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Within the study of the elasticity of matter, the modulus of volume is a constant that describes to what extent a substance is resistant to compression. It is defined as the ratio between the increase in pressure and the resulting decrease in volume of a material. Along with Young’s modulus, shear modulus, and Hooke’s law, bulk modulus describes a material’s response to stress or strain .
Usually, the bulk modulus is indicated by K or B in the equations and tables. It is most often used to describe the behavior of fluids, but it can be used to study the uniform compression of any substance. Some of its other uses are predicting compression, calculating density, and indirectly indicating the types of chemical bonds within a substance. The modulus of volume is considered a descriptor of elastic properties because a compressed material returns to its original volume once the pressure is released.
The units for the modulus of volume are pascals (Pa) or newtons per square meter (N/m 2 ) in the metric system, or pounds per square inch (PSI) in the English system.
Table of values of the volume modulus of various fluids
There are bulk modulus values for solids (for example, 160 GPa for steel; 443 GPa for diamond; 50 MPa for solid helium) and gases (for example, 101 kPa for air at constant temperature), but the most common tables list values for liquids. Below are representative values in both English and metric units:
English units Metric units
Acetone 1.34 0.92
Benzene 1.5 1.05
Carbon tetrachloride 1.91 1.32
Ethyl alcohol 1.54 1.06
Gasoline 1.9 1.3
Glycerin 6.31 4.35
Mineral oil ISO 32 2.6 1.8
Kerosene 1.9 1.3
Mercury 41.4 28.5
Paraffin 2.41 1.66
Petrol 1.55 – 2.16 1.07 – 1.49
Phosphate ester 4.4 3
SAE 30 oil 2.2 1.5
Sea water 3.39 2.34
Sulfuric acid 4.3 3.0
Water 3.12 2.15
Water – Glycol 5 3.4
Water – Oil emulsion 3.3 2.3
The value of B varies depending on the state of matter and in some cases on the temperature. In liquids, the amount of dissolved gas has a large impact on the value. A high value of B indicates that a material resists compression, while a low value indicates that the volume decreases appreciably under uniform pressure.
In general terms, solid matter can hardly be compressed, liquids can be compressed very little and it is only matter in a gaseous state that does not retain a certain volume and can be compressed. For example, in a butane bottle the gas is highly compressed.
Bulk Modulus Formulas
The bulk modulus of a material can be measured by powder diffraction, using X-rays, neutrons, or electrons directed at a powdered or microcrystalline sample. It can be calculated using the following formula:
Volumetric modulus (B) = volumetric stress / volumetric strain
This is the same as saying that it is equal to the pressure change divided by the volume change divided by the initial volume:
Volume modulus ( B ) = (p 1 – p 0 ) / [(V 1 – V 0 ) / V 0 ]
Here p 0 and V 0 are the initial pressure and volume, respectively, and p 1 and V1 are the pressure and volume measured after compression.
The elasticity of the bulk modulus can also be expressed in terms of pressure and density:
B = (p 1 – p 0 ) / [(ρ 1 – ρ 0 ) / ρ 0 ]
Here, ρ 0 and ρ 1 are the initial and final density values.
Calculation Example
The volume modulus can be used to calculate the hydrostatic pressure and density of a liquid. Consider, for example, the seawater in the deepest point of the ocean, the Mariana Trench. The base of the trench is 10,994 m below sea level.
The hydrostatic pressure in the Mariana Trench can be calculated as:
p 1 = ρ * g * h
Where p 1 is the pressure, ρ is the density of seawater at sea level, g is the acceleration due to gravity, and h is the height (or depth) of the water column.
p 1 = (1022 kg / m 3 ) (9.81 m / s 2 ) (10994 m)
p 1 = 110 x 10 6 Pa or 110 MPa
Knowing that the pressure at sea level is 105 Pa, the density of the water at the bottom of the trench can be calculated:
ρ 1 = [(p 1 – p) ρ + K * ρ) / K
ρ 1 = [[(110 x 10 6 Pa) – (1 x 10 5 Pa)] (1022 kg / m 3 )] + (2.34 x 10 9 Pa) (1022 kg / m 3 ) / (2, 34 x 10 9 PA)
ρ 1 = 1070 kg / m 3
What can you see from this? Despite the immense pressure on the water at the bottom of the Mariana Trench, it is not very compressed!
References
Espasa. (S/F). States of the material. Editorial Planet. Available at http://espasa.planetasaber.com/AulaSaber/ficha.aspx?ficha=16957
Ruiz, C. and Osorio Guillén, J. (2011). Theoretical study of the elastic properties of minerals. Engineering and Science. Available at file:///C:/Users/isabeljolie/Downloads/Dialnet-EstudioTeoricoDeLasPropiedadesElasticasDeLosMinera-3913114.pdf
Gilman, J. (1969). Flow micromechanics in solids. McGraw-Hill.
