Lesson 7: Properties of fluids: density
Recall that a fluid is a material such that relative positions of the elements of the material change
significantly when suitable chosen forces are applied. As a consequence of the constantly changing
shape and volume of a fluid, we prefer to measure the density of a fluid instead of its mass.
Density (  ) is defined as the mass per unit volume:

mass
Volume
And is usually represented with units of
the second unit,
g
kg
or 3 . Here we will almost exclusively work with
3
m
cm
kg
.
m3
Density is related to the specific volume,  , which is the volume per unit mass, as follows:
cm 3
m3
or
.
  . Specific volume has units of

g
kg
1
Equation of state: The functional relationship between the absolute pressure, specific volume and
temperature for a homogenous substance. The general equation of state is represented as
f  p, , T   0
(1)
Notice that an equation of state in this form is just a surface. Approximations and empirical
observations allow us to establish various analytic forms of the equation of state for liquids and
gasses. The important point to note about equation (1) is that the state of the system is completely
determined by any two of the three independent variables shown in equation (1). Further there is
nothing special about the choice of pressure, specific volume and temperature as our
thermodynamic state variables. There are many other variables we could have chosen to define the
state of the system such as entropy, Gibbs free energy, internal energy and so on. The decision of
what thermodynamic state variables to choose for the physical problem can make a significant
difference on the nature and complexity of the problem. It is up to the scientist to use his expertise
of the physical nature of the problem to justify his or her choice of the state variables.
Equation of state for Gasses:
Under the limit that p  0 , all gasses have been shown to exhibit the same analytic relationship.
It is under these circumstances that one obtains the most commonly recognized equation of state;
the ideal gas law:
pV  nR *T
(2)
1
The variables, and their units, in the ideal gas law are as follows:
 p  Pa
V  volume. V   m3
n  number of moles. n  moles
p  pressure.
R*  Universal gas constant. R*  8.3145 J K 1 mol 1
T  temperature. T   K
Notice that equation (2) is applicable for any gas and is applicable only under the limit p  0 . It
can be shown theoretically that this limit is the same as requiring that:
1. Molecules exhibit no potential forces between each other. They only interact through elastic
collisions like billiard balls.
2. Molecules occupy no volume. They are all treated as “point” masses
Each gas also exhibits its own unique equation of state based on its molecular mass as
p  RT
(3)
Where p is pressure in Pa,  is density in kg m-3, R is the gas constant at 287 J kg-1 K-1, and T is
temperature in K. To derive equation (3) from equation (2), we make use of the following
relationships:
n
R*
m
and R 
M
M
where m is the mass of the air in kg and M is the molecular weight of dry air, 0.02894 kg mol 1 .
The derivation follows these steps:
pV  nR *T
pV 
m *
RT
M
pV  m
pV

m
R*
T
M
m
R*
T
M
m
p  RT
p   RT
2
Equation of state for the atmosphere (moist air):
The atmosphere is a bit more complicated than an ideal gas in that it is composed of
multiple gasses and water vapor. Fortunately due to Dalton’s law of partial pressures, little
modification from equation (3) is necessary. Instead we use a modified form of temperature that
accounts for the effects for water vapor called the Virtual temperature. Thus the gas law for moist
air is of the form:
p  Rd T *
(4)
Where
P is pressure in Pascals
kg
kg
. Atmospheric density has values of ~ 1.2 3
3
m
m
Joules
is the specific gas constant for dry air
Rd  2.8705 102
kg  Kelvin
 is density measured in
T * is the virtual temperature in units of Kelvin and is defined as the temperature a parcel of dry air
would have if its pressure and density were those of the parcel of moist air. It is more easily
remembered mathematically in the following approximate form:
T* T 
W
6
Where W is the mixing ratio with units of gm/kg. T can have units of either C or K, and T * will
take the same unit.
Ocean Density1:
We can see from equation (4) that the state of the atmosphere is uniquely determined by
finding the pressure, temperature, density and moisture content of a parcel of air. Further, we have
a nice analytic representation to use for our calculations. Liquids clearly do not allow us to use as
simple a relationship due to the fact that molecular forces are unavoidable. One approach in
applying the state of a liquid is to create empirical surface charts called PT surfaces or tables for
the substance. We have an added difficulty for the ocean in that the salinity content has a major
impact on density as well as temperature. We can however use a Taylor series to represent small
changes in density from a reference state,  o   o To , S o , po  , in terms of small changes in salinity
and temperature. This leads to the following expression for the equation of state:
1
The background of most of the notes from this section are from:
Pond, S. and Pickard, G. L. Introductory Dynamical Oceanography, 2nd ed, Pergamon press, Oxford, UK. 1983.
3
 T , S , p    o 1   T T  To    S S  S o   K  p  po 
(5)
1 
1 
is the thermal expansion coefficient2 ,  S 
is the haline contraction
 T
 S
1 
coefficient for seawater and K 
 is the isothermal compressibility coefficient. Typical
 p
kg
values for the reference state are  o  10 3 3 , To  283K , S o  35 psu and po  1atm . Standard
m
values for the Thermal expansion and haline contraction coefficients are  T ~ 2  10 4 K 1 ,
 S ~ 2  10 4 psu 1 and K ~ 4.5  10 6 dbar 1 .
Equation (5) is a linear representation of a non-linear function and so one must use this
approximation with care. As with any linear estimate, the results are only valid for small changes
in state variables. This means that we should apply equation (5) only when considering local
variations in the domain of interest.
To obtain a more accurate representation of the equation of state that accounts for some of
the non-linear features, we can use empirical estimates of the secant bulk modulus K S , T , p . The
secant bulk modulus is an estimate of the compressibility of the equation of state. It is obtained by
taking the graph of pressure versus fractional change in volume and quantifying the slope of a
 
