Liquids  Densities vs. Pressure and Temperature Change
Densities and specific volume of liquids vs. pressure and temperature change.
The density of a liquid will change with temperature and pressure. The density of water versus temperature and pressure is indicated below:
See also Water  Density, Specific Weight and Thermal Expantion Coefficient , for online calculator, figures and tables showing changes with temperature.
Density
The density of a liquid can be expressed as
ρ = m / V (1)
where
ρ = density of liquid (kg/m ^{ 3 } )
m = mass of the liquid (kg)
V = volume of the liquid (m ^{ 3 } )
The inverse of density is specific volume:
v = 1 / ρ
= V / m (2)
where
v = specific volume (m ^{ 3 } /kg)
Volume and change in Temperature
When temperature increases  most liquids expands:
dV = V _{ 1 }  V _{0 }
= V _{0 } β dt
= V _{0 } β (t _{ 1 }  t _{0 } ) (3)
where
dV = V _{ 1 }  V _{0 } = change in volume  difference between final and initial volume (m ^{ 3 } )
β = volumetric temperature expansion coefficient (m ^{ 3 } /m ^{ 3 o } C)
dt = t _{ 1 }  t _{0 } = change in temperature  difference between final and initial temperature ( ^{o}C)
(3) can be modified to
V _{ 1 } = V _{0 } (1 + β (t _{ 1 }  t _{0 } )) (3b)
Density and change in Temperature
With (1) and (3b) the final density after a temperature change can be expressed as
ρ _{ 1 } = m / ( V _{0 } (1 + β (t _{ 1 }  t _{0 } ))) (4)
where
ρ _{ 1 } = final density (kg/m ^{ 3 } )
 or combined with (2)
ρ _{ 1 } = ρ _{0 } / (1 + β (t _{ 1 }  t _{0 } )) (4b)
where
ρ _{0 } = initial density (kg/m ^{ 3 } )
Volumetric Temperature Coefficients  β
 water : 0.0002 (m ^{ 3 } /m ^{ 3 o } C) at 20 ^{o}C
 ethyl alcohol : 0.0011 (m ^{ 3 } /m ^{ 3 o } C)
 volumetric expansion coefficient for some commonly used materials
Note!  volumetric temperature coefficients may vary strongly with temperature.
Density and change in Pressure
The influence of pressure on the volume of a liquid can be expressed with the three dimensional Hooke's law
E =  dp / (dV / V _{0 } )
=  (p _{ 1 }  p _{0 } ) / ((V _{ 1 }  V _{0 } ) / V _{0 } ) (5)
where
E = bulk modulus  liquid elasticity (N/m^{2})
The minus sign corresponds to the fact that an increase in the pressure leads to a decrease in volume.
With (5)  the final volume after pressure change can be expressed as
V _{ 1 } = V _{0 } (1  (p _{ 1 }  p _{0 } ) / E) (5b)
Combining (5b) with (1)  the final density can be expressed as:
ρ _{ 1 } = m / ( V _{0 } (1  (p _{ 1 }  p _{0 } ) / E)) (6)
 or combined with (2)  the final density can be expressed as
ρ _{ 1 } = ρ _{0 } / (1  (p _{ 1 }  p _{0 } ) / E) (6b)
Bulk Modulus Fluid Elasticity some common Fluids  E
 water : 2.15 10 ^{ 9 } (N/m^{2})
 ethyl alcohol : 1.06 10 ^{ 9 } (N/m^{2})
 oil : 1.5 10 ^{ 9 } (N/m^{2})
Note! Bulk modulus for liquids varies with pressure and temperature.
Bulk modulus for water  Imperial Units
Bulk modulus for Water  SI units
Density of a fluid changing both Temperature and Pressure
The density of a fluid when changing both temperature and pressure can be expressed combining (4b) and (6b) :
ρ _{ 1 } = ρ _{ 1(from eq.1) } / (1  (p _{ 1 }  p _{0 } ) / E)
= ρ _{0 } / (1 + β (t _{ 1 }  t _{0 } )) / (1  (p _{ 1 }  p _{0 } ) / E) (7)
Example  Density of Water at 100 bar and 20 ^{o}C
 density of water 0 ^{o}C : 999.8 (kg/m ^{ 3 } )
 expansion coefficient of water at 10 ^{o}C : 0.000088 ( m ^{ 3 } /m ^{ 3 } ^{o}C) (average value between 0 and 20 ^{o}C)
 bulk modulus of water: 2.15 10 ^{ 9 } (N/m^{2})
Density of water can be calculated with (3):
ρ _{ 1 } = (999.8 kg/m ^{ 3 } ) / (1 + (0.000088 m ^{ 3 } /m ^{ 3 } ^{o}C) ((20 ^{o}C)  (0 ^{o}C) )) / (1  ((100 10 ^{ 5 } Pa)  (1 10 ^{ 5 } Pa)) / ( 2.15 10 ^{ 9 } N/m^{2}) )
= 1002.7 (kg/m ^{ 3 } )
Related Topics

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Fluid Mechanics
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Thermodynamics
Work, heat and energy systems.
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