# Partial Molar Properties And Their Application

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18th May 2017 Environmental Sciences Reference this

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Thermodynamics deals with energy changes and its relationship with work. It is based on three laws of thermodynamics which are used as axioms just as Newton’s laws motion from the basis of classical mechanics. The first two laws are based on facts observed in every day life. The predictions based on these laws have been verified in most cases and so far no case has been reported where the laws break down. The laws can be stated in mathematical form. Hence, thermodynamics is an exact science. The thermodynamic theory can be developed without gaps in the argument using only moderate knowledge of mathematics.

Thermodynamic relations derived earlier are applicable to closed systems. In a system where not only the work and heat but also several kinds of matter are being exchanged, a multicomponent open system has to be considered. Here, the amounts of the various substances are treated as variables like any other thermodynamic variables. For example, the gibbs free energy of a system is a function not only of temperature and pressure , but also of the amount of each substance in the system,such that

G=f(T,p,n1,n2â€¦â€¦.nk)

Where n1,n2,â€¦.,nk represent the amounts of each of the K components in the system . for simplicity, let a system contain only two components. The total differential of G is

dG=(?G/?T)P,n1,n2 dT+(?G/?p)T,n1,n2 dp+(?G/?n1)T,p,n2 dn+(?G/?n2)T,p,n1 dn2

In this eq., the partial derivatives (?G/?n1)T,P,n2 and (?G/?n2)T,P,n1 are known as partial molar free energies of components one and two , respectively. In genral, the partial derivative of a thermodynamic function Y with respect to the amount of component i of a mixture when T,p and amounts of other constituents are kept constant , is known as the partial molar property of the ith component and is represented as Yi,pm. Thus

Yi,pm=(?Y/?ni)T,p,njs i=!j

## [C.]DEFINITION OF PARTIAL MOLAR PROPERTY:

The partial molar property may be defined in either of the following two ways:

1. it is the change in Y when 1 mole of component i is added to a system which is so large that this addition has a negligible effect on the composition of the system.

2. Let dY be the change in value of Y when an infinitesimal amount dni of component i is added to a sysem of definite composition. By an infinitesimal amount dni we mean that its addition does not cause any appreciable change in the composition of the system. If we divide dY by dni , we get the partial molar property (?Y/?ni). thus, the partial molar property of the component i may be defined as the change in Y per mole of component i when an infinitesimal amount of this component is added to a system of definite composition.

## [D.]TYPES OF MOLAR PROPERTIES:

(a.) Partial molar volume:

The partial molar volume is broadly understood as the contribution that a component of a mixture makes to the overall volume of the solution. However, there is rather more to it than this:

When one mole of water is added to a large volume of water at 25°C, the volume increases by 18cm3. The molar volume of pure water would thus be reported as 18cm3 mol-1. However, addition of one mole of water to a large volume of pure ethanol results in an increase in volume of only 14cm3. The reason that the increase is different is that the volume occupied by a given number of water molecules depends upon the identity of the surrounding molecules. The value 14cm3 is said to be the partial molar volume of water in ethanol.

In general, the partial molar volume of a substance X in a mixture is the change in volume per mole of X added to the mixture.

The partial molar volumes of the components of a mixture vary with the composition of the mixture, because the environment of the molecules in the mixture changes with the composition. It is the changing molecular environment (and the consequent alteration of the interactions between molecules) that results in the thermodynamic properties of a mixture changing as its composition is altered.

The partial molar volume, VJ, of any substance J at a general composition, is defined as:

Fig: the partial molar volumes of water and ethanol at 25degree C

where the subscript n’ indicates that the amount of all the other substances is held constant.

The partial molar is the slope of the plot of the total volume as the amount of J is changed with all other variables held constant:

Note that it is quite possible for the partial molar volume to be negative, as it would be at II in the above diagram. For example, the partial molar volume of magnesium sulphate in water is -1.4cm3 mol-1.

i.e. addition of 1 mol MgSO4 to a large volume of water results in a decrease in volume of 1.4 cm3. (The contraction occurs because the salt breaks up the open structure of water as the ions become hydrated.)

Once the partial molar volumes of the two components of a mixture at the composition and temperature of interest are known, the total volume of the mixture can be calculated from:

The expression may be extended in an analogous fashion to mixtures with any number of components.

The most common method of measuring partial molar volumes is to measure the dependence of the volume of a solution upon its composition. The observed volume can then be fitted to a function of the composition (usually using a computer), and the slope of this function can be determined at any composition of interest by differentiation.

(b.) Partial molar gibbs energies:

The concept of a partial molar quantity can be extended to any extensive state function. For a substance in a mixture, the chemical potential is a defined as the partial molar gibbs energy:

i.e. the chemical potential is the slope of a plot of the Gibbs energy of the mixture against the amount of component J, with all other variables held constant:

In the above plot, the partial molar Gibbs energy is greater at I than at II.