segment extending from the reference point, po  0, o
 0 , to some arbitrary value of
where  T  
o
pressure as shown in Figure 1.
p
Figure 1 – Representation showing the secant bulk modulus for the non-linear graph of pressure
versus specific volume.
From figure 1, we can see the linear relationship between the bulk modulus and pressure can be
expressed as
2
Notice the notation here: do not confuse the
specific volume.
T
for the thermal expansion coefficient with the  used to represent
4
K

  o
(6)
p
Solving equation (6) for the density at point 1 we obtain
 S , T , p  
 S , T , po 
(7)
p
1
K S , T , p 
For various values of temperature and salinity, the secant bulk modulus has also been empirically
fit to a quadratic function of pressure
K  K o S , T   AS , T  p  B( S , T ) p 2
(8)
Where K o S , T , S , T  and B ( S , T ) can be empirically fit to data. Substitution of equation (8) into
equation (7) with proper values for the coefficients leads to the International equation of state for
sea water.
 S , T , p  
 S , T ,0
1
p
K S , T , p 
For this equation, both   S , T ,0 and K  S , T , p  are empirically determined from laboratory
measurements. Their equations are long and complex functions of salinity S, temperature T, and
pressure p. (See next page).
5
K S , T , p  
19652.21
 148.4206T  2.327105T 2
 1.360477  10  2 T 3  5.155288  10 5 T 4
 3.239908 p  1.43713  10 3 Tp
 1.16092  10  4 T 2 p  5.77905  10 7 T 3 p
 8.50935  10 5 p 2  6.12293  10 6 Tp 2
 5.2787  10 8 T 2 p 2
 54.6746 S  0.603459TS
 1.09987  10  2 T 2 S  6.1670  10 5 T 3 S
 7.944  10  2 S 3 / 2  1.6483  10  2 TS 3 / 2
 5.3009  10  4 T 2 S 3 / 2  2.2838  10 3 pS
 1.0981  10 5 TpS  1.6078  10 6 T 2 pS
 1.91075  10  4 pS 3 / 2  9.9348  10 7 p 2 S
 2.0816  10 8 Tp 2 S  9.1697  10 10 T 2 p 2 S
and
 S , T ,0 
999.842594  6.793952  10  2 T
 9.095290  10 3 T 2  1.001685  10  4 T 3
 1.120083  10 6 T 4  6.536332  10 9 T 5
 8.24493  10 1 S  4.0899  10 3 TS
 7.6438  10 5 T 2 S  8.2467  10 7 T 3 S
 5.3875  10 9 T 4 S  5.72466  10 3 S 3 / 2
 1.0227  10  4 TS 3 / 2  1.6546  10 6 T 2 S 3 / 2
 4.8314  10  4 S 2
6
Laboratory versus in-situ measurement of density:
In experimental work, measurements are based on some reference condition. If
measurements are taken at depth, such as with a CTD or rawindosnde, then the point of reference is
at the depth or altitude of the measurement. If we take a sample and bring it to the surface then the
point of reference is at the surface or in the Laboratory. We must account for variations in density
due to the transport to the surface when considering the laboratory point of reference. One often
uses potential density in such circumstances where variations of density with depth are accounted
for due to changes in potential temperature,  . Recall  measure changes in temperature as a
result of adiabatic expansion or contraction. The equation for potential density is generally
expressed as
 pot   S ,  , p o ;
(9)
We must be careful in using potential density when examining the environmental conditions
in the ocean or atmosphere. For example, North Atlantic deep water has a larger potential density
than Antarctic Bottom water but observation show that the former is above the latter. The cause
for this discrepancy is that our point of reference is at the surface but non-linear variations of
density at such large depths actually indicate the ocean environment is neutrally stable.
7
Appendix:
Table of density values – the following are some values of density given by the International
equation of state for seawater 3
 
 kg 
3 
m 
S (psu)
T oC
p (bars)
 S , T , p  
K(S,T,p) (bars)
0
5
0
999.96675
20337.80375
0
5
1000
1044.12802
23643.52599
0
25
0
997.04796
22100.72106
0
25
1000
1037.90204
25405.09717
35
5
0
1027.67547
22185.93358
35
5
1000
1069.48914
25577.49819
35
25
0
1023.34306
23726.34949
35
25
1000
1062.53817
27108.94504
3
From:
Pond, S. and Pickard, G. L. Introductory Dynamical Oceanography, 2nd ed, Pergamon press, Oxford, UK. 1983, 311.
8