1. UNIT III
SOLUTION THERMODYNAMICS
LESSON 12:
PARTIAL MOLAR PROPERTIES
Objectives:
 ∂(nM ) 
THERMODYNAMICS OF BIOLOGICAL SYSTEMS
· To define partial molar properties M ≡  (1.1)
 ∂ni 
i
· To understand the concepts of chemical potential and
fugacity this equation defines partial molar property of species i in
· Study ideal and non-ideal solutions solution, where the generic symbol i may stand for the partial
· Analyse and apply various composition models molar internal energy , the partial molar enthalpy, the partial
molar entropy ,the partial molar Gibbs energy ,etc.It is a
We have learnt that the applications of chemical-engineering
response function, representing the change of total property nM
thermodynamics are often to systems wherein composition is a
due to addition at constant T and P of a differential amount of
primary variable. In the chemical, petroleum, and pharmaceutical
species i to a finite amount of solution.Comparison of
industries multicomponent gases or liquids commonly
Equations written for the Gibbs energy show that the chemical
undergo composition changes as the result of mixing and
potential and the partial molar Gibbs energy are identical; i.e.,
separation processes, the transfer of species from one phase to
another, or chemical reaction. Because the properties of such µ i ≡ Gi (1.2)
systems depend strongly on composition as well as on Equations Relating Molar and Partial Molar
temperature and pressure, our purpose in this chapter is to Properties
develop the theoretical foundation for applications of thermo-
The definition of a partial molar property, Eq. (1.1), provides
dynamics to gas mixtures and liquid solutions.
the means for calculation of partial properties from solution-
The theory is introduced through derivation of a fundamental property data. Implicit in this definition is another, equally
property relation for homogeneous solutions of variable important, equation that allows the reverse, i.e., calculation of
composition. Convenience here suggests the definition of a solution properties from knowledge of the partial properties.
fundamental new property called the chemical potential, upon The derivation of this equation starts with the observation that
which the principles of phase and chemical-reaction equilibrium the thermodynamic properties of a homogeneous phase are
depend. This leads to the introduction of a new class of functions of temperature, pressure, and the numbers of moles
thermodynamic properties known as partial properties. The of the individual species which comprise the phase. Thus for
mathematical definition of these quantities allows them to be thermodynamic property M:
interpreted a properties of the individual species as they exist in
solution. For example, in a liquid solution of ethanol and water nM = Μ (T , P, n1 n 2. .......ni .....)
the two species have partial molar properties whose values are
somewhat different from the molar properties of pure ethanol The total differential of nM is:
and pure water at the same temperature and pressure.
 ∂(nM )  ∂(nM 
)  ∂(nM ) 
∑i
∂P  T .n dP +  ∂T P,ndT +  ∂ni  P,T ,n dni
Property relations for mixtures of ideal gases are important as d (nM) =  
   
references in the treatment of real-gas mixtures, and they form
j
the basis for introduction of yet another important property, where subscript n indicates that all mole numbers are held
the fugacity. Related to the chemical potential, it is vital in the constant, and subscript nj that all mole numbers except nj are
formulation of both phase-and chemical-reaction-equilibrium held constant. Because the first two partial derivatives on the
relations. right are evaluated at constant n and because the partial deriva-
Finally, a new class of solution properties is introduced. Known tive of the last term is given by Eq. (1.1) this equation has the
as excess properties, they are based on an idealization of simpler form:
solution behavior called the ideal solution behavior called the  ∂M   ∂M 
ideal solution. Its role is like that of the ideal gas in that it d (nM) = n  ∂P  dP +n  ∂T  dT+ ∑iM i dni (1.3)
 T .x   P. x
serves as a reference for real-solution behavior. Of particular
interest is the excess Gibbs energy, a property which underlines Where subscript x denotes differentiation at constant composi-
the activity coefficient, introduced from a practical point of view tion.
in the preceding chapter. Since ni = xin
Partial Properties Dni = xi dn + n dxi
The definition of chemical potential as the mole number When dni is replaced by this expression, and d(nM) is replaced
derivative of nG suggests that other derivatives of this kind by the identity.
should prove useful in solution thermodynamics. d (nM) º n dM + M dn
Thus
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2. Equation (1.3) becomes: Equation (1.4) implies that a molar solution property is given
THERMODYNAMICS OF BIOLOGICAL SYSTEMS
as a sum of its parts and that is the molar property of species i
 ∂M 
ndM + Mdn  ∂P  = n  ∂T
 ∂M 
 dP +ndT + ∑M i ( xi dn + ndxi ) as it exists in solution. This is a proper interpretation provided
 T ,x   P, x i
one understands that the defining equation for Mi, Eq. (1.1), is
The terms containing n are collected and separated from those an apportioning formula which arbitrarily assigns to each species
containing dn to yield: i a share of the mixture property.
