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# 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

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.

### 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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