative density. This is found from
<math display="block">RD_\mathrm{V} = RD_\mathrm{A} - {\rho_\mathrm{a} \over \rho_\mathrm{w} }(RD_\mathrm{A}-1).</math>

Since the density of dry air at 101.325&nbsp;kPa at 20&nbsp;°C is<ref>DIN51 757 (04.1994): Testing of mineral oils and related materials; determination of density</ref> 0.001205&nbsp;g/cm<sup>3</sup> and that of water is 0.998203&nbsp;g/cm<sup>3</sup> we see that the difference between true and apparent relative densities for a substance with relative density (20&nbsp;°C/20&nbsp;°C) of about 1.100 would be 0.000120. Where the relative density of the sample is close to that of water (for example dilute ethanol solutions) the correction is even smaller.

The pycnometer is used in ISO standard: ISO 1183-1:2004, ISO 1014–1985 and [[ASTM International|ASTM]] standard: ASTM D854.

'''Types'''

* [[Gay-Lussac]], pear shaped, with perforated stopper, adjusted, capacity 1, 2, 5, 10, 25, 50 and 100 mL
* as above, with ground-in [[thermometer]], adjusted, side tube with cap
* Hubbard, for [[bitumen]] and [[heavy crude oil]]s, cylindrical type, [[ASTM]] D 70, 24 mL
* as above, conical type, ASTM D 115 and D 234, 25 mL
* Boot, with vacuum jacket and thermometer, capacity 5, 10, 25 and 50 mL

===Digital density meters===
''Hydrostatic Pressure-based Instruments'': This technology relies upon Pascal's Principle which states that the pressure difference between two points within a vertical column of fluid is dependent upon the vertical distance between the two points, the density of the fluid and the gravitational force. This technology is often used for tank gauging applications as a convenient means of liquid level and density measure.

''Vibrating Element Transducers'': This type of instrument requires a vibrating element to be placed in contact with the fluid of interest. The resonant frequency of the element is measured and is related to the density of the fluid by a characterization that is dependent upon the design of the element. In modern laboratories precise measurements of relative density are made using [[oscillating U-tube]] meters. These are capable of measurement to 5 to 6 places beyond the decimal point and are used in the brewing, distilling, pharmaceutical, petroleum and other industries. The instruments measure the actual mass of fluid contained in a fixed volume at temperatures between 0 and 80&nbsp;°C but as they are microprocessor based can calculate apparent or true relative density and contain tables relating these to the strengths of common acids, sugar solutions, etc.

''Ultrasonic Transducer'': Ultrasonic waves are passed from a source, through the fluid of interest, and into a detector which measures the acoustic spectroscopy of the waves. Fluid properties such as density and viscosity can be inferred from the spectrum.

''Radiation-based Gauge'': Radiation is passed from a source, through the fluid of interest, and into a scintillation detector, or counter. As the fluid density increases, the detected radiation "counts" will decrease. The source is typically the radioactive isotope [[caesium-137]], with a half-life of about 30 years. A key advantage for this technology is that the instrument is not required to be in contact with the fluid—typically the source and detector are mounted on the outside of tanks or piping.<ref>[http://www.ohmartvega.com/en/nuclear_density_DSG.htm Density – VEGA Americas, Inc]. Ohmartvega.com. Retrieved on 2011-09-30.</ref>

''Buoyant Force Transducer'': the buoyancy force produced by a float in a homogeneous liquid is equal to the weight of the liquid that is displaced by the float. Since buoyancy force is linear with respect to the density of the liquid within which the float is submerged, the measure of the buoyancy force yields a measure of the density of the liquid. One commercially available unit claims the instrument is capable of measuring relative density with an accuracy of ± 0.005 RD units. The submersible probe head contains a mathematically characterized spring-float system. When the head is immersed vertically in the liquid, the float moves vertically and the position of the float controls the position of a permanent magnet whose displacement is sensed by a concentric array of Hall-effect linear displacement sensors. The output signals of the sensors are mixed in a dedicated electronics module that provides a single output voltage whose magnitude is a direct linear measure of the quantity to be measured.<ref>[http://www.gardco.com/pages/density/electric_hydrometer.cfm Process Control Digital Electronic Hydrometer]. Gardco. Retrieved on 2011-09-30.</ref>

===Relative density in soil mechanics===
Relative density <math>D_\mathrm{R}</math> a measure of the current void ratio in relation to the maximum and minimum void rations, and applied effective stress control the mechanical behavior of cohesionless soil. Relative density is defined by <math display="block">D_\mathrm{R}=\frac{e_\mathrm{max}-e}{e_\mathrm{max}-e_\mathrm{min}} \times 100\%</math>
in which <math>e_\mathrm{max}, e_\mathrm{min}</math>, and <math>e</math> are the maximum, minimum and actual void rations.

