Lecture 6. NONELECTROLYTE SOLUTONS

Lecture 6. NONELECTROLYTE SOLUTONS

NONELECTROLYTE SOLUTIONS SOLUTIONS single phase homogeneous mixture of two or more components NONELECTROLYTES do not contain ionic species.

CONCENTRATION UNITS percent by weight wt. of wt. of solvent solute wt. of solute X100% wt of solute wt. of solution X100% mole fraction (x i ) no. of moles of component i no. of moles of all components

CONCENTRATION UNITS Molarity (M) no. of moles of solute L of solution Molality (m) no. of moles of wt. of solvent solute in kg

PARTIAL MOLAR VOLUME Imagine a huge volume of pure water at 25 C. If we add 1 mol H 2 O, the volume increases 18 cm 3 (or 18 ml). So, 18 cm 3 mol -1 is the molar volume of pure water.

PARTIAL MOLAR VOLUME Now imagine a huge volume of pure ethanol and add 1 mol of pure H 2 O it. How much does the total volume increase by?

PARTIAL MOLAR VOLUME When 1 mol H 2 O is added to a large volume of pure ethanol, the total volume only increases by ~ 14 cm 3. The packing of water in pure water ethanol (i.e. the result of H-bonding H interactions), results in only an increase of 14 cm 3.

PARTIAL MOLAR VOLUME The quantity 14 cm 3 mol -1 is the partial molar volume of water in pure ethanol. The partial molar volumes of the components of a mixture varies with composition as the molecular interactions varies as the composition changes from pure A to pure B.

V V n J p,t,n '

PARTIAL MOLAR VOLUME When a mixture is changed by dn A of A and dn B of B, then the total volume changes by: dv V n A p,t,n B dn A V n B p,t,n A dn B

PARTIAL MOLAR VOLUME for a two component system (A, B), the volume of the system at a particular composition is given by : V V A n A V B n B

PARTIAL MOLAR VOLUME How to measure partial molar volumes? Measure dependence of the volume on composition. Fit a function to data and determine the slope by differentiation.

PARTIAL MOLAR VOLUME Molar volumes are always positive, but partial molar quantities need not be. The limiting partial molar volume of MgSO 4 in water is -1.4 cm 3 mol -1, which means that the addition of 1 mol of MgSO 4 to a large volume of water results in a decrease in volume of 1.4 cm 3.

PARTIAL MOLAR GIBBS ENERGIES The concept of partial molar quantities can be extended to any extensive state function. For a substance in a mixture, the chemical potential, is defined as the partial molar Gibbs energy. G i μ i G n T,P,n

PARTIAL MOLAR GIBBS ENERGIES Using the same arguments for the derivation of partial molar volumes, G n A μ A n B μ B Assumption: Constant pressure and temperature

THERMODYNAMICS OF MIXING So we ve seen how Gibbs energy of a mixture depends on composition. We know at constant temperature and pressure systems tend towards lower Gibbs energy. When we combine two ideal gases they mix spontaneously, so it must correspond to a decrease in G.

THERMODYNAMICS OF MIXING

DALTON S LAW The total pressure is the sum of all the partial pressure. p j x j p p p ( x x ) p p A B A B

THERMODYNAMICS OF MIXING G m G m RT ln RT ln p p RT ln p p p mix G nrt( x A ln x A x B ln x B ) x A, x B n total number of moles are mole fractions of components A, B, respectively

Thermodynamics of mixing RT ln p G i n A A RT ln p n B B RT ln p G f n A A RT ln p A n B B RT ln p B mix G n A RT ln p A p n B RT ln p B p

Thermodynamics of mixing mix G n A RT ln p A p n B RT ln p B p p A p x A p A p x B mix G n A RT ln x A n B RT ln x B x A n n A x B n n B mix G nrt ( x A ln x A x B ln x B ) mix G 0

THERMODYNAMICS OF MIXING

SAMPLE PROBLEM: A container is divided into two equal compartments. One contains 3.0 mol H 2 (g) at 25 C; the other contains 1.0 mol N 2 (g) at 25 C. Calculate the Gibbs energy of mixing when the partition is removed.

Gibbs energy of mixing mix G nrt ( x A ln x A x B ln x B ) mix G 3.0(RT ln 3 4 ) 1.0(RT ln 1 4 ) p mix G 2.14 kj 3.43 kj mix G 5.6 kj p

ENTHALPY, ENTROPY OF MIXING mix G H mix T mix S for ideal solutions, H 0 mixg T mix S mix S nr( x A ln x B x B ln x B )

Other mixing functions Other mixing functions T) and p (constant 0 H S T H G ) x ln x x ln x ( nr S ) x ln x x ln x ( nrt G B B A A mix B B A A mix

IDEAL SOLUTIONS To discuss the equilibrium properties of liquid mixtures we need to know how the Gibbs energy of a liquid varies with composition. We use the fact that, at equilibrium, the chemical potential of a substance present as a vapor must be equal to its chemical potential in the liquid.

