# Density of Liquids versus change in Pressure and Temperature

## Density and specific volume of a liquid versus change in pressure and temperature

The density of a liquid will change with temperature and pressure. The density of water versus temperature and pressure is indicated below:

### Density

The density of a liquid can be expressed as

ρ = m / V                                (1)

where

ρ = density of liquid (kg/m3)

m = mass of the liquid (kg)

V = volume of the liquid (m3)

The inverse of density is specific volume:

v = 1 / ρ

= V / m                             (2)

where

v = specific volume (m3/kg)

### Volume and change in Temperature

When temperature increases -  most liquids expands:

dV = V1 - V0

= V0 β dt

= V0 β (t1 - t0)                               (3)

where

dV = V1 - V0 = change in volume - difference between final and initial volume (m3)

β = volumetric temperature expansion coefficient (m3/m3 oC)

dt = t1 - t0 = change in temperature - difference between final and initial temperature (oC)

(3) can be modified to

V1 = V0 (1 + β (t1 - t0))                           (3b)

### Density and change in Temperature

With (1) and (3b) the final density after a temperature change can be expressed as

ρ1 = m / (V0 (1 + β (t1 - t0)))                              (4)

where

ρ1 = final density (kg/m3)

- or combined with (2)

ρ1ρ0 / (1 + β (t1 - t0))                                  (4b)

where

ρ0 = initial density (kg/m3)

#### 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 / V0

= - (p1 - p0) / ((V1 - V0) / V0)                                   (5)

where

E = bulk modulus - liquid elasticity (N/m2)

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

V1 = V0 (1 - (p1 - p0) / E)                                     (5b)

Combining (5b) with (1) - the final density can be expressed as:

ρ1 = m / (V0 (1 - (p1 - p0) / E))                                  (6)

- or combined with (2) - the final density can be expressed as

ρ1 = ρ0 / (1 - (p1 - p0) / E)                                  (6b)

#### Bulk Modulus Fluid Elasticity some common Fluids - E

• water : 2.15 109 (N/m2)
• ethyl alcohol : 1.06 109 (N/m2)
• oil : 1.5 109 (N/m2)

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):

ρ1ρ1(from eq.1) / (1 - (p1 - p0) / E)

= ρ0 / (1 + β (t1 - t0)) / (1 - (p1 - p0) / E)                                (7)

### Example - Density of Water at 100 bar and 20oC

• density of water 0oC: 999.8 (kg/m3)
• expansion coefficient of water at 10oC: 0.000088 (m3/m3oC) (average value between 0 and 20oC)
• bulk modulus of water: 2.15 10(N/m2)

Density of water can be calculated with (3):

ρ1 = (999.8 kg/m3) / (1 + (0.000088 m3/m3oC) ((20 oC) - (0 oC))) / (1 - ((100 105 Pa) - (1 105 Pa)) / (2.15 109 N/m2)

= 1002.7 (kg/m3)

## Related Topics

• Fluid Mechanics - The study of fluids - liquids and gases. Involves velocity, pressure, density and temperature as functions of space and time
• Thermodynamics - Effects of work, heat and energy on systems
• Density - Density of different solid materials, liquids and gases. Definitions and convertion calculators.

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