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General Thermodynamic Relationships:
single component systems,
or
systems
of
constant com position
The relationships which follow are based on single component systems, or systems of
constant composition. These are a subset of the more general equations which can be
derived, and which allow for changes in composition. It will be shown in Chapter 12 that
if a change in composition occurs then another term defining the effect of this change is
required.
6.1 The Maxwell relationships
The concept of functional relationships between properties was introduced previously. For
example, the Second Law states that, for a reversible process,
T,
s,
u,
p and v are related
in the following manner
T
dS=dU+p dV
(6.1)
or, in specific (or molar) terms
T
ds=du+pdv
(6.la)
Rearranging eqn (6.la) enables the change of internal energy, du, to be written
du= T dsp dv
(6.2)
(6.2a)
It will be shown in Chapter 12 that, in the general case where the composition can change,
eqn (6.1) should be written
TdS
=
dU +p dV

Cpidn,
1
where
p,
=
chemical potential of component
i
n,
=
amount of component
i
(6.lb)
The chemical potential terms will be omitted in the following analysis, although similar
equations to those below can be derived by taking them into account.
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View Full Document The Maxwell relationships
101
It can be seen from eqns
(6.2)
and (6.2a) that the specific internal energy can be
represented by a threedimensional surface based on the independent variables of entropy
and specific volume.
If this surface is
continuous
then the following relationships can be
based on the mathematical properties of the surface. The restriction of a continuous
surface means that it is ‘smooth’. It can be seen from Fig
6.1
that the
pv
T surface for
water is continuous over most of the surface, but there are discontinuities at the saturated
liquid and saturated vapour lines. Hence, the following relationships apply over the major
regions of the surface,
but not across the boundaries.
For a continuous surface z
=
z(x,
y)
where z is a continuous function. Then
dz=

dx+

dy
(6.3)
(a
(3
Let
IV=($)~
and
.=(:)I
(6.4)
Then
dz
=
M
dx
+
N
dy
(6.5)
For continuous functions, the derivatives
aZz
d2Z
ax
ay
ay
ax

102
General thermodynamic relationships
are equal, and hence
(?)x=(:)y
Consider also the expressions obtained when
z
=
z(x,
y)
and
x
and
y
are
themselves related
to additional variables
u
and v, such that
x
=
x(u,
v) and
y
=
y(u,
v). Then
Let
z =
v, and
u
=
x,
then
x
=
x(z)
and
(2)
V
=0,
and
($)v=l
Hence
giving
(6.7)
These expressions will now be used to consider relationships derived previously. The
du=Tdspdv
(6.10)
dh
=
T ds
+
v dp
(6.1 1)
df
=
p
dv

s
dT
(6.12)
dg
=

s
(6.13)
following functional relationships have already been obtained
Consider the expression for du, given in eqn
(6.10)
then, by analogy with eqn
(6.3)
T
=
(
$)v,
p
=
(%),
and
(5).
=
(
$)"
(6.14)
In a similar manner the following relationships can be obtained,
for constant composition
or single component systems
T=($);
($)s=($)
P
P=(:);
s=($;
($)v=($)T
v=($);
.=($);
($)p=($)T
(6.15)
(6.16)
(6.17)
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View Full Document The Maxwell relationships
103
If the pairs of relationships for
T
are equated then
(z)v=($)p
and, shilarly
In addition to these equivalences, eqns (6.14) to (6.17) also show that
($l=
($)p
=
MT
Equations (6.19) are called the
Maxwell relationships.
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This note was uploaded on 03/09/2010 for the course MECHANICAL ME9802701 taught by Professor Prof.william during the Spring '10 term at Institut Teknologi Bandung.
 Spring '10
 Prof.William

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