THERMODYNAMICS OF SOLUTIONS Solutions quantification
Transcript Of THERMODYNAMICS OF SOLUTIONS Solutions quantification
THERMODYNAMICS OF SOLUTIONS
Solutions quantification ..................................................................................................................................... 1 Mixtures and solutions................................................................................................................................... 1 Concentration specification ........................................................................................................................... 4 Moles-of-solute variables .......................................................................................................................... 5 Mass-of-solute variables ............................................................................................................................ 6 Volume-of-solute variables ....................................................................................................................... 7 Concentration measurement .......................................................................................................................... 8 Solution preparation from pure solutes and concentrated solutions. Dilution............................................... 8 Dissolving rate ............................................................................................................................................... 9 Electrolytic solutions ..................................................................................................................................... 9
Solution properties........................................................................................................................................... 10 Phase diagram .............................................................................................................................................. 10 Thermodynamics of solubility..................................................................................................................... 11 Qualitative solubility. Solubility rules ..................................................................................................... 12 Solute equilibrium and its temperature variation..................................................................................... 12 Solubility equilibrium for gas solutes...................................................................................................... 13 Solubility equilibrium for solid solutes ................................................................................................... 17 Partition of a solute between two immiscible solvents............................................................................ 20 Supersaturation. Undercooling and overheating. Hydrates and clathrates .................................................. 20 Density......................................................................................................................................................... 22 Heat of solution............................................................................................................................................ 23 Freezing mixtures .................................................................................................................................... 24 Colligative properties................................................................................................................................... 25 Vapour-pressure depression..................................................................................................................... 25 Freezing-point depression........................................................................................................................ 25 Boiling-point increase.............................................................................................................................. 26 Osmotic pressure ..................................................................................................................................... 26
Properties of particular solutions ..................................................................................................................... 27 Thermochemical data of solutes ...................................................................................................................... 27 Solubility data for aqueous solutions............................................................................................................... 27 Density data for solutions ................................................................................................................................ 27 Heat of solution data ........................................................................................................................................ 27 Cryoscopic and ebullioscopic data .................................................................................................................. 27
Solutions quantification
Mixtures and solutions
A mixture is any multi-component system, i.e. one with several chemical species. The thermodynamics of
mixtures in general (gaseous, liquid or solid) has been considered under the heading Mixtures, mainly
devoted to ideal mixtures. The term 'solution' is mostly used for the special case of a mixture between very
dissimilar components, i.e. when a small amount of substance, called solute (solid, liquid or gas), dissolves
to a certain limit in a liquid or solid substance (pure, or a mixture itself) called the solvent. In many solutions
of interest, the sum of the mole fractions of the solutes is small compared to unity, and the solution is called
a dilute solution. We assume true solutions, i.e. homogeneous solutions, and do not consider colloids and
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suspensions, treated under the heading Mixture settling. Reacting mixtures are covered in Chemical reactions, aside.
Most solutions depart from the ideal-mixture-model developed in Mixtures, but it is important to recall the basic result from the multiphase equilibrium of ideal mixtures, i.e. Raoult's law for ideal vapour-liquidequilibrium (VLE):
xi,gas = pi*(T ) (for ideal mixtures)
(1)
xi,liq
p
i.e., for any component i, its molar fraction in the vapour phase, xi,gas, is to its molar fraction in the liquid phase, xi,liq, as its pure-component vapour-pressure at that temperature, pi*(T ) , is to the actual pressure, p.
If a solution behaves as an ideal mixture, Raoult's law for each component in a binary mixture of solute 's' in solvent 'dis', in terms of solute mole fractions would read:
= xs,gas p= s* (T ) and 1− xs,gas pd*is (T )
(2)
xs,liq
p
1 − xs,liq
p
which can be approximately solved in the following limits like that:
for p* >> p* (e.g. salts in water) = xs,gas
dis
s
xs,liq
p
* s
(T
)
<< 1
and=p
pd*is (T )
p* dis
(1
−
xs,liq
)
(3)
( ) for p* << p* (e.g. gases in water) = xs,liq
p* dis
(T
)
<< 1
and =p
p* 1− x
(4)
dis
s
xs , gas
p
* s
(T
)
s
s ,liq
with the following interpretation: • when a non-volatile solute dissolves in a liquid, (3), it lowers the solvent VLE-pressure, and its fraction on the vapour phase is proportional to its vapour pressure when pure (negligible); • when a very-volatile solute dissolves in a liquid (4), it lowers the solute VLE-pressure, and its fraction on the liquid phase is proportional to the vapour pressure of the pure solvent (small).
Solutions are everywhere: seawater (but also tap water), coffee, soda, wine, vinegar, gasoline, antifreeze, body fluids (e.g. human plasma is roughly an 8%wt proteins plus 1% salts in water, milk serum is roughly a 5%wt lactose plus 3% proteins in water, urine is roughly a 2.5%wt urea plus 2%wt salts in water), etc. Milk may appear to be a homogeneous mixture to the unaided eye, but the tiny oil and protein droplets in the system make milk to appear white (milk is a colloid). Sea water contains many ions in addition to sodium and chloride ions; when gradually evaporated, the first salt to precipitate is CaCO3 (present to the extent of 0.12 g/L), followed by CaSO4·H2O (1.75 g/L), then NaCl (29.7 g/L), MgSO4 (2.48 g/L), MgCl2 (3.32 g/L), NaBr (0.55 g/L), and KCl (0.53 g/L).
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Solvents should be inert (not react with solutes), volatile (low boiling point, to ease distillation), and recyclable. They are usually grouped as:
• Polar solvents (water, methanol, ammonia). Polar solvents are hydrophilic, usually H-bond donors (like water itself, the most important solvent by far), or through hydroxyl groups as in ethanol (CH3CH2OH), or through carboxylic groups like in acetic acid (CH3-COOH). Polarity is measured by the dielectric constant. Some polar solvents are organic, notably acetone (they are not H-bond donors).
• Non-polar solvents (benzene, toluene, ether, hexane, carbon tetrachloride, trichloromethane (chloroform), methylene chloride, gasoline, mineral oil). Most organic solvents, greatly used in the chemical and biochemical industries (food and pharmacy), are harmful to health and the environment, and are being replaced by supercritical fluids. 'Dry cleaning' is not dry at all; it is said dry because the liquid is not soaking water by a volatile solvent (usually 1,1,2,2-tetrachloroethene). Carbon dioxide is a non-polar fluid, but has poor solvent properties outside supercritical conditions.
• Supercritical fluids (carbon dioxide, ethane, trifluoromethane (fluoroform), sulfur hexafluoride, dinitrogen oxide (nitrous oxide), carbon disulphide). Supercritical fluids show a dissolving power quickly increasing with pressure (up to a limit). Pure supercritical carbon dioxide is a relatively nonpolar solvent, but has some limited affinity with polar molecules due to its large molecular quadripole, although modifiers (e.g. methanol, fluorinated hydrocarbons) can be added to improve the solubility of polar molecules. Supercritical water becomes totally miscible with many organic compounds like heptane and benzene. An advantage of supercritical solvents is that they yield the dry solute by simple evaporation of the solvent. The range for supercritical applications is p/pCR=1..2 and T/TCR=1.0..1.1.
