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VI. Adiabatic Transformations
VI.
Adiabatic Transformations
If a transformation of a closed system occurs in which there is no exchange of heat with the outside world, we have
dQ = 0
(6.1)
and the transformation is called adiabatic. In this case, the two laws of thermodynamics (3.7) and (3.14) yield, together with (3.4) and (3.16), two forms of the equation for adiabatic transformations of a closed system; namely,
d:/f' - V'dp' - V" dp" - GdO = 0
(6.2)
T dS + L..,y "" A dm r + L..,y "" A "dm y " = 0 •
(6.3)
and I
I
y
y
Note that the total entropy of the system is an invariant of the system for every transformation which is adiabatic (dQ = 0) and reversible (Ay' = 0, A/ = 0), for (6.3) then becomes
dS = O.
(6.4)
We recall that
V' =
V" = "" L.., m y "» y " y
(6.5)
y
where the specific volumes vy' and v/, defined in (5.10), have the values given by (5.11). Also :/f" = L.., "" m y 'h.y'I
:/f''' = L.., "" m y"h.y "
y
(6.6)
y
where the specific enthalpies ky' and h;", defined in (4.18), have the values given by (5.11). 49
50
VI. ADIABATIC TRANSFORMATIONS
We deduce from (5.11) that 2PAT a In! ' PA a In! ' dh ' = dh '* - - y dT - - T2d y Y Y My aT My aT'
(6.7)
2PAT a In! // PA a In! // dh // = dh //* - - y dT - - T2d y • Y Y MY aT MY
er
But ah '* ah '* dh '* = --Y-dT + --Y-dp' Y aT ap" ah //* dh //* = -Y-dT Y aT
ah //*
+ -Y-d ap//
P
",
(6.8)
.
It is easily shown, starting with the formulas for hy'*, hy"*, and vy'*, vy"* given by Prigogine and Defay (1954, Table 7.2), that equations (6.8) may be written dh Y'*
=
'*)
Y- d'l1' '* dT + ( v Y'* - Tav -aT .r:»
C Y
dh //* = c //*dT Y
Y
+
(v
av //* //* T -Y-)d'l1// Y aT r'
(6.9)
where cY'* and cy"* are the standard specific heats of component y in phases' and ", Since the system is closed, we have (6.10)
The first law (6.2) is then written, with the use of (2.6), (4.21), (4.38), and (6.6),
L: my' dhy'
+
Y
+
L: my" dhy"
+
Cpa dT - V'dp' - V//dp//
Y
bdO,
+
L: (liy' Y
hya)dm y'
+
L: (liy" -
hya)dmy" = O.
(6.11)
Y
Then, according to (5.11), (6.5), (6.7), and (6.9), Eq. (6.11) takes the following form:
51
VI. ADIABATIC TRANSFORMATIONS
"" y m y 'e '* (~ y
+~ "" y m He yH* + Qep a)dT + bdQ y
8v '* 8v H* - T"" m ,--y-dp' - T"" m H_y_d H y 8T . : y 8T P
c:y
y
' 2 m ' 8 Inf + ~T2 2 m ' 8 8T 8T2
[2~T
-y y My
21nf '] y
---.?:'.-
y
y
mH81nfH _ [ 2~T2-y y y My 8T
dT
My
"821fH + ~T2 2 m y n y ]dT y
8T2
My
m ' 8 In!y ' + ~T22-y m ' 82 In!y '] dp' - [ ~T2-y 8p'
y My
y
(6.12)
8p'8T
My
- [~T
" "] 2 m " 8 Inf + ~T2 2 m " 8 Inf H8T dpH 8p" 8p
- ~T2
22
2
-yY My
i
y
y
-y y My
Y
m ' 821nf' ---.?:'.- --1'.-dN/ My
8T8Ni '
m 82 1 f " _ ~T2 """" _ y n y dN H ~~ M y 8T8N" i i i y H
+ ~y "" (Ii '
- h y a)dm y '
+ "" (Ii " ~y
y
- h r a)dm y H = 0 ,
y
where i takes the values 2, 3, ... , c, one of the mole fractions being dependent on the others. Notice first that (Prigogine and Defay, 1954, Chapter XXI, Section 6), according to the Gibbs-Duhem equation, "" my' 8 In!r' ~M 8N' y y i ""
m" 8 Inf
0
"" Y ~M y y
'
m ' 821nf ' y
~M y
=
y
y-O
8T8N' i
,
8Ni
H
Y
H
=0,
m H8 21nfH "" -yy - 0 ~M 8T8N H . Y
Y
i
(6.13)
52
VI. ADIABATIC TRANSFORMATIONS
We write the first equation of (6.5), using (5.11), V' = '" m 'v '* + '" m ,fJiT a lnfy' .y: y y ~ yM y y o'P'
(6.14)
where the first sum gives the value which the volume would have if the mixture of components in the first phase were ideal. The second sum is called the excess volume of this mixture (cf. Prigogine and Defay, 1954, Chapter XXIV, Section 1). The excess volumes of the first and second phases are thus defined by the relations
,
Ve = fJiT
2: -my' ----'a lnfy' y
My
m " aIn!" V" = fJiT '" -yy . e ~ M y a'P" y
a" 'P
(6.15)
Analogous considerations lead us to define the excess heat capacities by the relations my' a lnfy' my' a2lnfy' Ope' = - 2fJiT - fJiT2 ----'y M y aT y M y aT2' (6.16) ",my" alnfy" ",my" a2 In f y" Ope" = - 2fJiT ~ - fJiT2 ~ -----'--y My aT y My aT2
2:
2:
Equations (4.21), (6.13), (6.15), and (6.16), yield the following form for Eq. (6.12): '" y m y 'c '* (~ y
+
+ '" m
"c ,,*
~yy
+
.Qcp a
+ 0 pe '+ 0 pe")dT
y
av '* b d.Q - T('" my'--y_
.: y
er
aV') + __ e dp' aT
a"* + __ av")dp" - T('"c: m"~ aT aT y
+
y
"'(h y"+ ~ m"rx y"- ha)dm y y"= ~ y
(6.17)
e
0.