IDEAL SOLUTIONS A * Chemical potential of vapor equals the chemical potential of the liquid at equilibrium. * A RT ln p * A denotes pure substance If another substance is added to the pure liquid, the chemical potential of A will change. A A RT ln p A A is the standard chemical potential at p 1 bar

Ideal Solutions A * A A A * RT ln p A * RT ln p A * A A RT ln p A A A * A A * RT ln p A * RT ln p A p A * RT ln p A

RAOULT S LAW p A x A p A * The vapor pressure of a component of a solution is equal to the product of its mole fraction and the vapor pressure of the pure liquid.

SAMPLE PROBLEM: Liquids A and B form an ideal solution. At 45 o C, the vapor pressure of pure A and pure B are 66 torr and 88 torr, respectively. Calculate the composition of the vapor in equilibrium with a solution containing 36 mole percent A at this temperature.

IDEAL SOLUTIONS: A A * p A x A p A * RT ln p A p A * A A * RT ln x A

Non-Ideal Solutions

Ideal-dilute dilute solutions Even if there are strong deviations from ideal behaviour, Raoult s law is obeyed increasingly closely for the component in excess as it approaches purity.

Henry s s law For real solutions at low concentrations, although the vapor pressure of the solute is proportional to its mole fraction, the constant of proportionality is not the vapor pressure of the pure substance.

Henry s s Law p B x B K B Even if there are strong deviations from ideal behaviour, Raoult s law is obeyed increasingly closely for the component in excess as it approaches purity.

Ideal-dilute dilute solutions Mixtures for which the solute obeys Henry s Law and the solvent obeys Raoult s Law are called ideal-dilute dilute solutions.

Properties of Solutions We ve looked at the thermodynamics of mixing ideal gases, and properties of ideal and ideal-dilute dilute solutions, now we shall consider mixing ideal solutions, and more importantly the deviations from ideal behavior.

G i n A A * Ideal Solutions n B B * * A A RT ln x A * * G f n A { A RT ln x A } n B { B RT ln x B } mix G n A RT ln x A n B RT ln x B mix G nrt { x A ln x A x B ln x B }

Ideal Solutions mix G nrt { x A ln x A x B ln x B } mix S nr { x A ln x A x B ln x B } mix H 0

Ideal Solutions mix G nrt { x A RT ln x A x B RT ln x B } G p T V mix G p T mix V 0

Real Solutions Real solutions are composed of particles for which A-A, A A, A-B A B and B-B B B interactions are all different. There may be enthalpy and volume changes when liquids mix. G= G=H-TS So if H H is large and positive or S S is negative, then G G may be positive and the liquids may be immiscible.

Excess Functions Thermodynamic properties of real solutions are expressed in terms of excess functions, X E. An excess function is the difference between the observed thermodynamic function of mixing and the function for an ideal solution.

Real Solutions S E G E mix S mix S ideal mix G mix G ideal H E mix H mix H ideal mix H V E mix V mix V ideal mix V

Real Solutions Benzene/cyclohexane Tetrachloroethene/cyclopentan e

Colligative Properties A colligative property is a property that depends only on the number of solute particles present, not their identity. The properties we will look at are: lowering of vapor pressure; the elevation of boiling point, the depression of freezing point, and the osmotic pressure arising from the presence of a solute. Only applicable to dilute solutions.

Colligative Properties All the colligative properties stem from the reduction of the chemical potential of the liquid solvent as a result of the presence of solute. A A * RT ln x A

Thermodynamics of mixing

Boiling Point Elevation How do we figure out where the new boiling point is when a solute is present? Look for the temperature at which at 1 atm,, the vapor of pure solvent vapor has the same chemical potential as the solvent in the solution.

Boiling Point Elevation Let s s denote solvent A and solute B. Equilibrium is established when: A ( g ) A * ( g ) RT ln x A A * ( l) RT ln x A See Justification 5.1 (Atkins) T Kx B K RT *2 vap H

Boiling Point Elevation T Kx B K RT *2 vap H T T makes no reference to the identity of the solute, only to its mole fraction. So the elevation of boiling point is a colligative property.

Boiling Point Elevation T Kx B For practical purposes : T K b b K b boiling point constant; b molality

Freezing Point Depression Let s s denote solvent A and solute B. Equilibrium is established when: A ( s) * A ( l) RT ln x A Same calculation as before (Justification 5.1) T K x B K RT *2 fus H

Freezing Point Depression T K x B For practical purposes : T K f b K f freezing point constant; b molality

Cryoscopy T K f b K f freezing point constant; b molality b n 1 kg solvent m M 1 kg solvent

Solubility Although solubility is not strictly a colligative property (because solubility varies with the identity of the solute), it may be estimated using the same techniques. When a solid solute is left in contact with a solvent, it dissolves until the solution is saturated with the dissolved solute.

