This journal is © The Royal Society of Chemistry 2000
Supplementary data
Apparent molar isentropic compressions and expansions of solutions
Michael J. Blandamer, Michael I. Davis, Gérard Douhéret and João Carlos R. Reis
Department of Chemistry, University of Leicester, Leicester LE1 7RH, England. E-mail:
mjb@le.ac.uk
Department of Chemistry, University of Texas at El Paso, El Paso,Texas 79968-0513, USA. Email: mdavis@utep.edu
Laboratoire de Thermodynamique des Solutions et des Polymères, CNRS UPRESA 6003,
Université Blaise Pascal, F-63177 Aubière Cedex, France. E-mail: gerard.douheret@univbpclermont.fr
Departamento de Quimica e Bioquimica, Centro de Electroquimica e Cinetica, Faculdade de
Ciencias, Universidade de Lisboa, P-1749-016 Lisboa, Portugal. E-mail: jreis@fc.ul.pt
Appendix
This Appendix reports derivations of several key equations cited in the original review. If you
use these derivations (some of which have not been previously published), please make reference
to the Chem. Soc. Rev.
A Calculus Operations
In many of the derivations we use two standard operations in differential calculus. Consider a
dependent variable u defined by the independent variables x and y; u  u[x, y]
Then, du  (u / x) y  dx  (u / y) x  dy
At constant u, du = 0; then (u / x) y   (u / y) x  (y / x) u
And, (y / x) u  (u / x) y  (y / u) x
..... (a)
Consider the function, z  z[x, y] . Then dz  (z / x) y  dx  (z / y) x  dy
At constant u,
(z / x ) u  (z / x) y  (z / y) x  (y / x ) u
Then, using equation (a),
(z / x) u  (z / x) y  (u / x) y  (y / u) x  (z / y) x
..... (b)
Another important operation takes the following form.
(x / y ) z  (x / q ) z  (q / y ) z
..... (c)
B Maxwell Relationships
From U=U[S,V],
Then
 2 U / S  V   2 U / V  S
( T / V) S  ( p / S) V
We invert the latter equation. Hence, E S  (V / T) S   (S / p ) V
Similarly,  2 G / T  p   2 G / p  T .
Hence
E p  (V / T) p   (S / p) T
The equations for Ep and ES are Maxwell equations. Two other Maxwell relations are worthy of
note
(T / p ) S  (V / S) p ; ( p / T) V  ( S / V) T
C Gibbs-Helmholtz Equation
We combine H  G  T  S
and
S  (G / T) p
Hence, the Gibbs-Helmholtz equation, H  G  T  (G / T) p
..... (a)
Also (H / T) p  C p  T  ( 2 G / T 2 ) p  T  (S / T) p
..... (b)
-1-
Also starting from H  G  T  S .
Then, ( H / p) T  ( G / p) T  T  ( S / p) T
Hence, using the Maxwell equation in Appendix B,
(H / p) T  V  T  (V / T) p
..... (c)
Similarly, ( U / T) V  C V  T  ( S / T) V
And (U / V) T   p  T  (V / T) p  (p / V) T
D Difference between Isothermal and Isentropic Compressibilities
From Appendix A, (V / p) S  (V / p) T  (V / T) p  (T / p) S
Or, (V / p) S  (V / p) T  (V / T) p  (S / p) T  (T / S) p
Using Appendix B, (V / p) S  (V / p) T  (V / T) p  (V / T) p  (T / S) p
Hence, KS  K T  (T / C p )  [(V / T) p ]2  K T  T  ( E p ) 2 / C p
Or,  S   T  (T / C p  V)  [(V / T) p ]2
Further,  S   T  ( V  T   2p / C p )
..... (a)
By definition,    T  S and  = C p / V
Then from equation (a),   T  ( p ) 2 / 
E Heat Capacities
From Appendix A,
(S / T) V  (S / T) p  (S / p) T  (p / V) T  (V / T) p
But (Appendix B) (S / p) T  (V / T) p
Hence, (S / T) V  (S / T) p  (V / T) p  (p / V) T  (V / T) p
Then from Appendix C, C V  C p  T  ( E p ) 2 / K T
F Ratio of Compressions
From Appendix A, we obtain equations relating isothermal and isentropic dependencies of
volume on pressure.
Thus, (V / p) T  (T / p) V  (V / T) p  (T / p) V  (V / S) p  (S / T) p
And, (V / p) S  (S / p) V  (V / S) p  (T / p) V  (V / S) p  (S / T) V
Then, (V / p) T / (V / p) S  (S / T) p / (S / T) V
We use Appendix C. Hence, K T / K S  C p / C V
-2-
G Differences between Expansions and Expansibilities
Using the procedure described in Appendix A, we contrast expansions at constant pressure and
constant entropy.
(V / T) S  (V / T) p  (S / T) p  (p / S) T  (V / p) T
The inverse of a Maxwell relation is (Appendix B), (p / S) T  (T / V) p
But isothermal compression, K T  ( V / p) T
Then using Appendix C, the isentropic expansion, E S  E p  K T  C p / T  E p
..... (a)
By definition,   C p / V and  p  E p / V
Hence, dividing equation (a) by V,  S   p   T   / T   p
..... (b)
By definition,    p   S
..... (c)
Then from equation (b)    T   / T   p
..... (d)
H Isochoric Thermal Pressure Coefficient
By definition,  V  (p / T) V
..... (a)
Then (Appendix A)  V  (V / T) p / (V / p) T
Or
V  E p / K T   p / T
..... (b)
From equation (a) using a Maxwell Relation, (Appendix B),
 V  ( S / V) T  ( S / T) V / ( V / T) S . But, ( S / T) V  C V / T
Then, V  CV / T  ES  CV / T  V  S
..... (c)
From equations (b) and (c), E p / K T  C V / T  E S
Or,  p /  T  C V / T  V   S
I Isentropic Thermal Pressure Coefficient
By definition,  S  (p / T) S
..... (a)
From equation (a) and equation (a) of Appendix A,
 S  (S / T) p / (S / p) T
From Appendices B and C,  S  C p / T  E p  ( C p / V) / (T  E p / V)
Then,  S   / T   p
..... (b)
Using equation (a) multiplied by (V / V) S
 S  ( p / V) S  ( V / T) S
or  S   E S / K S  ( E S / V) / ( K S / V)   S /  S
-3-
..... (c)
Then from above, E S / K S   C p / T  E p , or  S /  S   / T   p
J
..... (d)
L’Hospital’s Rule
In several important cases, analysis of the thermodynamic properties of solutions (and liquid
mixtures) requires consideration of a term having the general form, x  ln(x) where x is an
intensive composition variable; e.g. molality or mole fraction. The accompanying analysis
requires comment on the product in the limit that x tends to zero. But limit ( x  0) ln( x )   .
The key question is: does x tend to zero faster than ln(x) approaches 'minus infinity'? If the
answer is 'no', then the accompanying thermodynamic analysis has to take account of this feature.
In fact most accounts assume, without comment, that the answer to the above question is 'yes'.
Confirmation that the latter statement is correct emerges from application of L’Hospital’s Rule
(G.F.A. de l’Hospital, 1661-1704, marquis de Saint-Mesme). This rule allows the evaluation of
terms having indeterminate forms. In fact most applications of this method usually involve the
ratio of two terms, each term being a function of x.
If f(x)/F(x) approaches one of the forms 0/0 or / when x approaches a, and f ( x ) / F( x )
[where f ( x ) and F( x ) are the derivatives of f(x) and F(x)] approaches a limit as x approaches a,
then f(x)/F(x) approaches the same limit.
Example: If f(x) = x2 1 and F(x) = x1,
then
f ( x ) 2  x

