Dr. Cooper's comments: Remember to label equations. Also check your symbols. Please use the greek letters.

 

(Jason Drown) 

 

 

Mixing of Liquids:  Ideal Solutions vs. Real Solutions

 

Recall that for an ideal solution: 

 

     G_i = n_Aµ_A* + n_Bµ_B* 

 

Where G_i is the Gibbs energy prior to mixture, n_A is the number of moles of liquid A, and µ_A* is the chemical potential of pure liquid A.  When µ_A is expressed without the *, it denotes the chemical potential of liquid A in mixture.

 

In ideal mixtures, the liquids possess no intermolecular forces, or the intermolecular forces between the A-B molecules are exactly the same as the intermolecular forces between the A-A and B-B molecules, thus providing no difference in attraction or repulsion forces in the mixture from the two pure liquids.  In the latter case, non-ideal liquids can behave ideally.

 

We can relate an ideal chemical potential in solution to its pure chemical potential by the equation: 

   

 

     µ_A = µ_A* + RTln(χ_A)

 

 

Note that the difference in chemical potential for a liquid in mixture from its pure form is thus determined by temperature and molar fraction, or χ_A.  Thus the species of chemical present does not make any difference to its behavior in this respect. 

 

Thus, the Gibbs energy after mixing becomes:

 

 

     G_f = n_A{ µ_A* + RTln(χ_A)} + n_B{ µ_B* + RTln(χ_B)}

 

and the change in Gibbs energy, G_f – G_i simplifies to:

  

     Δ_mixG = n_ARTln(χ_A) + n_BRTln(χ_B)

     Δ_mixG = nRT{χ_Aln(χ_A) + χ_Bln(χ_B)}

 

For Ideal solutions, the following also hold true:

 

     Δ_mixS = -nR{χ_Aln(χ_A) + χ_Bln(χ_B)}

     Δ_mixH = 0

     (∂G / ∂p)_T = V

     (∂Δ_mixG / ∂p)_T = Δ_mixV = 0

 

Notice that ideally, a mixture of two solutions totaling a certain volume produces no change in that total volume over constant temperature.

 

Real solutions, however, have dominant intermolecular forces between A-A, A-B, and B-B particles.  These forces account for possible changes in enthalpy and volume upon mixing.  The change in Gibbs free energy is still defined as

 

 

     ΔG=ΔH-TΔS

 

Thus in a real mixture, if the resulting change in enthalpy (ΔH) is large and positive or entropy (ΔS) is negative, ΔG may be positive resulting in an immiscible mixture.

 

 

 

Excess Functions

 

Real solutions can be expressed by excess functions, X^E.  Excess functions are the difference between the thermodynamic properties of the mixing and the ideal solutions:

 

 

     S^E = Δ_mixS - Δ_mixS^ideal

     G^E = Δ_mixG - Δ_mixG^ideal

     H^E = Δ_mixH - Δ_mixH^ideal

     V^E = Δ_mixV - Δ_mixV^ideal

 

 

 

    

              (a) Benzene/cyclohexane             (b) Tetrachloroethene/cyclopentane

 

 

The figure above depicts the plots of H^E and V^E vs. molar fraction of two different mixtures.   For ideal mixtures, these excess functions would remain zero regardless of the solution’s dilution.  Intermolecular forces in the mixtures, however, cause divergence from ideality dependant on the molar fraction.  Note that as the solutions approach purity on either end, the excess functions approach zero.

 

 

 

Colligative Properties

 

 

Colligative properties are properties that depend only on the number of solute particles present, not their identity.

 

Examples of colligative properties important to us are the lowering of vapor pressure, the elevation of boiling point, the depression of freezing point, and the osmotic pressure from the presence of a solute.  Colligative properties are only applicable to dilute solutions.

 

These colligative properties are a result of the reduction of chemical potential as a result of the presence of solute:

 

 

     µ_A = µ_A* + RTln(χ_A)

 

 

Notice for any value of χ_A less than 1 (a mixture) ln(χ_A) is negative, and µ_A is thus less than µ_A*, representing a reduction in chemical potential from the pure liquid.  This is depicted in the graph below:

 

    

 

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(J.Dolyniuk)

Specific Colligative Properties:

Figure 1

Figure 1 above shows that changing (decreasing) the chemical potential of a pure liquid by adding a solvent will have a great effect on the freezing point and a lesser effect on the boiling point. The effect on freezing point is known as freezing point depression, since adding a solvent to pure liquid will cause the freezing point to decrease. The effect on boiling point is known as boiling point elevation, since adding a solvent to pure liquid will cause the boiling point to increase. Both are colligative properties so they only depend on the number of particles of solute and not on the solute's identity. They do, however, depend on the intrinsic properties of the solvent and different solvents will often yield different results in freezing point depression and boiling point elevation experiments.