The main solvent is water, and thus most solutions are aqueous solutions. Most solutes are solid powders (e.g. salt-water solutions). Water scarcely dissolves gases and organic liquids, and liquid-liquid solutions may be considered plain mixtures; that is why the basic understanding of ‘to dissolve’ is to untie a solid to become part of a liquid. It is the solvent that pulls the solute into solution, and its internal motion (diffusion and convection) what spreads the solute around. Some details on mixing can be found aside in Mass transfer and in Chemical kinetics.
Usually, solutes do not dissolve on the solvent vapour (i.e. they are usually non-volatile), neither on the solvent solid phase (they do not fit into the crystalline lattice). Both mentioned-processes can be used to get rid of solutes (e.g. salt water becomes fresh water when vaporised or solidified), at least when done carefully, but, in practice, solutes can be entrained by solvent-vapours (mainly as an aerosol of particles) and can be trapped within the solid solvent (at the seashore, one may feel salty lips just from the marine breeze, and some ice may taste salty too).
All solutions have some stratification of composition at equilibrium, as studied in Mixture settling. Besides, solution tend to change the composition with time in open systems, i.e. in solutions open to the atmosphere or when the solution is blooded someway, since the escape will not have the same average composition; e.g. the composition inside a liquefied-petroleum-gas bottle changes during use, becoming richer in the heavier components, as well as within a liquefied-natural-gas tanker, due to the boil-off.
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3
The relative amount of solute in a solution (its concentration) may be measured by different concentration variables related amongst them (but not always in a trivial manner); since all solutions depart from ideal (Raoult’s law) mixtures, the use of molar quantities for the solvent is rare (molar quantities were an advantage for modelling ideal mixtures because, in that case, the behaviour is just proportional to the molar composition, independently of the actual substances involved).
Concentration specification
One of the main difficulties in the study of solutions is the wide variety of variables and units used to specify concentration (in the case of ideal mixtures, only molar fractions, and sometimes mass fractions, were common). Generically speaking, concentrations in a mixture express the quantitative composition (although, under the SI standard, composition must refer to the unit volume of the system, i.e. mol/m3, kg/m3). The subject gets obscure also by the lack of a clear nomenclature, e.g. to distinguish with a letter from 'solution' (we will use subscript m, from mixture), 'solute' (we will use subscript s, from solute) and 'solvent' (we will use subscript dis, from dissolve); the standard labelling of '1' for solute and '2' for solvent, may be unclear when one wants to distinguish between solvent and solution.
To give an idea of the variety of variables used, consider the following values for oxygen-solubility in water (all of them valid under certain circumstances): 39 mg/kg, 0.039 kg/m3, 1.2 mol/m3, 22 ppm, 4.6 ppm, 0.26 mol/m3, 8.2 mg/L, 5.9 mL/L, 55 mL/L, to name a few. And, fortunately, oxygen solubility is so small that the value is the same by amount of solution than by amount of solvent. The explanation of the above figures is that 78 mg/kg is equal to 78 mg/L for aqueous solutions where 1 L has 1 kg; it is also equal to 78/32=2.4 mol/m3; and it is also equal to 22 parts per million molar, but these values refer to concentration of dissolved oxygen in equilibrium with pure oxygen at 25 ºC and 100 kPa (a common physical standard state). However, when the equilibrium is not with pure oxygen but with air at 25 ºC and 100 kPa, previous values have to be multiplied by xO2,air=0.21, giving 22·0.21=4.6 ppm, 39·0.21=8.2 mg/L, equal to 8.2 mg/kg for water, and equal to 1.2·0.21=0.26 mol/m3. Finally, the volumetric fractions refer to the volume at 0 ºC and 100 kPa that the full out-gassing of dissolved oxygen in one litre of solution at a given temperature, would occupy: 55 mL/L means that the oxygen dissolved in 1 L of solution would occupy 55 mL as a pure gas at 0 ºC and 100 kPa, a value that is correct for a solution at 25 ºC saturated against pure oxygen at 0 ºC and 100 kPa; if it were against air instead of pure oxygen, the value would be 55·0.21=11.6 mL/L for air at 0 ºC, and, as solubility of gases decreases with temperature and nearly halve from 0 ºC to 25 ºC, in equilibrium with air at 25 ºC and 100 kPa the value is 5.9 mL/L for air at 25 ºC).
We assume a single solute, s (of molar mass Ms) dissolved in a solvent, dis (of molar mass Mdis), but the generalisation to multiple solutes is trivial. Composition may be quantified by the ratio of solute to solution (e.g. moles of solute per litre of solution, grams of solute per kilogram of solution), or by the ratio of solute to solvent, but the former ratio will be favoured here, except in the case is molalities (commonly used to express salt solubilities, however).
A possible grouping for solution quantification may be:
• Moles-of-solute variables.
• Mass-of-solute variables.
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4
• Volume-of-solute variables.
WARNING. In the study of solutions, the litre, L (sometimes with a less distinguishable lower case l), is often used as volume-unit instead of the cubic meter (1 L ≡ 1 dm3), and the millilitre, mL, is also found in common usage, instead of the cubic centimetre (1 mL≡ 1 cm3).
Moles-of-solute variables • Molar fraction of solute, xs, i.e. moles of solute per mole of solution, either per-one, percent (%) or per million (ppm). Its definition, and relation with other variables, is:
ms
x ≡ ns =
Ms =
1 = c Mm
(5)
s ns + ndis ms + mdis 1+ 1 M s s ρm
M s M dis
ys M dis
where Mm is the molar mass of the mixture, Mm=xsMs+xdisMdis. For instance, when adding a 10 g bag of sugar (C12H22O11) to a 100 cm3 cup of boiling water (96 g of water, yx=10/(10+96)=9.4% by weight, having taken 958 kg/m3 for boiling water), one gets a solution with 0.010/0.342=0.030 moles of sucrose and 958·100·10-6/0.018=5.3 moles of water, i.e. xs=0.03/(0.03+5.3)=0.56%, and a concentration of 0.3 mol/L. A problem with the amount of substance is that it is not conserved in electrolytic solutions; e.g. seawater with 35 g/kg of NaCl has a molar fraction of dissolved particles of xs=0.022, mainly sodium cations and chlorine anions; i.e. 1% of the particles are Na+, 1% are Cl-, and the rest mainly water molecules; lab seawater may be prepared by adding 0.035 kg of NaCl (0.6 mol) to 1 kg of H2O (55 mol). • Molar ratio, rs≡ns/ndis, i.e. moles of solute per mole of solvent; rarely used. • Molar density, or concentration, cs, (sometimes written as [formula of solute], e.g. cN2≡[N2]) i.e. moles of solute per cubic meter of solution;
c ≡ n=s xsnm= x ρm= ms / M=s ρs
(6)
s Vm mm ρm s M m
Vm
Ms
e.g. seawater with 3.5%wt NaCl has a molar density of 600 moles of Na+ per cubic meter of solution 0.6 mol/L of Na+), and the sugar solution mentioned above (10 g sucrose in 100 cm3 of boiling water) has a concentration of 0.03/106·10-6=28 mol/m3 (0.28 mol/L, where the fact that adding 10 g sucrose increases 6.5 cm3 the volume of solution has been used). The problem with molar density is that it
changes with temperature even if the system is closed (due to overall volume change).