F( x )
1
Then limit(x1) f ( x ) / F( x )  2
2
Then limit ( x  1) x  1  2
x 1
This rule can be proved using three assumptions.
(i)
In the neighbourhood of x = a, F(x)  0, if x  a.
(ii)
f(x) and F(x) are continuous at the neighbourhood of x = a except perhaps at a.
(iii)
f ( x ) and F( x ) exist in some neighbourhood of x=a ( except perhaps at x=a) and do
not vanish simultaneously for x  a.
In the present case the term under consideration has a different form.
With reference to the term, x  ln( x) , f ( x )  ln( x ) and F(x) = 1/x.
-4-
Then f (x)  1 / x and F(x) = 1/x 2 . Hence, f ( x ) /F(x) =  x
Thus limit ( x  0)f ( x ) / F(x) = 0 . Hence, limit ( x  0) x  ln(x) = 0
K Apparent Molar Volumes
The starting point for an examination of the volumetric properties of aqueous solutions is the
density aq  of a solution having known composition and the density of the solvent 1* l at the
same temperature and presure.
1. Molality Scale
For a solution prepared using 1 kg of solvent, water, (at fixed T and p)
V(aq; w1  1 kg)  (1 / M1 )  V1* (l)  m j  (Vj )
(aq )  (1  M j  m j ) / V(aq; w 1  1 kg)
Density
Then
1
1
 [1  M j  m j ]  *  m j  ( Vj )
(aq )
 1 ( l)
or
Mj  mj
1
1