At a pressure of 1 atm, the new boiling point after addition of a solute to pure liquid can be determined by finding the temperature at which the vapor of pure solvent has the same chemical potential as the solvent in the solution.

If we say the solvent is A and the solute is B, equilibrium is found at a temperature where:

µ_A^*(g)= µ_A^*(l) + RT ln x_A                                                                                                        (Equation 5.32)

 

This implies that the presence of a solute at a mole fraction x_B causes an increase in normal boiling point from T* to T*+ ΔT (can you change D to a delta?)

ΔT =Kx_B, where K=RT^*2/Δ_vapH                                                                                             (Equation 5.33)

Since the mole fraction of B is proportional to its molality, b, this equation can be rewritten:

ΔT =K_bb, where K is the boiling-point constant specific to the solvent                      (Equation 5.34)

It is determined that the elevation of the boiling point is a colligative property since Equation 5.33 above only makes reference to the mole fraction of solute and says nothing about the identity of the solute.

NOTE: The value of ΔT does depend on the intrinsic properties of the solvent. Solvents with higher boiling points will often have a larger ΔT.

 Figure 2

Figure 2 above represents two seperate isolated systems. The left (a) is that of a pure liquid, and the right (b) is a solution with dark purple representing the solute. The entropy of liquid a is lower than that of liquid b since liquid a is very ordered, and the entropy of the vapor in liquid a is higher than that of liquid b. Since there is a higher entropy in the liquid in system b, vapor pressure is lowered from that of the pure liquid.

 

 

Freezing Point Depression

If we say the solvent is A and the solute is B, equilibrium is found at a temperature where:

µ_A^*(s)= µ_A^*(l) + RT ln x_A                                                                                                                        (Equation 5.35)

Since the equation above is equivalent to Equation 5.33:

ΔT =K’x_B, where K’=RT^*2/Δ_fusH                                                                                                            (Equation 5.36)

The freezing point depression is ΔT and Δ_fusH is the enthalpy of fusion of the solvent.

For dilute solutions, the mole fraction of B is proportional to its molality, b, this equation can be rewritten:

ΔT =K_fb, where K is the freezing-point constant specific to the solvent                                    (Equation 5.37)

Once again, it is determined that the depression of the freezing point is a colligative property since Equation 5.36 above only makes reference to the mole fraction of solute and says nothing about the identity of the solute.

NOTE: The value of ΔT does depend on the intrinsic properties of the solvent. Solvents with higher melting points and low enthalpies of fusion will often have a larger ΔT.

Cryoscopy is a technique used to determine the molar mass of a solute when freezing-point depression is known.

Reworking Equation 5.37:

ΔT/K_f=b and b=moles per 1kg of solvent, which is equivalent to mass/molar weight per 1kg solvent

Figure 3:

Figure 3 above shows a few examples of freezing-point constants and boiling-point constants for different solvents. The f stands for freezing and the b stands for boiling (it has nothing to do with molality b). 

 

 

Osmosis

This is the spontaneous passage of pure solvent into a solution separated from it by a semipermeable membrane, which is permeable only to the solvent.

Osmotic Pressure, ∏, refers to the pressure that must be applied to the solution to keep the solvent on the other side of the membrane (out of the solution).

           Figure 4 

Figure 4 above shows two seperate chambers seperated by a semipermeable membrane, permeable only to the solvent. This system is currently at equilibrium with constant pressure, P, applied to both sides and osmotic pressure, ∏,  applied only to the solution side. While pure solvent is being maintained in the left chamber, the right chamber does contain solvent.

Van’t Hoff Equation:

Since, at equilibrium, the chemical potentials of solvent on either side of the membrane must be equal, the following equation is implied for dilute solutions.

∏=[B]RT, where B=n_B/V      (the molar concentration of solute)                                                (Equation 5.40)

This can be expanded to Equation 5.41 to account for error. B is know as the osmotic virial coefficient.

