# Liquids - Densities 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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