53
VI. ADIABATIC TRANSFORMATIONS
Arguing as above, it can be shown that Eq. (6.3) expressing the second law can be written in the following form:
(~ m
+ ~I'I' ~ m "e ,,* +
'c '*
~I'I'
'Y
+ bdn
.: 'Y
~[T(8
~
Cpe ")dT
aV') L m / -avaTI'-'* + __ e dp' aT
- T(
_T(~m +
neP a + Cpe '+
'Y
'Y
av ,,* aT
"_'Y_ 'Y
'Y
+
aV")d er P"
_e
' + m'~fJ ' 'Y
8
a)
'Y
(6.18)
+ A I' 'Jdm 'Y'
'Y
"
+~ ~[T(8'Y +
~fJ m" " - 8 I' a) 'Y
+ A I'"]dm 'Y" =
0•
'Y
It follows immediately from (3.2), (3.4), (3.10), (3.27), (4.20), (4.21), and (4.24) that Eqs. (6.17) and (6.18) are identical. To obtain the equation for reversible transformations, we must set the Ay' and A/ equal to zero in (6.18). The fJl" and fJ/, as well as the rxy' and rx/, probably vanish at equilibrium, as do the 'Ay' and 'Ay", but this has not yet been proved.
Adiabatic Transformations
If a transformation of a closed system occurs in which there is no exchange of heat with the outside world, we have
dQ = 0
(6.1)
and the transformation is called adiabatic. In this case, the two laws of thermodynamics (3.7) and (3.14) yield, together with (3.4) and (3.16), two forms of the equation for adiabatic transformations of a closed system; namely,
d:/f' - V'dp' - V" dp" - GdO = 0
(6.2)
T dS + L..,y "" A dm r + L..,y "" A "dm y " = 0 •
(6.3)
and I
I
y
y
Note that the total entropy of the system is an invariant of the system for every transformation which is adiabatic (dQ = 0) and reversible (Ay' = 0, A/ = 0), for (6.3) then becomes
dS = O.
(6.4)
We recall that
V' =
V" = "" L.., m y "» y " y
(6.5)
y
where the specific volumes vy' and v/, defined in (5.10), have the values given by (5.11). Also :/f" = L.., "" m y 'h.y'I
:/f''' = L.., "" m y"h.y "
y
(6.6)
y
where the specific enthalpies ky' and h;", defined in (4.18), have the values given by (5.11). 49
50
VI. ADIABATIC TRANSFORMATIONS
We deduce from (5.11) that 2PAT a In! ' PA a In! ' dh ' = dh '* - - y dT - - T2d y Y Y My aT My aT'
(6.7)
2PAT a In! // PA a In! // dh // = dh //* - - y dT - - T2d y • Y Y MY aT MY
er
But ah '* ah '* dh '* = --Y-dT + --Y-dp' Y aT ap" ah //* dh //* = -Y-dT Y aT
ah //*
+ -Y-d ap//
P
",
(6.8)
.
It is easily shown, starting with the formulas for hy'*, hy"*, and vy'*, vy"* given by Prigogine and Defay (1954, Table 7.2), that equations (6.8) may be written dh Y'*
=
'*)
Y- d'l1' '* dT + ( v Y'* - Tav -aT .r:»
C Y
dh //* = c //*dT Y
Y
+
(v
av //* //* T -Y-)d'l1// Y aT r'
(6.9)
where cY'* and cy"* are the standard specific heats of component y in phases' and ", Since the system is closed, we have (6.10)
The first law (6.2) is then written, with the use of (2.6), (4.21), (4.38), and (6.6),
L: my' dhy'
+
Y
+
L: my" dhy"
+
Cpa dT - V'dp' - V//dp//
Y
bdO,
+
L: (liy' Y
hya)dm y'
+
L: (liy" -
hya)dmy" = O.