Solubility B ( s) B * ( l) RT ln x B See Justification 5.2 ln x B fus H R 1 T f 1 T

Osmosis Osmosis refers to the spontaneous passage of a pure solvent into a solution separated from it by a semi-permeable membrane. In this case, the membrane is permeable to the solvent but not to the solute.

Osmosis The osmotic pressure, is the pressure that must be applied to the solution to stop the influx of solvent. Examples of osmosis includes the transport of fluids across cell membranes and dialysis. See Justification 5.3

van t Hoff equation [ B ]RT [ B ] n B V For molecules of large molar mass, such as polymers or biological macromolecules, a viral- like expansion used to correct for non-ideality. B is the osmotic viral coefficient. [ B ]RT {1 B[B ]...}

Solvent Activities The general form of the chemical potential of a real or ideal solvent is given by: A A * RT ln * p A p A For an ideal solution, when the solvent obeys Raoult s law, then: A A * RT ln x A

Solvent Activities If a solution does not obey Raoult s law, we can still use a form of the chemical potential equation: A A * RT ln a A The quantity a A is the activity of A. It can be considered an effective mole fraction.

Solvent Activities Because this equation is true for real or ideal solvent, we can easily see that: a A p A p A *

Activities

Calculating solvent activity The vapor pressure of 0.500 M KNO 3 (aq) at 100 C C is 99.95 kpa so the activity of water in the solution at this temperature is a A a A p A p A * 99.95 kpa 101.325 kpa 0.9864

Solvent Activities Because all solvents obey Raoult s law increasingly close as the concentration of solute approaches zero, the activity of the solvent approaches the mole fraction as x A =1 p A p A * x A a A x A as x A 1

Solvent Activities A way of expressing this convergence is to introduce the activity coefficient, a A A x A A 1 as x A 1 A A * RT ln x A RT ln A

Solute Activities Ideal-dilute dilute solutions obeys Henry s s law has a vapor pressure given by p B =K B x B, where K B is Henry s s law constant. B B * RT ln p B p B * * B RT ln K B p B * RT ln x B B B B * B RT ln K B p B * RT ln x B

Solute Activities For real solutions we can replace x B with a B B B RT ln x B B RT ln a B a B p B K B a B B x B a B x B and B 1 as x B 0

B B Solute Activities The selection of a standard state is entirely arbitrary, so we are free to choose one that best suits our purpose and description of the composition of the system RT ln b B a B B b B b where B 1 as b B 0 B B RT ln a B

Activities of regular solutions Ignore section 5.8

Ion Activities If the chemical potential of a univalent M + is denoted + and that of a univalent anion X - is denoted -, the total molar Gibbs energy of the ions in the electrically neutral solution is the sum of these partial molar quantities. G m ideal ideal ideal

G m G m ideal G m ideal Ion Activities For a real solution of M + and X - of the same molality ideal RT ln RT ln RT ln

Ion Activities Because experimentally a cation cannot exist in solution without an anion, it is impossible to separate the product + - into contributions from each ion, we introduce the mean activity coefficient 1 2

Ion Activities The individual chemical potentials of the ions are: ideal ideal RT ln RT ln

Ion Activities If a compound M p X q that dissolves to give a solution of p cations and q anions. G m p q G m ideal G m prt ln qrt ln q p 1 s s p q u i u i ideal RT ln

Ion Activities G m p q u i u i ideal RT ln G m G m ideal prt ln qrt ln

Debye-Hückel theory The departure from ideal behavior in ionic solutions can be mainly attributed to the Coulombic interaction between positively and negatively charged ions. Oppositely charged ions attract one another. As a result anions are more likely found near cations in solution, and vice versa.

Debye-Hückel theory

Debye-Hückel theory Although overall the solution is neutrally charged, but near any given ion there is an excess of counter ions. Averaged over time this causes a spherical haze around the central ion, in which counter ions outnumbers ions of the same charge as the central ion, has a net charge the same but magnitude but opposite sign of the central ion, and is called the ionic atmosphere.

Debye-Hückel theory The chemical potential of any given central ion is lowered as a result of its electrostatic interaction with its ionic atmosphere. This lowering of chemical potential is due to the activity of the solute and can be identified with RTln ±.

Debye-Hückel theory log z z AI 1 2 A 0.509 for an aqueous solution at 25 z i At very low concentrations the activity coefficient can be calculated from the Debye-Hückel limiting law charge number of an ion I ionic strength of the solution o C

Debye-Hückel theory When the ionic strength is too high for the limiting law to be valid, the activity coefficient can be estimated from the Debye-Hückel extended law log Az z I 1 2 1 BI 1 2 CI

Debye-Hückel theory I 1 2 i z i 2 b i b

Debye-Hückel theory