 *  m j  ( Vj )
(aq )
(aq )
1 ( l)
Also
Then
1 mj  M j
(aq )

*
1 (l) 1  m j  1* (l)  ( Vj )
( Vj )  [ m j  (aq)  1* (l)]1  [1* (l)  1 (aq)]  M j  [(aq)]1
2. Concentration Scale
From Appendix L, 1 / m j  ( / c j )  M j , and
Using the conversion between mj and cj ,
or, ( Vj ) 
1
1
1

 *  ( Vj )
c j m j 1 (l)

1
1  (aq )
 * 
 M j   ( Vj )
c j 1 (l)  c j

1  (aq )  M j
 1  *   *
cj 
1 ( l)  1 ( l)
Hence, ( Vj )  [c j  1* (l)]1  [1* (l)  (aq)]  M j  [1* (l)]1
3. For a solution at fixed T and p, prepared using nj moles of solute j and n1 moles of water(l),
V aq   n1  V1* (l)  n j  ( V j )
Then at fixed T, p and n1 ,
 V 


  ( Vj )  n j
 n j  n (1)
-5-
 ( Vj ) 

 
 n j  n (1)
Vj (aq )  (V / n j ) n (1)
But
Hence, Vj (aq)  ( Vj )  n j  [( Vj ) / n j ] n (1)
L Conversions; Molality and Concentration
(1) A given aqueous solution has volume V and density (aq). The solution contains nj moles of
solute, chemical substance j, having molar mass Mj .
Mass of system, w  V   . Mass of solvent  V    n j  M j
Molality of solute, m j 
nj
V  n j  M j
or,
m j  [( V / n j )    M j ] 1
But concentration cj = nj/V. Then m j  [ / c j  M j ] 1 or 1 m j  ( / c j )  M j
A common approximation for dilute solutions assumes that  / c j  M j .
Then
c j   mj
(2) An elegant conversion between mj and cj incorporates the apparent molar volume of the
solute-j. Since V(aq )  n1  V1* (l)  n j  ( V j ) ,
then
n M
V(aq )
1
1

 ( V j )
 1 * 1  ( V j ) or,
c j m j  1* (l)
nj
n j  1 (l)
M Partial and Apparent Molar Volumes; Molalities
For a solution, m j  n j / w 1 . Then dn j  w 1  dm j
From Appendix K, Vj (aq)  (Vj )  m j  w1  [(Vj ) /  ( w1  m j )]
Hence, Vj (aq)  ( Vj )  m j  [( Vj ) / m j ]T,p
N Apparent Molar Isobaric Expansions from Molalities and (Vj)
The volume of an aqueous solution is related to the apparent molar volume of solute-j.
V(aq)  n1  V1* (l)  n j  (Vj )
 ( V j ) 
By definition ( E pj )  

 T  p
 V1* ( l) 
 V( aq ) 
Then at constant pressure, 
  n j  ( E pj )
  n1  
 T  p
 T 
p
Hence, following division by V(aq)
-6-
nj
n1
V * ( l)  V * ( l) 
1
 V( aq ) 

 1*   1  
 ( E pj )
 
