PROPERTIES OF SOLUTIONS

Dr. Sumitra Maurya Chemistry Department Brahmanand P. G. College, Kanpur Ideal solutions

An ideal mixture is one, for which there is no change in volume due to mixing. In other words for an ideal solution, partial molar volume of each component will be equal to its pure component volume at same temperature and pressure. --- Raoults law V V i i V  Vi xi   xiVi Ideal solutions are formed when similar components or adjacent groups of group are mixed

Vi Vi

Eg: Benzene-Toluene, Methanol- Ethanol, Hexane- Heptane

Solution undergo change in volume due to mixing are known as non ideal solutions

Vi Vi

Eg: Methanol-Water, Ethanol-water. Ideal solutions formed when the intermolecular force between like molecules and unlike molecules are of the same magnitude.

Non-ideal solutions are formed when intermolecular forces between like molecules and unlike molecules of different magnitude.

For ideal solutions t ------(1) V  niVi Vi is the molar volume of pure component i V t  Vi     Vi ni  T ,P ,n j

------(2) The residual volume for the pure component is RT   V  i p Therefore we know from reduced properties

f 1 P  RT  ln i  V  dP   i  ----(3) P RT 0  P  For component i in terms of partial molar properties

P fi 1  RT  ln   V i  dP ----(4) Pi RT 0  P 

Subtracting equation (3) from (4)

P fi P 1 ln  V i V dP   i  ----(5) fi Pi RT 0

we know that Pi  yi P equation (5) reduces to P fi 1 ln  V i V dP   i  fi yi RT 0 comparing equation (2)

f i  1 fi  fi yi fi yi

--Lewis Randal rule Lewis Randal rule is applicable for evaluating fugacity of components in gas mixture.

Lewis Randal rule is valid for 1. At low pressure when gas behaves ideally. 2. When Physical properties are nearly same. 3. At any pressure if component present in excess. Henry’s Law This law is applicable for small concentration ranges. For ideal solution Henry’s law is given as

fi  xi ki

Pi  xiki ki- Henrys Constant, -Partial molar fugacity, fi -Partial pressure of component i. Pi Non Ideal solutions

For ideal solutions fi  xi ki ------(a)

For non ideal solutions fi   i xiki ------(b)

 i - is an activity coefficient of component i

Comparing equation (a) and (b) fi nonideal  i  fi ideal For both ideal and Non ideal solutions the fugacity of solution is given by equation

fi ln f   xi ln xi

ln   xi lni Activity in Solutions: The activity of a component in a solution is defined as the ratio of fugacity of a component in the solution in a given condition to the fugacity of that component in the standard state.

f i ai  o f i The fugacities are related to chemical potential as --(1) i  RT ln f i  C

at standard state o o -----(2) i  RT ln f i  C Subtracting equation 2 from 1

f i i  RT ln  RT lnai f i

o i  i  i Is the increase in chemical potential of species i when it is brought into solution from its standard state

For ideal solutions f i  xi f i

f i ai  o f i

a i  x i (for ideal solutions) Activity coefficients: The calculation of fugacity of a component in a real solution should take into account of the degree of the departure from ideal behavior

For real solutions the partial molar fugacites .is given by the relation o f i   i xi f i o   1 for ideal solutions f i  xi f i i Here we have observed two types of ideal behavior o --- Lewis Randall rule or Raoults law f i  f i o ----- Henrys law f i  K i In case of Lewis Randall rule or Raoults law

o f i   i xi f i

f a   i  i i o ai   i xi xi f i xi For Henrys Law

f i   i xi K i Pi   i xi K i Effect Pressure on Activity Coefficients

Effect of pressure on fugacity

  ln f  V  i   i ------(A)  P T RT In terms of Partial molar properties   ln f  V  i   i ------(B)  P  RT  T Equation B-A   ln( f / f ) V  V f  i i   i i i   x  P  RT i i  T f i   ln x  V  V  i i   i i  P T RT

Mole fraction is independent of pressure  ln xi     0  P    ln  V  V V - Partial molar volume  i   i i i P RT  T Vi – Pure component volume Effect of Temperature on activity Coefficients The effect of temperature on fugacity of a pure substance was given by the equation   ln f  H o  H  i   i i ------(C)  T  RT 2   P Similarly for the substance in the solution (Partial molar properties)

o   ln f  H  H i  i   i ------(D)  T  RT 2   P

Equation D-C   ln i  H i  H i    2  T  P RT

( H i  H i ) - Partial heat of mixing of component i from its pure state of given composition both in the same state of aggregation and pressure. PROBLEM

The fugacity of component 1 in binary liquid mixture of component 1 and 2 at 298K and 20bar is given by