(6.11)
Y
Then, according to (5.11), (6.5), (6.7), and (6.9), Eq. (6.11) takes the following form:
51
VI. ADIABATIC TRANSFORMATIONS
"" y m y 'e '* (~ y
+~ "" y m He yH* + Qep a)dT + bdQ y
8v '* 8v H* - T"" m ,--y-dp' - T"" m H_y_d H y 8T . : y 8T P
c:y
y
' 2 m ' 8 Inf + ~T2 2 m ' 8 8T 8T2
[2~T
-y y My
21nf '] y
---.?:'.-
y
y
mH81nfH _ [ 2~T2-y y y My 8T
dT
My
"821fH + ~T2 2 m y n y ]dT y
8T2
My
m ' 8 In!y ' + ~T22-y m ' 82 In!y '] dp' - [ ~T2-y 8p'
y My
y
(6.12)
8p'8T
My
- [~T
" "] 2 m " 8 Inf + ~T2 2 m " 8 Inf H8T dpH 8p" 8p
- ~T2
22
2
-yY My
i
y
y
-y y My
Y
m ' 821nf' ---.?:'.- --1'.-dN/ My
8T8Ni '
m 82 1 f " _ ~T2 """" _ y n y dN H ~~ M y 8T8N" i i i y H
+ ~y "" (Ii '
- h y a)dm y '
+ "" (Ii " ~y
y
- h r a)dm y H = 0 ,
y
where i takes the values 2, 3, ... , c, one of the mole fractions being dependent on the others. Notice first that (Prigogine and Defay, 1954, Chapter XXI, Section 6), according to the Gibbs-Duhem equation, "" my' 8 In!r' ~M 8N' y y i ""
m" 8 Inf
0
"" Y ~M y y
'
m ' 821nf ' y
~M y
=
y
y-O
8T8N' i
,
8Ni
H
Y
H
=0,
m H8 21nfH "" -yy - 0 ~M 8T8N H . Y
Y
i
(6.13)
52
VI. ADIABATIC TRANSFORMATIONS
We write the first equation of (6.5), using (5.11), V' = '" m 'v '* + '" m ,fJiT a lnfy' .y: y y ~ yM y y o'P'
(6.14)
where the first sum gives the value which the volume would have if the mixture of components in the first phase were ideal. The second sum is called the excess volume of this mixture (cf. Prigogine and Defay, 1954, Chapter XXIV, Section 1). The excess volumes of the first and second phases are thus defined by the relations
,
Ve = fJiT
2: -my' ----'a lnfy' y
My
m " aIn!" V" = fJiT '" -yy . e ~ M y a'P" y
a" 'P
(6.15)
Analogous considerations lead us to define the excess heat capacities by the relations my' a lnfy' my' a2lnfy' Ope' = - 2fJiT - fJiT2 ----'y M y aT y M y aT2' (6.16) ",my" alnfy" ",my" a2 In f y" Ope" = - 2fJiT ~ - fJiT2 ~ -----'--y My aT y My aT2
2:
2:
Equations (4.21), (6.13), (6.15), and (6.16), yield the following form for Eq. (6.12): '" y m y 'c '* (~ y
+
+ '" m
"c ,,*
~yy
+
.Qcp a
+ 0 pe '+ 0 pe")dT
y
av '* b d.Q - T('" my'--y_
.: y
er
aV') + __ e dp' aT
a"* + __ av")dp" - T('"c: m"~ aT aT y
+
y
"'(h y"+ ~ m"rx y"- ha)dm y y"= ~ y
(6.17)
e
0.
53
VI. ADIABATIC TRANSFORMATIONS
Arguing as above, it can be shown that Eq. (6.3) expressing the second law can be written in the following form:
(~ m
+ ~I'I' ~ m "e ,,* +
'c '*
~I'I'
'Y
+ bdn
.: 'Y
~[T(8
~
Cpe ")dT
aV') L m / -avaTI'-'* + __ e dp' aT
- T(
_T(~m +
neP a + Cpe '+
'Y
'Y
av ,,* aT
"_'Y_ 'Y
'Y
+
aV")d er P"
_e
' + m'~fJ ' 'Y
8
a)
'Y
(6.18)
+ A I' 'Jdm 'Y'
'Y
"
+~ ~[T(8'Y +
~fJ m" " - 8 I' a) 'Y
+ A I'"]dm 'Y" =
0•
'Y
It follows immediately from (3.2), (3.4), (3.10), (3.27), (4.20), (4.21), and (4.24) that Eqs. (6.17) and (6.18) are identical. To obtain the equation for reversible transformations, we must set the Ay' and A/ equal to zero in (6.18). The fJl" and fJ/, as well as the rxy' and rx/, probably vanish at equilibrium, as do the 'Ay' and 'Ay", but this has not yet been proved.