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

Densities
Densities of solids, liquids and gases. Definitions and convertion calculators. 
Fluid Mechanics
The study of fluids  liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time. 
Thermodynamics
Work, heat and energy systems.
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