2 3 f 1  50 x1  80 x1  40 x1

Where f is in bar and x1 is the mole fraction of component 1 Determine (a) The fugacity f of pure component 1 1 1 (b) The fugacity coefficient  1

(c) The Henrys law constant K1 (d) The activity coefficient Gibbs Duhem Equation Consider a multi component solution having ni moles of component i , let the property of solution be M in terms of partial molar properties

t M  nM  nM i ------(1). Where n is the total no of moles of solution Differentiating eq (1) we get ------(2) d( nm )  nid Mi   Midni We know that nM  f T ,P,n1 ,n2    

( nM ) ( nM ) ( nM ) ( nM )    dT    dP    dn1  T  P ,n ,n   P T ,n ,n  n1 1 2 1 2  T ,P1 ,n2.n3 ( nM )    dn2      n2  T ,P ,n1 ,n3 

n( M ) n( M ) ( nM ) ( nM )    dT    dP     dni  T  P ,n ,n   P T ,n ,n  ni 1 2 1 2  T ,P ,n j From the definition of partial molar properties n( M ) n( M ) ( nM )  dT  dP  M dn  T   P   i i --(3)   P ,n1 ,n2   T ,n1 ,n2 

Subtracting equation (2) from equation (3)

n( M ) n( M ) dT  dP  nid Mi  0 ------(4)  T   P     P ,n1 ,n2   T ,n1 ,n2 

Equation (4) is the fundamental form of Gibbs Duhem equation Special Case At constant temperature and pressure, dT and dP are equal to zero. The equation becomes

 xid Mi  0

For binary solution at constant temperature and pressure the equation becomes

x1d M1  x2d M 2  0

x1d M1  ( 1  x1 )d M 2  0 ------(5) dividing equation(5) by dx1

d M 1 d M 2 x1  ( 1  x1 )  0 dx1 dx1

The above equation is Gibbs Duhem equation for binary solution at constant temperature and pressure in terms of Partial molar properties.

Any Data or equation on Partial molar properties must satisfy Gibbs Duhem equation. Problem: Find weather the equation given below is thermodynamically consistent

G  100 x1  150 x2  x1 x2 (10 x1  x2 )

G 3 2 G1  G  ( 1 x1 ) G1  18x1  35 x1  16 x1  101 x1

G G  18x 3  8x 2  150 G2  G  x1 2 1 1 x1 dG1 2  54 x1 70 x1  16 dx1

dG 2  54 x 2  16 x dx 1 1 G D equation 1

d M 1 d M 2 x1  ( 1  x1 )  0 dx1 dx1

2 2 x1 ( 54 x1 70 x1  16 ) ( 1  x1 )( 54 x1  16 x1 )  0 It satisfies the GD equation, the above equation is consistent. Property Changes of Mixing:

For ideal solution For any thermodynamic property

V   xiVi M   xi M i

For non ideal solutions, we apply a correction term M known as property change of mixing. For non ideal solutions

M   xi M i  M M is the difference in the property of the solution M and sum of the properties of the pure components that make it up all at the same temperature and pressure as the solution thus

M  M   xi Mi in terms of volume V   xiVi  V

For ideal solutionsV  0 More general definition of o M  M   xi M i Properties change in terms Partial molar o M   xi ( M i  M i ) o Mi  Mi -change in property of component i when one mole of pure i in its in its standard state is brought to the solution of given composition at same T and P.

In terms of Volume

o V   xi (Vi  Vi )

o G   xi (Gi  Gi ) Activity and property change of mixing

Free energy change of mixing (G): Using the definition of fugacity the change of free energy of substance when it is brought from the standard state to the solution, it can be written as

o f i Gi  Gi  RT ln o  RT lnai f i We know that o G   xi (Gi  Gi )

G  x lna RT  i i Volume change of Mixing V: The partial molar free energy(chemical potential) varies with pressure as

 G   i   V  P  i  T ,x Similarly for pure substance, we have

 G o   o   i  o (G i  Gi )  Vi V  x    P   i P  T ,x   T ,x o V   xi (Vi  Vi ) o (G i  G ) V  x  i   i P   T ,x

Result of the above equation

d lnai  V  RT xi    P  T ,x The above equation can be rewritten in the form

PV d lnai    xi   RT  d lnP  T ,x Enthalpy change of mixing H: The Gibbs Helmoltz equation is given as

(G / T ) H o i   i   2  T  P T

in terms of partial molar properties (G / T ) H  i    i T T 2   P o The enthalpy of mixing H   xi (H i  Hi ) Using the above equations

   G  G o  H  T 2 x   i i   i T  T     P ,x

2    H  T R xi  lnai  T  P ,x

H  lnai    xi   RT   lnT  P ,x