The total Gibbs energy of a binary mixture is given by:

The above expression may be generalised quite trivially to a mixture with an arbitrary number of components:

where the sum is across all the different substances present in the mixture, and the chemical potentials are those at the composition of the mixture.

This indicates that the chemical potential of a substance in a mixture is the contribution that substance makes to the total Gibbs energy of the mixture.

In general, the Gibbs energy depends upon the composition, pressure and temperature. Thus G may change when any of these variables alter, so for a system that has components A, B, etc, it is possible to rewrite the equation dG = Vdp – SdT (which is a general result that was derived here) as follows:

which is called the fundamental equation of chemical thermodynamics.

At constant temperature and pressure, the equation simplifies to:

Under these conditions, dG = dwn,max (as was demonstrated here), where the n indicates that the work is non-expansion work. Therefore, at constant temperature and pressure:

The idea that the changing composition of a system can do work should be familiar – this is what happens in an electrochemical cell, where the two halves of the chemical reaction are separated in space (at the two electrodes) and the changing composition results in the motion of electrons through a circuit, which can be used to do electrical work.

On a final note, it is possible to use the relationships between G and H, and G and U, to generate the following relations:

Note particularly the conditions (the variables that must be held constant) under which each relation applies.

Fig: the partial molar volumes of water and ethanol at 25degree C

where the subscript n’ indicates that the amount of all the other substances is held constant.

The partial molar is the slope of the plot of the total volume as the amount of J is changed with all other variables held constant:

Note that it is quite possible for the partial molar volume to be negative, as it would be at II in the above diagram. For example, the partial molar volume of magnesium sulphate in water is -1.4cm3 mol-1. i.e. addition of 1 mol MgSO4 to a large volume of water results in a decrease in volume of 1.4 cm3. (The contraction occurs because the salt breaks up the open structure of water as the ions become hydrated.)

Once the partial molar volumes of the two components of a mixture at the composition and temperature of interest are known, the total volume of the mixture can be calculated from:

The expression may be extended in an analogous fashion to mixtures with any number of components.

The most common method of measuring partial molar volumes is to measure the dependence of the volume of a solution upon its composition. The observed volume can then be fitted to a function of the composition (usually using a computer), and the slope of this function can be determined at any composition of interest by differentiation.

(C.)PARTIAL MOLAR THERMAL PROPERTIES:

1. Partial molar heat capacities: the heat capacity at constant pressure Cp of a solution containing n1 moles of solvent and n2 moles of solute is given by

Cp=(?H/?T)P,N â€¦â€¦â€¦â€¦â€¦â€¦eq(1)

The pressure and compostion being constant. Upon differentiation with respect to n1,maintaining n2 constant,it follows that

CP1=(?CP/?n1)T,P,n2

=?H/?T?n1 ……………….eq(2)

Where Cp1 is the partial molar heat capacity,at constant pressure,of the constituent 1 of the given solution. The partial molar heat constant H1 of this constituent is defined by

H1=(?H/?n1)T,P,n2

And hence differentiation with respect to temp. gives

(?H1/?T)P,N=?H/?T?n1 =CP1 â€¦â€¦â€¦â€¦â€¦â€¦.eq(3)

The result being identical with CP1 by eq.(3).

The partial molar heat capacity of the solvent is any particular solution thus be defined by either eq(1) and eq(2).

Similarly,i.e.,constituent 2,

Cp2=(?CP/?n2)T,P,n1

=(?H2/?T)P,N â€¦â€¦â€¦â€¦â€¦â€¦â€¦..eq(4)

We know,

Li=H1-H10

Is differentiated with respect to temp.,at constant pressure and composition,it follows that

(?L1/?T)P,N=(?H1/?T)P,N-(?H10/?T)P,N

= Cp1-Cp10 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦eq(5)

Where Cp1,identical with Cp1 or Cp1o, is the molar heat capacity of the pure solvent or the partial molar heat capacity of the solvent in a solution at infinite dilution. Thus, Cp10 may be regarded

as an experimental quantity, and if the variation of the relative partial molar heat content of the solvent with temperature,i.e. (?L1/?T)P,N, is known , it is possible to determine Cp1 at the corresponding composition of the solution. The necessary data are rarely available from direct thermal measurements of L1, such as thus described in 44f,at several temperatures, but the information can often be obtained, although not very accurately from E.M.F measurements.