V( aq )  T  p V( aq ) V1 ( l)  T  p V( aq )
nj
n 1  V1* ( l) *
or,  p ( aq) 
  p 1 ( l) 
 ( E pj )
V( aq)
V( aq)
The latter equation contains the (isobaric equilibrium) expansivities of the solution and the
solvent,  p ( aq) and  *p1 ( l) respectively together with the apparent molar isobaric expansion
(Epj ). Thus (Epj ) is expressed in m3 K 1 mol 1 .
Then
V( aq)   p ( aq)  n1  V1* ( l)   *p1 ( l)  n j  ( E pj )
But V(aq)  n1  V1* (l)  n j  (Vj )
Hence,  p ( aq) [ n1  V1* ( l)  n j  ( Vj )]  n1  V1* ( l)   *p1 ( l)  n j  ( E pj )
or
( E pj ) 
 p ( aq )  n 1  M 1
n j   1* ( l)

n 1  M 1   *p1 ( l)
n j   1* ( l)
  p ( aq )  ( Vj )
Hence, ( E pj )  [ m j  1* ( l)] 1  [ p ( aq)   *p1 ( l)]   p ( aq)  ( Vj )
O Apparent Molar Isobaric Expansions from Molalities
We start with the equation derived in Appendix N and substitute for (Vj ) .
 V(aq )  n 1  V1* (l) 
Hence, ( E pj )  [ m j  1* (l)] 1  [ p (aq )   *p1 (l)]   p (aq )  

nj


or
or
1
n  M1 
( E pj )  [ m j   1* ( l)] 1  [ p ( aq )   *p1 ( l)]   p ( aq )    *1

 c j  1 ( l)  n j 
(E pj)  [m j  1* (l)]1  [ p (aq)  *p1 (l)]
  p (aq)
 p (aq)  M j
 p (aq) 



 *
(aq)
1 (l)  m j 
 m j  (aq)
or
( E pj ) 
 p (aq )
m j   ( l)
*
1

 *p1 (l)
m j   ( l)
*
1

 p (aq )
m j  (aq )

 p (aq )  M j

(aq )
Then

( E pj )  [ m j  (aq )  1* (l)]1   p (aq )  1* (l)   *p1 (l)  (aq )
  p (aq )  M j  [(aq )] 1
-7-

 p (aq )
1* (l)  m j
P Apparent Molar Isobaric Expansions from Concentrations
(aq ) 
For a given aqueous solution,
or
Then
V(aq )  (aq)  n j  M j
M1
n1  M1  n j  M j
V(aq )
 n1
 V(aq )  (aq )  n j  M j 
V(aq )   p (aq )  V1* (l)   *p1 (l)  
  n j  ( E pj )
M1


or
( E pj ) 
V(aq )   p (aq )
nj
 p ( aq )
 *p1 ( l)  ( aq )
M1  n j
+
V1* (l)   *p1 (l)  M j
M1
 *p1 ( l)  M j
Then
( E pj ) 
Hence
( E pj )  [c j  1* (l)]1  [ p (aq)  1* (l)   *p1 (l)  (aq)] +  *p1 (l)  M j  [1* (l)]1
Thus
(Epj ) is related to cj and p(aq).
cj


V1* (l)   *p1 (l)  V(aq )  (aq )
 1* ( l)  c j
+
 1* ( l)
Q Apparent Molar Isobaric Expansions from Concentrations and (Vj)
We start with the equation for (Epj ) in terms of (Vj ) and mj ; Appendix N.
( E pj )  [ m j  1* ( l)] 1  [ p ( aq)   *p1 ( l)]   p ( aq)  ( Vj )
Then using conversion (2) from Appendix L,
 1* ( l)

( E pj )  
 1* ( l)  ( Vj )  [1* ( l)] 1  [ p ( aq)   *p1 ( l)]   p ( aq)  ( Vj )
 c j

Hence
( E pj )  (c j ) 1  [ p (aq)   *p1 (l)]   p (aq)  ( Vj )  ( Vj )   *p1 (l)   p (aq)  ( Vj )
or
( E pj )  (c j ) 1  [ p (aq)   *p1 (l)]   *p1 (l)  ( Vj )
R Apparent Molar Isentropic Expansions
From,
(E Sj ; def ) 
E s (aq; w1  1 kg)  (1/M 1 )  E*S1 (l)
mj
( E Sj ; def ) 
E s (aq; w 1  1 kg)
1

 E *S1 (l)
mj
M1  m j
-8-
or
(E Sj ; def ) 
 S (aq)  V(aq; w1  1 kg)
1

  *S1 (l)  V1* (l)
mj
M1  m j
or
(E Sj ; def ) 
 S (aq)
1
 [(1 / M 1 )  V1* (l)  m j  (Vj )] 
  *S1 (l)  V1* (l)
mj
M1  m j
or
( E Sj ; def ) 
V1* (l)
 [ S (aq )   *S1 (l)]   S (aq )  (Vj )
m j  M1
or
( E Sj ; def )  [ m j  1* (l)]1  [ S (aq)   *S1 (l)]   S (aq)  ( Vj )
From Appendix L,
Then
1
1