∏=[B]RT{1+B[B]..}                                                                                                                                (Equation 5.41)

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Solvent Activities:

It is possible to substitute an activity for concentration, since activity is essentially

the effective concentration, or more specifically the effective mole fraction.

The general form of the chemical potential of a real or ideal solvent is given by:

          Eqn 5.23: m_A=m_A* + RTln(P_A/P_A^*)

The chemical potential for an ideal solution, when the solvent obey’s Rault’s Law:

          Eqn 5.25: m_A=m_A* + RTlnx_A

The chemical potential if solution doesn’t obey Raoult’s Law (not ideal):

          Eqn 5.42: m_A=m_A* + RTlna_A    -because equation 5.23 is true for both ideal and real solutions.

               where a_A=activity, which is an effect of mole fraction

          Eqn 5.43: a_A=(P_A/P_A^*) the vapor pressure of pure solvent divided by the vapor pressure of the

               solution, this makes sense because if the solution where behaving ideally, the vapor pressures

               of the pure solvent and the solution would be the same and activity would equal 1,

               which would indicate a pure liquid, even though it’s a mixture. However, for a

               real solution, the vapor pressures would not be the same and activity accounts for

               that, giving an equation for chemical potential that takes into account derivations

               from ideality.

Example Problem: The vapor pressure of 0.500 M KNO3(aq) at 100°C is 99.95kPa, so the activity of water in the solution at this temperature is?

          a_A=(P_A/P_A^*) = 99.95kPa/101.325kPa (atmospheric pressure) = 0.9864

 

We can see from Eqn 5.42 that activity also has some relation to mole fraction, particularly as the solutions gets closer and closer to ideal. We know that all solvents obey Raoult’s law (which would imply that a_A=x_A or P_A/P_A^*=x_A) as the concentration of the solute approaches zero. Therefore as the solution gets more and more diluted (closer to ideal), and the mole fraction approaches 1 (pure solvent) the activity of the solution approaches ideality (activity of an ideal solution would equal 1). At ideal a_A=x_A so a_A®x_A as x_A®1.

 

This convergence can be expressed by an activity coefficient, g, (can you use gamma instead of g) which expresses deviations from ideality.

          Eqn 5.45 a_A=g_Ax_A and g_A®1 as x_A®1 by definition.

When this is accounted for in equation 5.25 we get

          Eqn 5.46 m_A=m_A* + RTlnx_A + RTlng_A

               where g is a correction factor, it takes into account all deviations from ideal in one number.

 

 

Solute Activites:

A solute that obey’s Henry’s law has a vapor pressure that is given by P_B=K_BX_B, where K_B is Henry’s law constant.

The chemical potential of the solute is m_B=m_B* + RTln(P_B/P_B^*)

using Henry’s law constant (the empirical constant) we get

          m_B=m_B* + RTln(K_B/P_B^*) + RTlnx_B.

Since this equation depends only on properties of the solute, the first and second terms can be combined to give a new standard chemical potential, which is simply just a function of properties of the solute.

          Eqn 5.47 m_B^q=m_B* + RTln(K_B/P_B^*)

substitute this in for the equation for chemical potential and we get

          Eqn 5.48: m_B=m_B^q+ RTlnx_B for an ideal solution.

 

For a real solute, we substitute mole fraction for activity

          Eqn 5.49: m_B=m_B^q+ RTlna_B.

               Where a_B=(P_B/K_B) and the activity coefficient can be introduced by a_B=g_Bx_B.

As the concenration approaches 0, the solute obey’s Henry’s law and a_B®x_B and g_B®1 as x_B®0.

This is different than with solvents because in order to be ideal we want the mole fraction of the solvent to be 1 (pure) and the mole fraction of the solute to be 0.

          Eqn 5.53: m_B=m_B^q+ RTlnb_B

               where b_B is the molality, which we want to replace with activity

          a_B=g_Bx_B and a_B=g_B(b_B/b^q)

               where b_B is the molality, and b^qis standard molality.

               This is used simply to get rid of the units of molality since activity has no units.

g_B®1 as b_B®0, meaning as we decrease molality, the activity coefficient is going to approach ideality (approaching 1=activity coefficient of a pure solute=no deviations from ideal). Finally,

          Eqn 5.55: m_B=m_B^q+ RTlna_B

               which is the equation for the chemical potential of a real solute at any molality.