Molarity, is just the common way to express molar density, cs, but taking as unit 1 mol/L instead of 1 mol/m3, and given it a special, not recommended, symbol: 1 M ≡ 1 mol/L e.g. seawater with 3.5%wt
NaCl has approximately 0.6 mol of NaCl per litre of aqueous solution (referred too as 0.6 M NaCl
solution (but notice that it is actually a 1.2 mol/L solution since each mole of NaCl yields two moles
in solution), and the sugar solution mentioned above is a 0.28 M sucrose aqueous solution (0.28
mol/L solution). WARNING. Some authors use M as the symbol for the mole, instead of the SI-
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5
recommended 'mol', and thus write e.g. that seawater is a 0.6 M/L solution, further contributing to the already wide confusion in nomenclature. Normality, is the mole-equivalent amount of solute (moles divided by valence or replaceable H atoms) per litre of solution; i.e., a normalised molar density, cs. Often, a special symbol, not recommended, is used for normality: 1 N ≡ 1 mol/L e.g., since NaCl has valence equal to 1, seawater is approximately a 0.6 N Na+ and 0.6 N Cl- water solution (in this case 1 N is also 1 M). Normality of a closed system changes with temperature as molarity (molar density) does. • Molality, bs, is the amount of solute per mass of solvent, not of solution. Molality is an easy way to specify recipes (the ingredients, instead of the final result), and does not change with temperature in a closed system. Molality is not a recommended composition descriptor; sometimes m is used as symbol for the molality variable, and, to add confusion, sometimes 'm' is used as a molality unit, 1 m ≡ 1 mol/kg; e.g. seawater with 3.5%wt NaCl is approximately a 0.6 m (pronounced 0.6 molal) NaCl solution in water, i.e. adding 0.6 mol of NaCl to one kilogramme of water produces lab seawater. Notice that, although sometimes molality is expressed in mol/L, it is a liberty for the case of water where 1 kg roughly occupies 1 L, but molality is always based on mass of the solvent and does not change with temperature; furthermore, for dilute aqueous solutions, molality values in mol/L (of solvent), practically coincide with molarity values in mol/L (of solution).
b ≡ ns = ns = ms / M s= ws
(7)
s mdis ρ V dis dis
mdis
Ms
Mass-of-solute variables Mass-of-solute (or weight) variables have the advantage over molar quantities in that the molecular structure has not to be known; e.g. how many moles are there in one litre of gasoline?):
• Mass fraction of solute, ys, i.e. kilograms of solute per kilogram of solution, either per-one, percent (%) or per million (ppm); e.g. seawater has a 3.5% mass fraction of dissolved salts (the majority NaCl). WARNING. The recommended symbol under SI-conventions for the mass fraction is w instead of y (but we use w for mass ratio, here).
y ≡ = ms ms=
xs M s
(8)
s mm mdis + ms xdis M dis + xs M s
• Mass ratio, ws, usually in units of grams of solute per kilogramme of solvent (absolute humidity is such a mass-ratio function, widely used in Humid air). It is directly related to molality.
w ≡ ms = nsM s = b M =
ms
= ys
(9)
s mdis mdis
s s mdis + ms − ms 1− ys
• Solute density in the solution, ρs, or mass concentration, usually in units of grams per litre of solution, g/L (many times in mg/L, and the most times in Pharmacy and Medicine in mg/dL (milligrams per decilitre of solution).
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6
ρ ≡ ms = y ρ = c M = x ρ M s
(10)
s Vm
sm
ss
s m Mm
• Mass of solute per unit volume of solvent, e.g. in grams of solute per decilitre of solvent (e.g. g/dL means now milligrams per decilitre of solvent), used sometimes to specify solubility data for solids. It can be easily converted to mass ratio by knowing the density of the solvent, ρdis, by (9).
= ms Vdis
m= s mdis ρdis
w= s ρdis
cs M s ρdis
(11)
Volume-of-solute variables • Volume fraction, vf, or volume percent (the unit usually being mL/L, or %vol, or % (v/v)), is the volume of pure solute added divided by the sum of solute and solvent volumes before mixing (or after separation, i.e. not by the volume of the solution prepared, although the difference may be negligible for dilute solutions of condensed solutes); i.e. it refers to the recipe to prepare the solution, not to what is obtained. Volume percent, or sometimes volume per volume, is often used when the solute is liquid itself; e.g. ethanol 96% means an aqueous solution that, when separated, corresponds to 96 volumes of ethanol and 4 volumes of water; e.g. a 12% ethanol in water (v/v) solution of 1 L is obtained by mixing 120 cm3 pure ethanol with 880 cm3 pure water.
v ≡ Vs =
1=
1=
1
(12)
f
Vm + Vdis
1+ ρs / ρdis
1+
ρs
1
−1
1+
ρs
M dis
1
−1
ms / mdis
ρdis ys
ρdis M s xs
• Notice however that there are occasions were the inverse volume fraction is used. Volume ratio, vr, also named Bunsen solubility coefficient, is the volume of pure solute divided by the solvent volumes before mixing (or after separation, i.e. not by the volume of the solution prepared, although the difference may be negligible); i.e. it refers to the recipe to prepare the solution, not to what is obtained. Notice however that there are occasions were the inverse volume fraction is used, as when saying that oxygenated water has 10 volumes of H2O per one of H2O2. For solid and liquid solutes forming dilute solutions, volume ratio and volume fraction practically coincide since Vs,pure<
When specifying concentrations in dimensional units, temperature must be specified also if it affects densities, and, if gases are involved, pressure too must be specified.
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7
Concentration measurement
The experimental finding of qualitative or quantitative composition in a mixture is known as chemical analysis, or simply analysis. We focus here on concentration analysis, assuming the substances involved are already known.
Most methods of concentration analysis are based on measuring solution density (provided the density dependence on solute concentration, ρm=ρ m(T,p,xi), is known beforehand by calibration), by one of the different techniques:
• Gravimetry. Weighting a known volume of liquid. This is perhaps the easiest and quickest method to measure solution concentration, but requires sampling (e.g. milk producers are paid according to milk density).
• Refractometry. By ray tracing on a sample or in a process flow. Refractive index varies almost linearly with density. This is the easiest way for well-known solutions.
• Resonant vibration. The natural frequency of an encapsulated liquid sample precisely metered depends on its mass. May be applied to a liquid flowing along a bend in a pipe (see Coriolis flowmeter).
• Sonic velocimetry. Density is obtained from ρ =E/c2, where E is the bulk modulus of the solution and c the sound speed through it (see Ultrasonic flowmeter).