  (V j )
*
m j   1 (l ) c j
1

 (E Sj ; def )    (Vj )  [ S (aq )   *S1 (l )]   S (aq )   (Vj )
 c j

( E Sj ; def )  [c j ]1  [ S (aq)   *S1 (l)]  ( Vj )   S (aq)  ( Vj )   *S1 (l)   S (aq)  ( Vj )
Hence
( E Sj ; def )  [c j ]1  [ S (aq)   *S1 (l)]   *S1 (l)  ( Vj )
Thus ( E Sj ; def ) is expressed in m3 K1 mol1.
S Apparent Molar Isothermal Compressions
From Appendix K, V(aq)  n1  V1* (l)  n j  (Vj )
..... (a)
Then the isothermal differential dependence of volume V(aq) on pressure is given by the
following equation.
 (Vj ) 
 V1* (l) 
 V(aq) 

n


  nj 


1 
 p  T
 p  T
 p  T
By definition the apparent molar (isothermal) compression of the solute,
 (Vj ) 
(K Tj )  

 p  T
For the solution the (equilibrium ) isothermal compressibility,
 T (aq )  
 V(aq ) 
1


V(aq )  p  T
1  V1* (l) 
Similarly for the solvent,  (l)   *  

V1 (l)  p  T
*
T1
-9-
Hence V(aq)   T (aq)  n1  V1* (l)   *T1 (l)  n j  (K Tj )
(1) Using equation (a),
[n1  V1* (l)  n j  (Vj )]   T (aq)  n1  V1* (l)  *T1 (l)  n j  (K Tj )
n 1  V1* (l)   T (aq ) n 1  V1* (l) *

  T1 (l)  (Vj )   T (aq )
nj
nj
Hence
(K Tj ) 
or
(K Tj )  [m j  1* (l)]1  [ T (aq)   *T1 (l)]   T (aq)  (Vj )
(2) From the latter equation, we substitute for (Vj ) .
 V  n 1  V1* (l) 
Hence, (K Tj )  [m j  1* (l)] 1  [ T (aq )   *T1 (l)]   T (aq )  

nj


or
1
n  M1 
(K Tj )  [m j  1* (l)] 1  [ T (aq )   *T1 (l)]   T (aq )    *1

 c j 1 (l)  n j 
or
  (aq )
 T (aq )  M j
 (aq ) 
( K Tj )  [ m j  1* (l)]1  [ T (aq )   *T1 (l)]   T

 *T

(aq )
1 (l)  m j 
 m j  (aq )
 T (aq)
(K Tj ) 
Then
(K Tj )  [m j  (aq )  1* (l)]1  [  T (aq )  1* (l)  *T1 (l)  (aq)]
m j  1* (l)

 T (aq)  M j
 T (aq)
 (aq)

 *T
m j  (aq)
(aq )
1 (l)  m j
or
m j  1* (l)

*T1 (l)
  T (aq )  M j  [(aq )]1
(3) From the equation,
V(aq)   T (aq)  n1  V1* (l)   *T1 (l)  n j  (K Tj )
But in terms of the mass of the solution,
V(aq )  (aq )  n 1  M1  n j  M j
or, n 1  [V(aq)  (aq)  n j  M j ] / M1
Then
 V(aq )  (aq )  n j  M j 
V(aq )   T (aq )  V1* (l)   *T1 (l)  
  n j  ( K Tj )
M1


With c j  n j / V(aq)
 (aq ) V1* (l)   *T1 (l)  (aq ) V1 (l)   T1 (l)  M j
(K Tj )  T


cj
c j  M1
M1
*
or
Then
(K Tj ) 
*
*
 T (aq )  *T1 (l)  (aq )  T1 (l)  M j


cj
c j  1* (l)
1* (l)
(K Tj )  [c j  1* (l)]1  [  T (aq)  1* (l)  *T1 (l)  (aq)]  *T1 (l)  M j  [1* (l)]1
(4) From ( KTj )  [ m j  1* (l)]1  [ T (aq)   *T1 (l)]   T (aq)  ( Vj )
- 10 -
We substitute for (1/mj ).
 1* (l)

