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7
Equations
of
State
The properties of fluids can be defined in two ways, either by the use of tabulated data
(e.g. steam tables) or by state equations (e.g. perfect gas law). Both of these approaches
have been developed by observation of the behaviour of fluids when they undergo simple
processes. It has also been possible to model the behaviour of such fluids from ‘molecular’
models, e.g. the kinetic theory of gases. A number of models which describe the
relationships between properties for single component fluids, or constant composition
mixtures, will be developed here.
7.1 Ideal gas law
The ideal and perfect gas laws can be developed from a number of simple experiments, or
a simple molecular model. First the experimental approach will be considered.
If a fixed mass of a single component fluid is contained in a closed system then two
processes can be proposed:
(i)
the volume of the gas can be changed by varying the pressure, while
maintaining the temperature constant;
(ii)
the volume of the system can be changed by varying the temperature, while
maintaining the pressure constant.
The first process is an isothermal one, and is the experiment proposed by Boyle to
define Boyle’s law (also known as Mariotte’s law in France). The second process is
an
isobaric one and is the one used to define Charles’ law (also known as GayLussac’s law in
France).
The process executed in (i) can be described mathematically as
v
=
v(
PIT
(7.1)
while the second one, process (ii), can be written
v
=
v(T),
Since these processes can be undergone independently then the relationship between the
three properties is
(7.3)
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Equations of state
Equation
(7.3)
is a functional form of the
equation ofstare
of a single component fluid.
It can be seen to obey the twoproperty rule, which states that any property of a single
component fluid or constant composition mixture can be defined
as
a function of
two
independent properties. The actual mathematical relationship has to be found from
experiment (or a simulation of the molecular properties of the gas molecules), and
this
can
be derived by knowing that, if the property,
v,
is a continuous function of the other
properties,
p
and T, as discussed in Chapter
6,
then
dv
=
(
$)T
dp
+
(
$)p
dT
(7.4)
Hence, if the partial derivatives
(av/ap),
and (av/aT), can be evaluated then the gas law
will be defined. It is possible to evaluate the first derivative by a Boyle’s law experiment,
and the second one by a Charles’ law experiment. It is found from Boyle’s law that
pv
=constant
giving
P
=
V
Similarly, it is found from Charles’ law that
V
_

constant
T
V
($)p
=r
Substituting eqns
(7.6)
and
(7.8) into
eqn
gives
V
V
=


dp
+

dT
P
T
which may be integrated to give
=
T
(7.5)
(7.7)
(7.8)
(7.9)
(7.10)
is known as the
ideal
gas
law.
This equation contains no information
about the internal energy of the fluid, and does not define the specific heat capacities. If
the specific heat capacities
are
not functions of temperature then the gas is said to obey the
perfect gas
law: if the specific heat capacities are functions of temperature (Le. the
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 Spring '10
 Prof.William

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