# Density of a Liquid - when Changing Pressure and Temperature

## Density and specific volume of a liquid when changing pressure or temperature

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The density of liquids changes 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/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})) *

* = 998.0 / 0.995 *

* = 1002.7 (kg/m ^{3})*

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