(K Tj )  
 1* (l)  (Vj )  [1* (l)]1  [  T (aq )  *T1 (l)]  (Vj )   T (aq )
 c j

( K Tj ) 
or
1
 [ T (aq)   *T1 (l)]  ( Vj )   T (aq)  ( Vj )   *T1 (l)  ( Vj )   T (aq)
cj
( KTj )  [c j ]1  [ T (aq)   *T1 (l)]   *T1 (l)  ( Vj )
Then (KTj ) is expressed in m3 Pa1 mol1.
T Apparent Molar Properties: Collective Expressions
A common pattern emerges from the equations in Appendices N to S relating apparent molar
properties of solutes to the solute molality or concentration. This pattern suggests that similar
equations can be written relating apparent molar properties (Qj), and Q, (an extensive property
equalling Ep, ES, KT, KS or Cp) using the corresponding volume intensive property q = Q/V with
respect to q = p, S, T, S and . We will show that the following equations are the practical
equivalent forms of the equation used to define an apparent molar property.
(Q j )  (q  q 1* )  ( m j  1* ) 1  q  ( Vj )
..... (a)
(Q j )  (q  q 1* )  (c j ) 1  q 1*  ( Vj )
..... (b)
(Q j )  (q  1*  q 1*  )  ( m j    1* ) 1  q  M j  1
..... (c)
(Q j )  (q  1*  q 1*  )  (c j  1* ) 1  q 1*  M j  (1* ) 1
..... (d)
At the outset (Qj) is defined by the following basic equation.
Q  n1  Q1*  n j  (Q j )
Then we shift to volume intensive properties
V  q  n1  V1*  q 1*  n j  (Q j )
..... (e)
We express the volume using the equation, V  n1  V1*  n j  (Vj ) and solve equation (e) for
(Qj).
n 1  V1*  q
n 1  V1*  q 1*
 q  ( Vj ) 
Hence, (Q j ) 
nj
nj
- 11 -
But
n1  V1* / n j  (m j  1* ) 1 . Then equation (f) is obtained.
(Q j )  (q  q 1* )  ( m j  1* ) 1  q  ( Vj )
..... (f)
In the latter equation molalities are converted to concentrations using the following equation
(Appendix L); ( m j  1* ) 1  (c j ) 1  ( Vj ) .
Then
(Q j )  (q  q 1* )  (c j ) 1  ( Vj )  (q  q 1* )  q  ( Vj )
Or
(Q j )  (q  q 1* )  (c j ) 1  q 1*  ( Vj )
..... (g)
We return to equation (e) and express the volume using the following equation.
V
n1  M 1  n j  M j

Then we solve for (Qj).
n 1  M 1  q M j  q n 1  V1*  q 1*
( Q j ) 


nj 

nj
But
n1  M1 / n j  ( m j ) 1 and n1  V1* / n j  (m j  1* ) 1
Then
( Q j ) 
Or
(Q j )  (q  1*  q 1*  )  ( m j    1* ) 1  q  M j  1
Mj q
q 1*
q


mj 

m j  1*
..... (h)
For converting equation (h) as an equation using concentrations, we use (Appendix L).
1/ mj   / c j  M j
Thus
*
*
q  1*  q 1*   M j  (q  1  q 1  ) q  M j
( Q j ) 



c j  1*
1*  
Or
(Q j )  (q  1*  q 1*  )  (c j  1* ) 1  q 1*  M j  (1* ) 1
..... (i)
Since the equations used for converting molalities to concentrations are exact (Appendix L), no
approximations are involved. Therefore equations (a) to (i) are rigorously equivalent.
- 12 -
U Desnoyers-Philip Equation
From Appendix D,    T   S  T  ( p ) 2 / 
Hence, for an aqueous solution,  (aq )  T  [ p (aq )] 2 / (aq )
..... (a)
For water at the same T and p,  1* ( l)  T  [ *p1 ( l)]2 /  1* ( l)
..... (b)
We formulate an equation for the difference (aq)  1* (l)
(aq)  1* (l)  T  [ p (aq)]2 / (aq)  T  [ *p1 (l)]2 / 1* (l)]
..... (c)
We add and subtract the same term. With some slight reorganisation,
(aq )  1* (l) T  [ p (aq )]2 / (aq ) T  [ p*1 (l)]2 / 1* (aq )
 T  [ p*1 (l)]2 / 1* (l)  T  [ p*1 (l)]2 / (aq )
(aq )  1* (l) T  [(aq)]1  {[  p (aq )]2  [ p*1 (l)]2 }