• Electric conductivity. This is the best method for very low concentration of electrolytic solutions (e.g. the preferred method for salt-water solutions). The measuring electrodes may be generic, or selective for some specific ion (e.g. Ca2+, NH4+, Cl-, NO3-).
Solution preparation from pure solutes and concentrated solutions. Dilution
Solutions can be prepared by simple dilution, either from pure solutes, or from concentrated solutions. Preparing solutions based on molality specifications (you say dissolving with, e.g. dissolving 300 g of NaCl with 1 kg of water) is easy: just measure, add and stir, whereas preparing solutions based on molarity specifications (you say dissolving in), is not so easy: you start with less that a litre of water, then add the solute, stir and keep adding water until a full litre of solution is obtained. To make it more clear, it is difficult to dissolve 300 g of NaCl with 1 kg of water, but it is impossible to dissolve 300 g of NaCl in 1 kg of water solution, since the solubility limit at 15 ºC is 26.4% by weight of NaCl in solution, or 360 g of NaCl with 1 kg of solvent).
Maintaining pure solutes may be tricky, particularly if they are readily oxidised, or hygroscopic, or very volatile or even gaseous. Many salts and hydroxides (e.g. NaOH(s)) are hygroscopic, and will absorb enough water from the air to change its apparent mass if left exposed to the air for long.
Many times, a desired solution is not prepared with the pure solute but from a more concentrated solution (to do it from a more diluted solution you might let some solvent to evaporate, but this is not practical). The basic rule to remember is that adding a concentrated solution to pure solvent maintains the amount of solute, thus ns=csVm=cs,finalVm,final=cs,initialVm,initial, from where the appropriate amount of concentrated solution needed, Vm,initial, may be found.
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A basic precaution for the preparation of well-defined aqueous solutions is to use distillated water, instead of tap water, and keep the solution in a closed container.
Dissolving rate
Dissolving takes time, since it only takes place at the interface solute/solvent, driven by the chemical potential difference of the solving species i, i.e. to the forcing gradient ∇µi. The dissolving of a solid solute into a liquid solvent is driven by the solvent-molecules pulling out solute-molecules at the surface of the solid and surrounding them in a cluster of solvent-molecules (solvation). Solution equilibrium can be viewed as balanced state where dissolving rate equals segregation rate (crystallization rate, outgassing rate).
Dissolving rate is always proportional to interface area A, and it usually increased by stirring or shaking, and by heating, but that only applies to solid solutes; Table 1 summarises the effect of different action on dissolving rate.
Table 1. Effect of different action on dissolving rate.
Action
Dissolving rate of a solid solute Dissolving rate of a gaseous solute
stirring
increases (A↑)
decreases
temperature increase
increases (∇µi↑)
decreases
pressure increase
has no effect
increases proportionally
crumbling
increases (A↑)
Not applicable
The effect of temperature on dissolving rates of solid solutes can be modelled by an Arrhenius law, so that the plot of the logarithm of the time to dissolve a certain amount (in [s]), versus the inverse of temperature, 1/T (in [K]), is a straight line of slope −Ea/Ru, what yields the dissolving activation energy, Ea (in [J/mol]), after multiplication by Ru=8.3 J/(mol·K).
Electrolytic solutions
When a solute dissolves in a liquid, its molecules always get apart from each other to be surrounded by solvent molecules (that is ‘to dissolve’), but sometimes solute molecules get split into ions, yielding an electrolytic solution, also known as ionic solution or simply an electrolyte. Most electrolytic solutions come from ionic solutes (e.g. NaCl), but there are molecular solutes that yield electrolytic solutions (e.g. HCl(g), NH3(g)).
An electrolyte is a volumetric ionic system; i.e. a system that transport electricity in relatively slow-moving particles (ions, from Gr. ienai to go) instead of by rapid moving electrons in a solid conductor. Electrolytes can be solid (like the polymer membranes used in PEM-fuel-cells or the oxides used in SO-fuel-cells), or they can be liquid (either a high-temperature melt like NaCl above 804 ºC, or a room-temperature solution in a given solvent, like NaCl(aq); we here deal only with the latter). Water is the most important solvent for electrolytes (ethanol, ammonia, and acetic acid are some of the non-aqueous solvents that are able to dissolve electrolytes). Electrolytes are electrically neutral, i.e. they have the same number of positive and negative charges. Electric conduction within an electrolyte implies electrochemical reactions at some chargebalancing electrodes (i.e. a closed electrical circuit).
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9
Solution electrolytes may be salts, acids or bases; e.g. NaCl(aq), HCl(aq) and NaOH(aq). Weak electrolytes partially dissociate (e.g. acetic acid and ammonia; acetic acid 0.001 M only dissociate up to 12% in water at 25 ºC), whereas strong electrolytes dissociate almost completely (e.g. NaCl(s), HCl(g), NaOH(s), H2SO4(l); NaCl(s) dissociate more than 99%). All electrolytes dissociate to some degree, but when the dissociation is very large they are called strong electrolytes (i.e. the proportion of solute molecules to ions is negligible). At small concentrations, the electrical conductivity of electrolytes is proportional to ion concentration.
When a ionic compound dissolves, ions get free from their partners, but get surrounded by the polar molecules of the solvent, an arrangement that has always less energy than the isolated ion, what implies there is a positive solvation energy (hydration energy for water solutions).
Solution properties
To begin with, notice that ,although most properties are associated to the solute, and so labelled (e.g. the solubility of common salt is 26%wt in solution), and some other properties just to the solvent (e.g. freezingpoint depression), all solution properties depend to some extent on the solute considered, the solvent, other solutes present, concentrations, temperature and pressure.
One of the key properties of a solution is solubility, i.e. how much solute the solvent can dissolve, best seen on a phase diagram, although phase diagrams are usually quite complex and just a table of solubilities, or at most a solubility-vs-temperature curve, is presented.
Besides the phase diagram for the mixture (or just vapour-liquid equilibrium data (VLE), or a simple solubility limit at room temperature), other properties of solutions are of interest, here grouped as:
• Solubility dependence with temperature, and other phase-diagram related data, as melting point, boiling point, vapour pressure, formation of compounds (e.g. hydrate, clathrate), etc.
• Density dependence on solute concentration. This is usually the basic measure in solutions. • Heat of solution. This is only important for large-size industrial systems, or when the application is
based on that (as for freezing mixtures), or when, jointly with entropy of mixing, properties are to be deduced from thermochemical data (see Solute equilibrium, below). • Other properties: thermal capacity, thermal conductivity, viscosity, diffusivity, refractive index, electrical conductivity, etc.
Phase diagram
Solutions, as for general mixtures, are homogeneous systems (more or less stratified in the presence of a force field), that may be in contact with other phases, forming multiphase systems. While phase diagrams for pure components are presented as p-T diagrams (see Potentials and properties), phase diagram for a mixture, are usually based on a T-x diagram (temperature versus concentration, at ambient pressure), where one may see at a glance the main thermodynamic characteristics of the system: solubility, melting and boiling points, and their temperature dependence.
