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 \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}}. 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{\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.
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: RD = \frac{W_\mathrm{air}}{W_\mathrm{air} - W_\mathrm{water}}, where •
Wair is the weight of the sample in air (measured in
newtons,
pounds-force 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 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 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 (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, •
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. • 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
ρref
Vg. Setting these equal, we have mg = \rho_\mathrm{ref}Vg or just {{NumBlk||m = \rho_\mathrm{ref} V.|}} Exactly the same equation applies when the hydrometer is floating in the liquid being measured, except that the new volume is (see note above about the sign of Δ
x). Thus, {{NumBlk||m = \rho_\mathrm{new} (V - A \Delta x).|}} Combining () and () yields {{NumBlk||RD_\mathrm{new/ref} = \frac{\rho_\mathrm{new}}{\rho_\mathrm{ref}} = \frac{V}{V - A \Delta x}.| }} But from () we have . Substituting into () gives {{NumBlk||RD_\mathrm{new/ref} = \frac{1}{1 - \frac{A \Delta x}{m} \rho_\mathrm{ref}}.|}} 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
first-order approximation of the
geometric series equation () can be written as: RD_\mathrm{new/ref} \approx 1 + \frac{A \Delta x}{m} \rho_\mathrm{ref}. This shows that, for small Δ
x, changes in displacement are approximately proportional to changes in relative density.
Pycnometer A
pycnometer (from ), 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 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, using an
analytical balance. 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 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 F_\mathrm{b} = g\left(m_\mathrm{b} - \rho_\mathrm{a}\frac{m_\mathrm{b}}{\rho_\mathrm{b}}\right), where
mb is the mass of the bottle and
g the
gravitational acceleration at the location at which the measurements are being made.
ρa is the density of the air at the ambient pressure and
ρb 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 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: 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). If we subtract the force measured on the empty bottle from this (or tare the balance before making the water measurement) we obtain. F_\mathrm{w,n} = gV( \rho_\mathrm{w} - \rho_\mathrm{a}), 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: F_\mathrm{s,n} = gV(\rho_\mathrm{s} - \rho_\mathrm{a}), where
ρs is the density of the sample. The ratio of the sample and water forces is: 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}}. 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
RDA depend on the actual volume of the pycnometer. Further manipulation and finally substitution of
RDV, the true relative density (the subscript V is used because this is often referred to as the relative density ), for
ρs/
ρw gives the relationship between apparent and true relative density: 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}}}. In the usual case we will have measured weights and want the true relative density. This is found from RD_\mathrm{V} = RD_\mathrm{A} - {\rho_\mathrm{a} \over \rho_\mathrm{w} }(RD_\mathrm{A}-1). Since the density of dry air at 101.325 kPa at 20 °C is 0.001205 g/cm3 and that of water is 0.998203 g/cm3 we see that the difference between true and apparent relative densities for a substance with relative density (20 °C/20 °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 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 oils, 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 °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.
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.
Relative density in soil mechanics Relative density D_\mathrm{R} a measure of the current void ratio in relation to the maximum and minimum void ratios, and applied effective stress control the mechanical behavior of cohesionless soil. Relative density is defined by D_\mathrm{R}=\frac{e_\mathrm{max}-e}{e_\mathrm{max}-e_\mathrm{min}} \times 100\% in which e_\mathrm{max}, e_\mathrm{min}, and e are the maximum, minimum and actual void ratios.
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. It also assumes materials are incompressible, which isn't true for gasses or some liquids under varying pressures. It doesn't provide detailed information about a material’s composition or properties beyond density. Errors can also occur due to impurities, incomplete mixing, or air
bubbles in liquids, which can skew results. ==Examples==