 T  [ p*1 (l)]2  *1  1 
  p1 (l)  p (aq ) 
We identify t he term {[  p (aq )]2  [ p*1 (l)]2 } as ' a square minus a square'.
(aq )  1* (l) T  [(aq)]1  [ p (aq )   p*1 (l)]  [ p (aq )   p*1 (l)]

T  [ p*1 (l)]2
 p (aq )   p*1 (l)
 [(aq)  1* (l)]
..... (d)
We use equations (a) for (aq) and (b) for 1* (l) to remove explicit reference to temperature in
equation (d).
(aq)  1* (l)  (aq)  [ p (aq)]2  [ p (aq)  *p1 (l)]  [ p  *p1 (l)]
 1* (l)  [(aq)]1  [(aq)   1* (l)]
..... (e)
Returning to the required proof, we insert equation (e) for the difference (aq)  1* (l) into
equation (d).
Thus
(K Tj )  (K Sj ; def )

(aq)  [ p (aq)   p*1 (l)] [ p (aq)   p*1 (l)]

cj
[ p (aq)]2

..... (f)
1* (l) [(aq)  1* (l)]

 (Vj )  1* (l)
(aq)
cj
But we showed (Appendix Q),
( E pj )  [c j ]1  [ p (aq)  1* (l)]   *p1 (l)  ( Vj )
- 13 -
..... (g)
We identify the difference [ p (aq)  1* (l)].
Then
( E pj )   *p1 (l)  ( Vj )  [c j ]1  [ p (aq)   *p1 (l)]
..... (h)
Similarly
(C pj)  1* (l)  (Vj )  [c j ]1  [(aq )  1* (l)]
..... (i)
Then from equations (f), (g), (h) and (i),
( K Tj )  ( K Sj ; def ) 
(aq )  [ p (aq )   *p1 (l)]
[ p (aq )]
2
 [( E pj )   *p1 (l)  ( Vj )]
1* (l)

 [(Cpj)  1* (l)  (Vj )]  (Vj )  1* (l)
(aq)
..... (j)
We collect the (Vj) terms.
( K Tj )  ( K Sj ; def ) 
 *p1 (l) 
 1* (l)
(aq ) 


 1 


(
E
)

 (C pj )

pj
 p (aq ) 

(
aq
)

(
aq
)

p


*
*
2


1* (l)  1* (l)
 (aq)   p1 (l) (aq)  [ p1 (l)]
* 
 




(
l
)
  (Vj )
1
2
 p (aq)
(aq)
[

(
aq
)]


p


..... (k)
We note that in the second {----} bracket, the product term of ( Vj ), by using equations (a) for
( aq ) and (b) for 1* (l) the second and third terms are together equal to zero.
Hence
 *p1 (l) 
 1* (l)
(aq ) 


( K Tj )  ( K Sj ; def ) 
 1 
 (C pj )
  ( E pj ) 
 p (aq ) 
(aq )
  p (aq ) 


(aq )   *p1 (l) 


  1* (l) 
   ( Vj )

(
aq
)


p


..... (l)
The latter is the full Desnoyers–Philip Equation .
But limit (c j  0) p (aq)  *p1 (l), (E pj)  (E pj)  , (C pj)  (C pj)  , (aq)  1* (l)
and (aq )   1* (l).
Hence


(K Tj )  (K Sj ; def ) 

 2  (E pj)
1* (l)  
*

  p1 (l)
- 14 -

(C pj)  

 *

1 (l) 

V Equation for the difference, (Epj )  (ES j; def )
From Appendix G for a solution,    p   S   T   / T   p
..... (a)
In order to simplify the algebra we omit (aq) and (l) when describing to the properties of an
aqueous solution and the pure liquid respectively. Superscript '*' identifies the pure solvent.
*  *p  *S  *T  * / T  *p
Hence,   * 
..... (b)
  [     *  * ]  *  [  * ]
T
T
p
p
T  
p
..... (c)
The latter equation is effectively an identity. According to equation (c)
  *   
  *  *  *  *  *
T   T
p p
..... (d)
We use equations (a) and (b) in the second and fourth terms on the rhs of the latter equation.
*  *p  T  