There are functions that may be deduced from the phase diagram, as the enthalpy of solution (related to solubility dependence on temperature), but some others are usually plotted aside, like density variation with
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Solutions quantification ..................................................................................................................................... 1 Mixtures and solutions................................................................................................................................... 1 Concentration specification ........................................................................................................................... 4 Moles-of-solute variables .......................................................................................................................... 5 Mass-of-solute variables ............................................................................................................................ 6 Volume-of-solute variables ....................................................................................................................... 7 Concentration measurement .......................................................................................................................... 8 Solution preparation from pure solutes and concentrated solutions. Dilution............................................... 8 Dissolving rate ............................................................................................................................................... 9 Electrolytic solutions ..................................................................................................................................... 9
Solution properties........................................................................................................................................... 10 Phase diagram .............................................................................................................................................. 10 Thermodynamics of solubility..................................................................................................................... 11 Qualitative solubility. Solubility rules ..................................................................................................... 12 Solute equilibrium and its temperature variation..................................................................................... 12 Solubility equilibrium for gas solutes...................................................................................................... 13 Solubility equilibrium for solid solutes ................................................................................................... 17 Partition of a solute between two immiscible solvents............................................................................ 20 Supersaturation. Undercooling and overheating. Hydrates and clathrates .................................................. 20 Density......................................................................................................................................................... 22 Heat of solution............................................................................................................................................ 23 Freezing mixtures .................................................................................................................................... 24 Colligative properties................................................................................................................................... 25 Vapour-pressure depression..................................................................................................................... 25 Freezing-point depression........................................................................................................................ 25 Boiling-point increase.............................................................................................................................. 26 Osmotic pressure ..................................................................................................................................... 26
Properties of particular solutions ..................................................................................................................... 27 Thermochemical data of solutes ...................................................................................................................... 27 Solubility data for aqueous solutions............................................................................................................... 27 Density data for solutions ................................................................................................................................ 27 Heat of solution data ........................................................................................................................................ 27 Cryoscopic and ebullioscopic data .................................................................................................................. 27
Solutions quantification
Mixtures and solutions
A mixture is any multi-component system, i.e. one with several chemical species. The thermodynamics of
mixtures in general (gaseous, liquid or solid) has been considered under the heading Mixtures, mainly
devoted to ideal mixtures. The term 'solution' is mostly used for the special case of a mixture between very
dissimilar components, i.e. when a small amount of substance, called solute (solid, liquid or gas), dissolves
to a certain limit in a liquid or solid substance (pure, or a mixture itself) called the solvent. In many solutions
of interest, the sum of the mole fractions of the solutes is small compared to unity, and the solution is called
a dilute solution. We assume true solutions, i.e. homogeneous solutions, and do not consider colloids and
Thermodynamics of solutions
1
suspensions, treated under the heading Mixture settling. Reacting mixtures are covered in Chemical reactions, aside.
Most solutions depart from the ideal-mixture-model developed in Mixtures, but it is important to recall the basic result from the multiphase equilibrium of ideal mixtures, i.e. Raoult's law for ideal vapour-liquidequilibrium (VLE):
xi,gas = pi*(T ) (for ideal mixtures)
(1)
xi,liq
p
i.e., for any component i, its molar fraction in the vapour phase, xi,gas, is to its molar fraction in the liquid phase, xi,liq, as its pure-component vapour-pressure at that temperature, pi*(T ) , is to the actual pressure, p.
If a solution behaves as an ideal mixture, Raoult's law for each component in a binary mixture of solute 's' in solvent 'dis', in terms of solute mole fractions would read:
= xs,gas p= s* (T ) and 1− xs,gas pd*is (T )
(2)
xs,liq
p
1 − xs,liq
p
which can be approximately solved in the following limits like that:
for p* >> p* (e.g. salts in water) = xs,gas
dis
s
xs,liq
p
* s
(T
)
<< 1
and=p
pd*is (T )
p* dis
(1
−
xs,liq
)
(3)
( ) for p* << p* (e.g. gases in water) = xs,liq
p* dis
(T
)
<< 1
and =p
p* 1− x
(4)
dis
s
xs , gas
p
* s
(T
)
s
s ,liq
with the following interpretation: • when a non-volatile solute dissolves in a liquid, (3), it lowers the solvent VLE-pressure, and its fraction on the vapour phase is proportional to its vapour pressure when pure (negligible); • when a very-volatile solute dissolves in a liquid (4), it lowers the solute VLE-pressure, and its fraction on the liquid phase is proportional to the vapour pressure of the pure solvent (small).
Solutions are everywhere: seawater (but also tap water), coffee, soda, wine, vinegar, gasoline, antifreeze, body fluids (e.g. human plasma is roughly an 8%wt proteins plus 1% salts in water, milk serum is roughly a 5%wt lactose plus 3% proteins in water, urine is roughly a 2.5%wt urea plus 2%wt salts in water), etc. Milk may appear to be a homogeneous mixture to the unaided eye, but the tiny oil and protein droplets in the system make milk to appear white (milk is a colloid). Sea water contains many ions in addition to sodium and chloride ions; when gradually evaporated, the first salt to precipitate is CaCO3 (present to the extent of 0.12 g/L), followed by CaSO4·H2O (1.75 g/L), then NaCl (29.7 g/L), MgSO4 (2.48 g/L), MgCl2 (3.32 g/L), NaBr (0.55 g/L), and KCl (0.53 g/L).
Thermodynamics of solutions
2
Solvents should be inert (not react with solutes), volatile (low boiling point, to ease distillation), and recyclable. They are usually grouped as:
• Polar solvents (water, methanol, ammonia). Polar solvents are hydrophilic, usually H-bond donors (like water itself, the most important solvent by far), or through hydroxyl groups as in ethanol (CH3CH2OH), or through carboxylic groups like in acetic acid (CH3-COOH). Polarity is measured by the dielectric constant. Some polar solvents are organic, notably acetone (they are not H-bond donors).
• Non-polar solvents (benzene, toluene, ether, hexane, carbon tetrachloride, trichloromethane (chloroform), methylene chloride, gasoline, mineral oil). Most organic solvents, greatly used in the chemical and biochemical industries (food and pharmacy), are harmful to health and the environment, and are being replaced by supercritical fluids. 'Dry cleaning' is not dry at all; it is said dry because the liquid is not soaking water by a volatile solvent (usually 1,1,2,2-tetrachloroethene). Carbon dioxide is a non-polar fluid, but has poor solvent properties outside supercritical conditions.
• Supercritical fluids (carbon dioxide, ethane, trifluoromethane (fluoroform), sulfur hexafluoride, dinitrogen oxide (nitrous oxide), carbon disulphide). Supercritical fluids show a dissolving power quickly increasing with pressure (up to a limit). Pure supercritical carbon dioxide is a relatively nonpolar solvent, but has some limited affinity with polar molecules due to its large molecular quadripole, although modifiers (e.g. methanol, fluorinated hydrocarbons) can be added to improve the solubility of polar molecules. Supercritical water becomes totally miscible with many organic compounds like heptane and benzene. An advantage of supercritical solvents is that they yield the dry solute by simple evaporation of the solvent. The range for supercritical applications is p/pCR=1..2 and T/TCR=1.0..1.1.