The constituents of a solution are in fact intimately intermixed,
  ∂M   ∂M  
 dT − ∑ M i dxi  n
and owing to molecular interactions cannot have private
 dM −   dP −  properties of their own. Nevertheless, partial molar properties,

  ∂P  T , x  ∂T  P , x i 
 as defined by Eq. (1.1), have all the characteristics of properties
of the individual species as they exist in solution. Thus for
 
+ M − ∑ x i M i  dn = 0 practical purposes they may be assigned as property values to
 i  the individual species.
The symbol M may express solution properties on a unit-mass
In application, one is free to choose a system of any size, as basis as well as on a mole basis. Property relations are the same
represented by n, and to choose any variation in its size, as in form on either basis; one merely replaces n, the number of
represented by dn. Thus n and dn are independent and arbitrary. moles, by m, representing mass, and speaks of partial specific
The only way that the left side of this equation can then, in properties rather than of partial molar properties. In order to
general, be zero is for each term in brackets to be zero. There- accommodate either, we generally speak simply of partial
fore, properties.
 ∂M   ∂M  The molar (or unit-mass) properties of solutions are repre-
dM=   dP +   dT + ∑ M i dx i sented by the plain symbol M. Partial properties are denoted by
 ∂P  T , x  ∂T  P , x i an overbar, with a subscript to identify the species; the symbol
is therefore. In addition, properties of the individual species as
and
they exist in the pure state at the T and P of the solution are
M= ∑x Mi
i i (1.4)
identified by only a subscript, and the symbol is . In summary,
the three kinds of properties used in solution thermodynamics
are distinguished by the following symbolism:
Multiplication of Eq. (1.4) by n yields the alternative expression:
Solution properties M
nM = ∑n i
i M i (1.5) Partial properties Mi
Equations (1.4) and (1.5) are new and vital. Known as summabil- Pure-species properties M
ity relations, they allow calculation of mixture properties from Partial Properties in Binary Solution
partial properties, playing a role opposite to that of Eq. (1.1), Equations for partial properties can always be derived from an
which provides for the calculation of partial properties from equation for the solution property as a function of composi-
mixture properties. tion by direct application of Eq. (1.1). For binary systems,
Since Eq. (1.4) is a general expression for M, differentiation however, an alternative procedure may be more convenient.
yields a general expression for d M: Written for a binary solution, the summability relation, Eq.
(1.4), becomes:
dM= ∑ x d M + ∑ M dx
i
i i
i
i i
M = x1 M 1 + x 2 M 2 (A)
Another general equation for d M, yields the Gibbs/Duhem dM = x1 d M 1 + M 1 dx1 + x 2 d M 2 + M 2 dx 2 (B)
equation:
2
Gibbs/Duhem equation is Eq. (1.6), expressed here as:
 ∂M   ∂M 
  dP +   dT − ∑ x i d M i = 0 x1 d M 1 + x 2 d M 2 = 0 (C)
 ∂P  T , x  ∂T  P , x i
Since, it follows that eliminating in favor of in Eq. (B) and
This equation must be satisfied for all changes in P, T, and the combining the result with Eq. (C) gives:
caused by changes of state in a homogeneous phase. For the
important special case of changes at constant T and P, it dM= M 1 dx1 − M 2 dx1
simplifies to:
or
∑ x dM i i = 0 (constant T,P) (1.6S) dM
i = M1 − M 2 (D)
dx1
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3. Elimination first of and then of from Eqs. (A) and (D) yields: but dU=TdS - PdV + S m IdnI, is one of fundamental equa-
THERMODYNAMICS OF BIOLOGICAL SYSTEMS
tions. Subtracting two equations yield:
dM
M1 = M + x (1.7)
S m IdnI = -Sdt + Vdp, similarly one can write the Gibbs-
dx1 Duhem equation for
dU=TdS - PdV + S m IdnI , dH=TdS - VdP + S m IdnI and
dM dF=-SdT - pdV + S m IdnI at equilibrium.