By differentiating the expression for the relative partial molar heat content of the solute it is found, in an exactly similar manner to that used above , that

(?L2/?T)P,N=(?H2/?T)P,N-(?H02/?T)P,N

=CP2-CP20 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦eq(6)

In this expression,Cp20 is the partial molar heat capacity of the solute in the infinitely dilute solution. Although the experimentel significance of the quantity is not immediately obvious.thus from a knowledge of the variation of L2, the partial molar heat content of the solute with temprature it should be possible to derive, with the aid of equation(6) , the partial molar heat capacity of the solute Cp2 at the given composition.

## [E.]Determination of partial molar properties:

1.Direct method:

in view of the definition of the partial molar properties Gi as

Gi=(?G/?ni)T,P,n1,â€¦.. â€¦â€¦â€¦â€¦â€¦.eq(1)

An obvious method ffor its determination is to plot the value of the extensive properties G,at constant temperature and pressure, for various mixtures of the two components against the number of moles,e.g.,n2,of the one of them,the value of n1 being kept constant. The slope of the curve at any particular composition,which maybe determined by drawing a tengent to the curve, gives the value of G2 at that comoposition. Since the molality of a solution represents the number of moles of solute associated with a constant mass,and hence a constant number of moles,the plot of the property G against the molality can be used for the evaluation of the partial molar property of the solute. Once G2 at any composition has been determined, the corresponding value of G1 is readily derived by means of the relationship,

G=n1G1+n2G2

In view of the difficulty of determining the exact slope of the curve at all points, it is preferable to use an analytical procedure instead of the graphical one just described. The property G is then expressed as a function of the number of moles of one component,e.g.,the molality, associated with a constant amount of the other component. Upon differentiation with respect to n,i.g.,the molality, an expression for the partial molar property is obtained.

2.from apparent molar properties:

a method that is often more convenient and accuarate than that described above,makes use of the apparent molar property.

We know

G-n1G1=n2ö2

If n1 is maintained constant,so that n1G1 is constant, differentiation with respect to n2 , constant temp. and pressure being understood,gives

G2 =(?G/?n2)n1 = (?öG/?n2)n1 + öG â€¦â€¦â€¦â€¦â€¦â€¦â€¦eq(2)

G2 = ((?öG/? ln n2)n1+ öG â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..eq(3)

Since the molality m is equivalent to n2, with n1 constant, eq(2) and eq(3) may be written as

G2= m (d öG/dm)+ öG â€¦â€¦â€¦â€¦â€¦eq(4)

G2=( d öG/d ln m)+ öG â€¦â€¦â€¦â€¦â€¦..eq(5)

Respectively. If the apparent molar property öG is determined for various values of n2 , with n1 constant , or at various molalities, the partial molar property G2 can be calculated from the slope, at any given composition, of the plot of öG against n2 or against ln n2. The method based on the use of eq’s(3)&(5) is usally more accurate than that involving the logarithmic plot,since it does not give undue importance to result obtained in dilute solutions. An analytical method can, of course, be used in place of the graphical procedure if öG can be expressed as a function of n2 or of the molality.

For use in a later connection, an alternative form of eq(4) is required and it will be derived here. The right hand side of this equation is equivalent to d(m öG)/dm, that is,

m (d öG/dm)= G2

and upon integration, m varying between the limits of zero and m, and mdöG between zero and möG, it is found that

möG=?0m G2 dm

öG=1/m?0m G2 dm

for dilute solutions,the molality is proportional to the molar concentration c, and hence it is permissible to put this result in the form

öG=1/c?0m G2 dm

## [F.] APPLICATION OF PARTIAL MOLAR PROPERTIES:

These properties are very useful since each and every reaction in chemistry occurs at a constant temperature and pressure and under these conditions we can determine these with the help of partial molar properties. They are highly useful when specific properties of pure substances and

properties of mixing are considered. By definition, properties of mixing are related to those of the pure substance by:

Here * denotes the pure substance

M the mixing property

z corresponds to the specific property

From the definition of partial molar properties,

substitution yields:

Hence if we know the partial molar properties we can derive the properties of mixing.For the internal energy U, enthalpy H, Helmholtz free energy A, and Gibbs free energy G, the following hold:

whereP is the pressure

V is the volume

T is temperature

S is the entropy

## [G.] BIBLIOGRAPHY:

1. THERMODYNAMICS AND CHEMICAL EQUILIBRIUM

AUTHOR: K L KAPOOR

2. THERMODYNAMICS FOR CHEMISTS

AUTHOR: SAMUEL GLASSTONE

3. http://www.everyscience.com/Chemistry/Physical/Mixtures/a.1265.php

4. http://www.everyscience.com/Chemistry/Physical/Mixtures/b.1266.php

5. http://www.chem.ntnu.no/nonequilibrium-thermodynamics/pub/192-Inzoli-etal.pdf

7. http://www.chem.boun.edu.tr/webpages/courses/chem356/EXP5-Determination%20of%20Partial%20Molar%20Quantities.pdf

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