The main solvent is water, and thus most solutions are aqueous solutions. Most solutes are solid powders (e.g. salt-water solutions). Water scarcely dissolves gases and organic liquids, and liquid-liquid solutions may be considered plain mixtures; that is why the basic understanding of ‘to dissolve’ is to untie a solid to become part of a liquid. It is the solvent that pulls the solute into solution, and its internal motion (diffusion and convection) what spreads the solute around. Some details on mixing can be found aside in Mass transfer and in Chemical kinetics.
Usually, solutes do not dissolve on the solvent vapour (i.e. they are usually non-volatile), neither on the solvent solid phase (they do not fit into the crystalline lattice). Both mentioned-processes can be used to get rid of solutes (e.g. salt water becomes fresh water when vaporised or solidified), at least when done carefully, but, in practice, solutes can be entrained by solvent-vapours (mainly as an aerosol of particles) and can be trapped within the solid solvent (at the seashore, one may feel salty lips just from the marine breeze, and some ice may taste salty too).
All solutions have some stratification of composition at equilibrium, as studied in Mixture settling. Besides, solution tend to change the composition with time in open systems, i.e. in solutions open to the atmosphere or when the solution is blooded someway, since the escape will not have the same average composition; e.g. the composition inside a liquefied-petroleum-gas bottle changes during use, becoming richer in the heavier components, as well as within a liquefied-natural-gas tanker, due to the boil-off.
Thermodynamics of solutions
3
The relative amount of solute in a solution (its concentration) may be measured by different concentration variables related amongst them (but not always in a trivial manner); since all solutions depart from ideal (Raoult’s law) mixtures, the use of molar quantities for the solvent is rare (molar quantities were an advantage for modelling ideal mixtures because, in that case, the behaviour is just proportional to the molar composition, independently of the actual substances involved).
Concentration specification
One of the main difficulties in the study of solutions is the wide variety of variables and units used to specify concentration (in the case of ideal mixtures, only molar fractions, and sometimes mass fractions, were common). Generically speaking, concentrations in a mixture express the quantitative composition (although, under the SI standard, composition must refer to the unit volume of the system, i.e. mol/m3, kg/m3). The subject gets obscure also by the lack of a clear nomenclature, e.g. to distinguish with a letter from 'solution' (we will use subscript m, from mixture), 'solute' (we will use subscript s, from solute) and 'solvent' (we will use subscript dis, from dissolve); the standard labelling of '1' for solute and '2' for solvent, may be unclear when one wants to distinguish between solvent and solution.
To give an idea of the variety of variables used, consider the following values for oxygen-solubility in water (all of them valid under certain circumstances): 39 mg/kg, 0.039 kg/m3, 1.2 mol/m3, 22 ppm, 4.6 ppm, 0.26 mol/m3, 8.2 mg/L, 5.9 mL/L, 55 mL/L, to name a few. And, fortunately, oxygen solubility is so small that the value is the same by amount of solution than by amount of solvent. The explanation of the above figures is that 78 mg/kg is equal to 78 mg/L for aqueous solutions where 1 L has 1 kg; it is also equal to 78/32=2.4 mol/m3; and it is also equal to 22 parts per million molar, but these values refer to concentration of dissolved oxygen in equilibrium with pure oxygen at 25 ºC and 100 kPa (a common physical standard state). However, when the equilibrium is not with pure oxygen but with air at 25 ºC and 100 kPa, previous values have to be multiplied by xO2,air=0.21, giving 22·0.21=4.6 ppm, 39·0.21=8.2 mg/L, equal to 8.2 mg/kg for water, and equal to 1.2·0.21=0.26 mol/m3. Finally, the volumetric fractions refer to the volume at 0 ºC and 100 kPa that the full out-gassing of dissolved oxygen in one litre of solution at a given temperature, would occupy: 55 mL/L means that the oxygen dissolved in 1 L of solution would occupy 55 mL as a pure gas at 0 ºC and 100 kPa, a value that is correct for a solution at 25 ºC saturated against pure oxygen at 0 ºC and 100 kPa; if it were against air instead of pure oxygen, the value would be 55·0.21=11.6 mL/L for air at 0 ºC, and, as solubility of gases decreases with temperature and nearly halve from 0 ºC to 25 ºC, in equilibrium with air at 25 ºC and 100 kPa the value is 5.9 mL/L for air at 25 ºC).
We assume a single solute, s (of molar mass Ms) dissolved in a solvent, dis (of molar mass Mdis), but the generalisation to multiple solutes is trivial. Composition may be quantified by the ratio of solute to solution (e.g. moles of solute per litre of solution, grams of solute per kilogram of solution), or by the ratio of solute to solvent, but the former ratio will be favoured here, except in the case is molalities (commonly used to express salt solubilities, however).
A possible grouping for solution quantification may be:
• Moles-of-solute variables.
• Mass-of-solute variables.
Thermodynamics of solutions
4
• Volume-of-solute variables.
WARNING. In the study of solutions, the litre, L (sometimes with a less distinguishable lower case l), is often used as volume-unit instead of the cubic meter (1 L ≡ 1 dm3), and the millilitre, mL, is also found in common usage, instead of the cubic centimetre (1 mL≡ 1 cm3).
Moles-of-solute variables • Molar fraction of solute, xs, i.e. moles of solute per mole of solution, either per-one, percent (%) or per million (ppm). Its definition, and relation with other variables, is:
ms
x ≡ ns =
Ms =
1 = c Mm
(5)
s ns + ndis ms + mdis 1+ 1 M s s ρm
M s M dis
ys M dis
where Mm is the molar mass of the mixture, Mm=xsMs+xdisMdis. For instance, when adding a 10 g bag of sugar (C12H22O11) to a 100 cm3 cup of boiling water (96 g of water, yx=10/(10+96)=9.4% by weight, having taken 958 kg/m3 for boiling water), one gets a solution with 0.010/0.342=0.030 moles of sucrose and 958·100·10-6/0.018=5.3 moles of water, i.e. xs=0.03/(0.03+5.3)=0.56%, and a concentration of 0.3 mol/L. A problem with the amount of substance is that it is not conserved in electrolytic solutions; e.g. seawater with 35 g/kg of NaCl has a molar fraction of dissolved particles of xs=0.022, mainly sodium cations and chlorine anions; i.e. 1% of the particles are Na+, 1% are Cl-, and the rest mainly water molecules; lab seawater may be prepared by adding 0.035 kg of NaCl (0.6 mol) to 1 kg of H2O (55 mol). • Molar ratio, rs≡ns/ndis, i.e. moles of solute per mole of solvent; rarely used. • Molar density, or concentration, cs, (sometimes written as [formula of solute], e.g. cN2≡[N2]) i.e. moles of solute per cubic meter of solution;
c ≡ n=s xsnm= x ρm= ms / M=s ρs
(6)
s Vm mm ρm s M m
Vm
Ms
e.g. seawater with 3.5%wt NaCl has a molar density of 600 moles of Na+ per cubic meter of solution 0.6 mol/L of Na+), and the sugar solution mentioned above (10 g sucrose in 100 cm3 of boiling water) has a concentration of 0.03/106·10-6=28 mol/m3 (0.28 mol/L, where the fact that adding 10 g sucrose increases 6.5 cm3 the volume of solution has been used). The problem with molar density is that it
changes with temperature even if the system is closed (due to overall volume change).