*
*
*
 
   T  p   
T  
T  
*
or
..... (e)
  *    *
Further, as an identity,
 T     *T   *    ( T   *T )   *T  (   * )
..... (f)
Then from equation (c)
   *T

*
*
 
 [ T   T ] 
 (   * ) 
 [ p   *p ]
T
T 
p
*
 S   *S
  *S   ( Vj )
But (Appendix R)  (E Sj ; def ) 
cj
..... (g)
..... (h)
The analogue for (Epj) is the following equation (Appendix Q).
 (E pj ) 
Hence
 p   *p
cj
  *p   (Vj )
 (E pj )   (E Sj ; def ) 
  *
  *   (Vj )
cj
..... (i)
From equation (g), dividing by cj ,
*
   *    1  [    * ]     T  1  [   * ]
T
T
cj
T c j
T   c j

 1
  [ p   *p ]
p c j
..... (j)
- 15 -
But from equation (i)
  *
  (E pj )   (E Sj ; def )   *   (Vj )
cj
..... (k)
Equations having similar form for (  T  *T ), (  * ) and ( p  *p ) are readily generated
(Appendix T).
Hence
  *T
(E pj)  (ESj ; def )    [(K Tj )  *T  (Vj )] 
 [(Cpj)  *  (Vj )]
T
T  
*
   [(E pj)  *p  (Vj )]  *  (Vj )
p
..... (l)
Therefore
*
  *T
(E pj)  (E Sj ; def )     (E pj)    (K Tj ) 
 (C pj)
p
T
T  
   *    * 

p
T
  *  1 

 1  *   (Vj )
 T   
   p 


..... (m)
In the limit of infinite dilution,
(E pj)   (ESj ; def ) 
1* (l)

(E pj) 
*p1(l)

(K Tj ) 
*T1(l)

(C pj) 
1* (l)
W Equation relating (ES j; def ) to (K S j; def )
Again we simplify the algebra by omitting the descriptors (aq) and (l) in the following equations.
The starting point is the following equation.
 S  *S 
S
*
 ( S    *S  * )  S  ( p  *p )
S  
p
..... (a)
The latter equation is effectively an identity. Thus from equation (a)
 S  *S
S
*S *
*
*
*
 S 
 S     S 

S  
p p
But from equation(d) in Appendix I.
 S   S   / T   p and  *p   *S   *S   * / T
Then from (b) ,
- 16 -
..... (b)
S  
 *S   *
*
*
S    S   
 S   
T  p  S  
p T
*
S
or
*
S
 S   *S   S   *S
But as an identity, S    *S  *    ( S  *S )  *S  (  * )
..... (c)
Then from equations (a) and (c),
 S   *S 
S
  *

 ( S   *S )  S S  (   * )  S  ( p   *p )
S
S  
p
But (Appendix R), ( E Sj ; def )  [c j ]1  ( S   *S )   *S  ( Vj )
Hence
( E Sj ; def ) 
S
  *
 [c j ]1  ( S   *S )  S S  [c j ] 1  (   * )
S
S  

 S  [c j ] 1  ( p   *p )   *S  ( Vj )
p
We note that for the isobaric heat capacities (Appendix T),
(C pj )  [c j ]1  (   * )   *  ( Vj )
Also
Hence
( KSj ; def )  [c j ]1  ( S   *S )   *S  ( Vj )
S
 S   *S
*
( E Sj ; def ) 
 [( K Sj ; def )   S  ( Vj )] 
 [(C pj )   *  ( Vj )]
S
S  

 S  [( E pj )   *p  ( Vj )]   *S  ( Vj )
p
Then with a little reorganisation,
( E Sj ; def )  
 *S

  *
 ( E pj )  S  ( K Sj ; def )  S S  (C pj )
p
S
S  

 *p
  *
* 
  *S  (1 
)  S S  (1  )   ( Vj )
p
S
 

Hence, in the limit of infinite dilution,
(ESj ; def ) 
*S1 (l)

(E pj) 
*p1 (l)

(K Sj ; def ) 
*S1 (l)

- 17 -
(C pj; def ) 
1* (l)
- 18 -