M 2 = M − x1 (1.8)
dx1 S m IdnI = -Sdt + Vdp=0 (T, P constant). Gibb-Duhem
equation can be used to calculate the chemical potential of the
Thus for binary systems, the partial properties are readily solute from that of the solvent in a binary ideal system.
calculated directly from an expression for the solution property
S m IdnI = -Sdt + Vdp= nd m + n2dm2 =0 where as symbol
as a function of composition at constant T and P. The corre-
without subscripts refer to the solvent and the one with
sponding equations for multi component systems are much
subscripts 2 refer to the solute.
more complex, and are given in detail by Van Ness and Abbott.
dm2 =-(n/n2)dm2 = -(x/x2)dm we also have m = m 0 + RTlnx ,
Relation among Partial Properties therefore dm2 =-RT(dx/x2) and x1 + x2 = 1, so dx + dx2 = 0 or
Now we,ll learn how practical properties are related to one dx = dx2
another. Since by Eq we derive the following equation Therefore, dm2 =RT(dx2/x2) after integrated and find m2 where
we have constant C but it is in term of T and P therefore we can
d (nG ) = (nV )dP − (nS )dT + ∑ G i dni (1.9) write m2 = m20 + RT(lnx2)
i
Finally we can define a Raoult’s law as P2 = x2 P20
Application of the criterion of exactness, equation a.1 yields the
For n moles nH = nU + P(nV)
Maxwell relation,
Differentiations with respect to ni at constant T,P and nj yields:
 ∂V   ∂S  By Eq. (1.1) this becomes:
  = −  (a.1)
 ∂T  P , n  ∂P  T , n
 ∂G   ∂G 
d Gi =  i  dP +  i  dT
plus the two additional equations:  ∂P   ∂T 
 T , x   P,x
 ∂ Gi 
 ∂G1 
 
 ∂ (nS )    =  ∂ (nV )  In a constant-composition solution, is a function of P and T,
 ∂T  = −  ∂n   ∂P   
  P,n  i  P ,T , n j
  T , n  ∂ni  P ,T .n j and therefore :
These examples illustrate the parallelism that exists between
where subscript n indicates constancy of all nj and therefore of equations for a constant-composition solution and the
composition, and subscript nj indicates that all mole numbers corresponding equations for the partial properties of the species
except the i th are held constant. In view of Eq.(1.1.), the last in solution. We can therefore write simply by analogy many
two equations are most simply expressed: equations that relate partial properties.
From Van Ness and Abbott: “The partial property is inter-
 ∂G i 
  = −Si preted as the value of the property of species as is exists in
 ∂T  solution. However, each species in a solution is an intimate part
  P,x of the solution, and cannot actually have identifiable separate
properties of its own. Nevertheless, we may view the definition
 ∂G i  of as a formula which also defines how a solution property is
  = Vi apportioned among its constituent species, and on this basis treat
 ∂P 
 T , x partial properties as though they represent values of properties
of the individual species in solution. Partial properties lend
These equations allow calculation of the effect of temperature themselves completely to this interpretation, and one can always
and pressure on the partial Gibbs energy (or chemical potential reason logically to correct conclusions from this point of view”
potential).
Every equation that provides a liner relation among thermody-
namic properties of a constant-composition solution has as its
counterpart an equation connecting the corresponding partial
properties of each species in the solution.
We demonstrate this by example. Consider the equation that Example of computing partial molar properties. Compute
defines enthalpy: given the binary correlation
H=U+PV
We can write the change in Gibbs Free Energy in term number
of mole as dG = S(nIdmI + m idnI),
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4. THERMODYNAMICS OF BIOLOGICAL SYSTEMS
Recall that
and
as,
J = n1 + n2
where n1 and n2 refer to the amount of substance of
component 1 and 2 respectively.
3. Define what is meant by the term partial molar property and
The above definition of does not take into consideration that describe how a partial molar property can be determined
th are not all independent, but are related by experimentally.
The volume of aqueous NaOH solutions containing 1000.0
g of H2O is given by,
V/cm3 = 1001.53 - 4.31m + 1.54m2
where m is the molality.
If we consider this then we get 4. Derive an expression for the partial molar volume of NaOH
and calculate the partial molar volume of NaOH in an
infinitely dilute solution and at a molality of 1.0 mol kg-1.
Calculate the partial molar volumes of H2O in these two
solutions.
[MW of NaOH = 40.0 a.m.u; H2O = 18.0 a.m.u.]
Notes
Both of these expressions are used in texts and both give the
same answer. Which you use is a matter of convenience.
Using the first equation we get:
Using the second equation we substitute to get:
This is exactly the same as obtained with the first equation.
Questions
1. The molar volume of pure methanol is 40 cc/mole. Also,
the volume of a solution containing 1000 g of water and n
moles of methanol is given by:
Calculate the partial molar volume for methanol when the
molality of the solution is 0 and also when the molality is 1.
2. For a general two-component system, any extensive
thermodynamic property J may be written in terms of the
partial molar quantities
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