Molarity, is just the common way to express molar density, cs, but taking as unit 1 mol/L instead of 1 mol/m3, and given it a special, not recommended, symbol: 1 M ≡ 1 mol/L e.g. seawater with 3.5%wt
NaCl has approximately 0.6 mol of NaCl per litre of aqueous solution (referred too as 0.6 M NaCl
solution (but notice that it is actually a 1.2 mol/L solution since each mole of NaCl yields two moles
in solution), and the sugar solution mentioned above is a 0.28 M sucrose aqueous solution (0.28
mol/L solution). WARNING. Some authors use M as the symbol for the mole, instead of the SI-
Thermodynamics of solutions
5
recommended 'mol', and thus write e.g. that seawater is a 0.6 M/L solution, further contributing to the already wide confusion in nomenclature. Normality, is the mole-equivalent amount of solute (moles divided by valence or replaceable H atoms) per litre of solution; i.e., a normalised molar density, cs. Often, a special symbol, not recommended, is used for normality: 1 N ≡ 1 mol/L e.g., since NaCl has valence equal to 1, seawater is approximately a 0.6 N Na+ and 0.6 N Cl- water solution (in this case 1 N is also 1 M). Normality of a closed system changes with temperature as molarity (molar density) does. • Molality, bs, is the amount of solute per mass of solvent, not of solution. Molality is an easy way to specify recipes (the ingredients, instead of the final result), and does not change with temperature in a closed system. Molality is not a recommended composition descriptor; sometimes m is used as symbol for the molality variable, and, to add confusion, sometimes 'm' is used as a molality unit, 1 m ≡ 1 mol/kg; e.g. seawater with 3.5%wt NaCl is approximately a 0.6 m (pronounced 0.6 molal) NaCl solution in water, i.e. adding 0.6 mol of NaCl to one kilogramme of water produces lab seawater. Notice that, although sometimes molality is expressed in mol/L, it is a liberty for the case of water where 1 kg roughly occupies 1 L, but molality is always based on mass of the solvent and does not change with temperature; furthermore, for dilute aqueous solutions, molality values in mol/L (of solvent), practically coincide with molarity values in mol/L (of solution).
b ≡ ns = ns = ms / M s= ws
(7)
s mdis ρ V dis dis
mdis
Ms
Mass-of-solute variables Mass-of-solute (or weight) variables have the advantage over molar quantities in that the molecular structure has not to be known; e.g. how many moles are there in one litre of gasoline?):
• Mass fraction of solute, ys, i.e. kilograms of solute per kilogram of solution, either per-one, percent (%) or per million (ppm); e.g. seawater has a 3.5% mass fraction of dissolved salts (the majority NaCl). WARNING. The recommended symbol under SI-conventions for the mass fraction is w instead of y (but we use w for mass ratio, here).
y ≡ = ms ms=
xs M s
(8)
s mm mdis + ms xdis M dis + xs M s
• Mass ratio, ws, usually in units of grams of solute per kilogramme of solvent (absolute humidity is such a mass-ratio function, widely used in Humid air). It is directly related to molality.
w ≡ ms = nsM s = b M =
ms
= ys
(9)
s mdis mdis
s s mdis + ms − ms 1− ys
• Solute density in the solution, ρs, or mass concentration, usually in units of grams per litre of solution, g/L (many times in mg/L, and the most times in Pharmacy and Medicine in mg/dL (milligrams per decilitre of solution).
Thermodynamics of solutions
6
ρ ≡ ms = y ρ = c M = x ρ M s
(10)
s Vm
sm
ss
s m Mm
• Mass of solute per unit volume of solvent, e.g. in grams of solute per decilitre of solvent (e.g. g/dL means now milligrams per decilitre of solvent), used sometimes to specify solubility data for solids. It can be easily converted to mass ratio by knowing the density of the solvent, ρdis, by (9).
= ms Vdis
m= s mdis ρdis
w= s ρdis
cs M s ρdis
(11)
Volume-of-solute variables • Volume fraction, vf, or volume percent (the unit usually being mL/L, or %vol, or % (v/v)), is the volume of pure solute added divided by the sum of solute and solvent volumes before mixing (or after separation, i.e. not by the volume of the solution prepared, although the difference may be negligible for dilute solutions of condensed solutes); i.e. it refers to the recipe to prepare the solution, not to what is obtained. Volume percent, or sometimes volume per volume, is often used when the solute is liquid itself; e.g. ethanol 96% means an aqueous solution that, when separated, corresponds to 96 volumes of ethanol and 4 volumes of water; e.g. a 12% ethanol in water (v/v) solution of 1 L is obtained by mixing 120 cm3 pure ethanol with 880 cm3 pure water.
v ≡ Vs =
1=
1=
1
(12)
f
Vm + Vdis
1+ ρs / ρdis
1+
ρs
1
−1
1+
ρs
M dis
1
−1
ms / mdis
ρdis ys
ρdis M s xs
• Notice however that there are occasions were the inverse volume fraction is used. Volume ratio, vr, also named Bunsen solubility coefficient, is the volume of pure solute divided by the solvent volumes before mixing (or after separation, i.e. not by the volume of the solution prepared, although the difference may be negligible); i.e. it refers to the recipe to prepare the solution, not to what is obtained. Notice however that there are occasions were the inverse volume fraction is used, as when saying that oxygenated water has 10 volumes of H2O per one of H2O2. For solid and liquid solutes forming dilute solutions, volume ratio and volume fraction practically coincide since Vs,pure<
When specifying concentrations in dimensional units, temperature must be specified also if it affects densities, and, if gases are involved, pressure too must be specified.
Thermodynamics of solutions
7
Concentration measurement
The experimental finding of qualitative or quantitative composition in a mixture is known as chemical analysis, or simply analysis. We focus here on concentration analysis, assuming the substances involved are already known.
Most methods of concentration analysis are based on measuring solution density (provided the density dependence on solute concentration, ρm=ρ m(T,p,xi), is known beforehand by calibration), by one of the different techniques:
• Gravimetry. Weighting a known volume of liquid. This is perhaps the easiest and quickest method to measure solution concentration, but requires sampling (e.g. milk producers are paid according to milk density).
• Refractometry. By ray tracing on a sample or in a process flow. Refractive index varies almost linearly with density. This is the easiest way for well-known solutions.
• Resonant vibration. The natural frequency of an encapsulated liquid sample precisely metered depends on its mass. May be applied to a liquid flowing along a bend in a pipe (see Coriolis flowmeter).