===Limitations===
Specific gravity (SG) is a useful concept but has several limitations. One major issue is its sensitivity to temperature since the density of both the substance being measured and the reference changes with temperature, affecting accuracy.<ref>{{cite journal |last=Shaw |first=P.E. |year=1916 |title=The newtonian constant of gravitation as affected by temperature |journal=Philosophical Transactions of the Royal Society of London |volume=216 |issue=538–548 |pages=349–392 |doi=10.1098/rsta.1916.0007|doi-access=free }}</ref> It also assumes materials are incompressible, which isn't true for gasses or some liquids under varying pressures.<ref>{{cite book |title=Engineering Design Handbook:Hydraulic Fluids |publisher=US Army Materiel Command |year=1971 |chapter=Chapter 3-Specific Gravity |page=3-38 |chapter-url=https://books.google.com/books?id=2gVTWYmMuZwC&dq=Specific+gravity+sensitive+to+temperature&pg=RA2-SA3-PA38 |access-date=Sep 15, 2024}}</ref> It doesn't provide detailed information about a material’s composition or properties beyond density.<ref>{{cite web |url=https://www.samaterials.com/specific-gravity-liquids-gases-and-solids.html |title=Specific Gravity: Liquids, Gases, and Solids |website=Stanford Advanced Materials |last=Trento |first=Chin |date=Jan 5, 2024 |access-date=Sep 15, 2024}}</ref> Errors can also occur due to impurities, incomplete mixing, or air [[Bubble (physics)|bubbles]] in liquids, which can skew results.<ref>{{cite patent |country=WO |number=1992005422A1 |status=patent}}</ref>

==Examples==
{| class="wikitable sortable"
|-
! Material !! Specific gravity
|-
| [[Balsa wood]] || 0.2
|-
| [[Oak]] wood || 0.75
|-
| [[Ethanol]] || 0.78
|-
|[[Olive oil]] || 0.91
|-
| [[Water]] || 1
|-
|[[Olea capensis|Ironwood]] || 1.5
|-
|[[Graphite]] || 1.9–2.3
|-
| [[Table salt]] || 2.17
|-
| [[Aluminium]] || 2.7
|-
| [[Cement]] || 3.15
|-
| [[Iron]] || 7.87
|-
| [[Copper]] || 8.96
|-
| [[Lead]] || 11.35
|-
| [[Mercury (element)|Mercury]] || 13.56
|-
| [[Depleted uranium]] || 19.1
|-
| [[Gold]] || 19.3
|-
| [[Osmium]] || 22.59
|}

(Samples may vary, and these figures are approximate.)

Substances with a relative density of 1 are neutrally buoyant, those with RD greater than one are denser than water, and so (ignoring [[surface tension]] effects) will sink in it, and those with an RD of less than one are less dense than water, and so will float.

Example:
<math display="block">RD_\mathrm{H_2O} = \frac{\rho_\mathrm{Material}}{\rho_\mathrm{H_2O}} = RD,</math>

[[Helium]] gas has a density of 0.164&nbsp;g/L;<ref name="UCSB">{{cite web|title=Lecture Demonstrations|website=physics.ucsb.edu| url=http://web.physics.ucsb.edu/~lecturedemonstrations/Composer/Pages/36.39.html}}</ref> it is 0.139 times as dense as [[air]], which has a density of 1.18&nbsp;g/L.<ref name="UCSB"/>

*[[Urine]] normally has a specific gravity between 1.003 and 1.030. The Urine Specific Gravity diagnostic test is used to evaluate renal concentration ability for assessment of the urinary system.<ref name=":0">{{Cite book|title=Medical-surgical nursing : assessment and management of clinical problems|last1=Lewis|first1=Sharon Mantik|last2=Dirksen|first2=Shannon Ruff| last3=Heitkemper|first3=Margaret M.|last4=Bucher|first4=Linda|last5=Harding|first5=Mariann|date=5 December 2013| isbn=978-0-323-10089-2|edition=9th|location=St. Louis, Missouri|oclc=228373703}}</ref> Low concentration may indicate [[diabetes insipidus]], while high concentration may indicate [[albuminuria]] or [[glycosuria]].<ref name=":0" />
*[[Blood]] normally has a specific gravity of approximately 1.060.<ref>{{cite web | url=https://hypertextbook.com/facts/2004/MichaelShmukler.shtml | title=Density of blood | first=Michael | last=Shmukler | year=2004 | website=The Physics Factbook | editor-last=Elert | editor-first=Glenn | accessdate=2022-01-23 }}</ref>
*[[Vodka]] 80° proof (40% v/v) has a specific gravity of 0.9498.<ref name="Cocktail Specific Gravities">{{cite web| url=http://www.goodcocktails.com/bartending/specific_gravity.php|title=Specific Gravity of Liqueurs|website=Good Cocktails.com}}</ref>

==See also==
{{Div col|colwidth=20em}}
*[[API gravity]]
*[[Baumé scale]]
*[[Buoyancy]]
*[[Fluid mechanics]]
*[[Gravity (beer)]]
*[[Hydrometer]]
*[[Jolly balance]]
*[[Plato scale]]
{{Div col end}}

==References==
{{reflist|30em}}

==Further reading==
*''Fundamentals of Fluid Mechanics'' Wiley, B.R. Munson, D.F. Young & T.H. Okishi
*''Introduction to Fluid Mechanics'' Fourth Edition, Wiley, SI Version, R.W. Fox & A.T. McDonald
*''Thermodynamics: An Engineering Approach'' Second Edition, McGraw-Hill, International Edition, Y.A. Cengel & M.A. Boles
*{{cite book |last=Munson |first=B. R.|author2=D. F. Young |author3=T. H. Okishi  |title=Fundamentals of Fluid Mechanics |year=2001 |publisher=Wiley|edition = 4th|isbn=978-0-471-44250-9}}
*{{cite book |last=Fox |first=R. W.|author2=McDonald, A. T.  |title=Introduction to Fluid Mechanics |year=2003 |publisher=Wiley |edition = 4th |isbn=0-471-20231-2}}

==External links==
*[https://web.archive.org/web/20060522133528/http://www.reade.com/particle_briefings/spec_gra2.html Specific Gravity Weights Of Materials] (archived 22 May 2006)

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{{DEFAULTSORT:Relative Density}}
[[Category:Mass density]]
[[Category:Dimensionless quantities]]
[[Category:Ratios]]