• Sonic velocimetry. Density is obtained from ρ =E/c2, where E is the bulk modulus of the solution and c the sound speed through it (see Ultrasonic flowmeter).
• Electric conductivity. This is the best method for very low concentration of electrolytic solutions (e.g. the preferred method for salt-water solutions). The measuring electrodes may be generic, or selective for some specific ion (e.g. Ca2+, NH4+, Cl-, NO3-).
Solution preparation from pure solutes and concentrated solutions. Dilution
Solutions can be prepared by simple dilution, either from pure solutes, or from concentrated solutions. Preparing solutions based on molality specifications (you say dissolving with, e.g. dissolving 300 g of NaCl with 1 kg of water) is easy: just measure, add and stir, whereas preparing solutions based on molarity specifications (you say dissolving in), is not so easy: you start with less that a litre of water, then add the solute, stir and keep adding water until a full litre of solution is obtained. To make it more clear, it is difficult to dissolve 300 g of NaCl with 1 kg of water, but it is impossible to dissolve 300 g of NaCl in 1 kg of water solution, since the solubility limit at 15 ºC is 26.4% by weight of NaCl in solution, or 360 g of NaCl with 1 kg of solvent).
Maintaining pure solutes may be tricky, particularly if they are readily oxidised, or hygroscopic, or very volatile or even gaseous. Many salts and hydroxides (e.g. NaOH(s)) are hygroscopic, and will absorb enough water from the air to change its apparent mass if left exposed to the air for long.
Many times, a desired solution is not prepared with the pure solute but from a more concentrated solution (to do it from a more diluted solution you might let some solvent to evaporate, but this is not practical). The basic rule to remember is that adding a concentrated solution to pure solvent maintains the amount of solute, thus ns=csVm=cs,finalVm,final=cs,initialVm,initial, from where the appropriate amount of concentrated solution needed, Vm,initial, may be found.
Thermodynamics of solutions
8
A basic precaution for the preparation of well-defined aqueous solutions is to use distillated water, instead of tap water, and keep the solution in a closed container.
Dissolving rate
Dissolving takes time, since it only takes place at the interface solute/solvent, driven by the chemical potential difference of the solving species i, i.e. to the forcing gradient ∇µi. The dissolving of a solid solute into a liquid solvent is driven by the solvent-molecules pulling out solute-molecules at the surface of the solid and surrounding them in a cluster of solvent-molecules (solvation). Solution equilibrium can be viewed as balanced state where dissolving rate equals segregation rate (crystallization rate, outgassing rate).
Dissolving rate is always proportional to interface area A, and it usually increased by stirring or shaking, and by heating, but that only applies to solid solutes; Table 1 summarises the effect of different action on dissolving rate.
Table 1. Effect of different action on dissolving rate.
Action
Dissolving rate of a solid solute Dissolving rate of a gaseous solute
stirring
increases (A↑)
decreases
temperature increase
increases (∇µi↑)
decreases
pressure increase
has no effect
increases proportionally
crumbling
increases (A↑)
Not applicable
The effect of temperature on dissolving rates of solid solutes can be modelled by an Arrhenius law, so that the plot of the logarithm of the time to dissolve a certain amount (in [s]), versus the inverse of temperature, 1/T (in [K]), is a straight line of slope −Ea/Ru, what yields the dissolving activation energy, Ea (in [J/mol]), after multiplication by Ru=8.3 J/(mol·K).
Electrolytic solutions
When a solute dissolves in a liquid, its molecules always get apart from each other to be surrounded by solvent molecules (that is ‘to dissolve’), but sometimes solute molecules get split into ions, yielding an electrolytic solution, also known as ionic solution or simply an electrolyte. Most electrolytic solutions come from ionic solutes (e.g. NaCl), but there are molecular solutes that yield electrolytic solutions (e.g. HCl(g), NH3(g)).
An electrolyte is a volumetric ionic system; i.e. a system that transport electricity in relatively slow-moving particles (ions, from Gr. ienai to go) instead of by rapid moving electrons in a solid conductor. Electrolytes can be solid (like the polymer membranes used in PEM-fuel-cells or the oxides used in SO-fuel-cells), or they can be liquid (either a high-temperature melt like NaCl above 804 ºC, or a room-temperature solution in a given solvent, like NaCl(aq); we here deal only with the latter). Water is the most important solvent for electrolytes (ethanol, ammonia, and acetic acid are some of the non-aqueous solvents that are able to dissolve electrolytes). Electrolytes are electrically neutral, i.e. they have the same number of positive and negative charges. Electric conduction within an electrolyte implies electrochemical reactions at some chargebalancing electrodes (i.e. a closed electrical circuit).
Thermodynamics of solutions
9
Solution electrolytes may be salts, acids or bases; e.g. NaCl(aq), HCl(aq) and NaOH(aq). Weak electrolytes partially dissociate (e.g. acetic acid and ammonia; acetic acid 0.001 M only dissociate up to 12% in water at 25 ºC), whereas strong electrolytes dissociate almost completely (e.g. NaCl(s), HCl(g), NaOH(s), H2SO4(l); NaCl(s) dissociate more than 99%). All electrolytes dissociate to some degree, but when the dissociation is very large they are called strong electrolytes (i.e. the proportion of solute molecules to ions is negligible). At small concentrations, the electrical conductivity of electrolytes is proportional to ion concentration.
When a ionic compound dissolves, ions get free from their partners, but get surrounded by the polar molecules of the solvent, an arrangement that has always less energy than the isolated ion, what implies there is a positive solvation energy (hydration energy for water solutions).
Solution properties
To begin with, notice that ,although most properties are associated to the solute, and so labelled (e.g. the solubility of common salt is 26%wt in solution), and some other properties just to the solvent (e.g. freezingpoint depression), all solution properties depend to some extent on the solute considered, the solvent, other solutes present, concentrations, temperature and pressure.
One of the key properties of a solution is solubility, i.e. how much solute the solvent can dissolve, best seen on a phase diagram, although phase diagrams are usually quite complex and just a table of solubilities, or at most a solubility-vs-temperature curve, is presented.
Besides the phase diagram for the mixture (or just vapour-liquid equilibrium data (VLE), or a simple solubility limit at room temperature), other properties of solutions are of interest, here grouped as:
• Solubility dependence with temperature, and other phase-diagram related data, as melting point, boiling point, vapour pressure, formation of compounds (e.g. hydrate, clathrate), etc.
• Density dependence on solute concentration. This is usually the basic measure in solutions. • Heat of solution. This is only important for large-size industrial systems, or when the application is
based on that (as for freezing mixtures), or when, jointly with entropy of mixing, properties are to be deduced from thermochemical data (see Solute equilibrium, below). • Other properties: thermal capacity, thermal conductivity, viscosity, diffusivity, refractive index, electrical conductivity, etc.
Phase diagram
Solutions, as for general mixtures, are homogeneous systems (more or less stratified in the presence of a force field), that may be in contact with other phases, forming multiphase systems. While phase diagrams for pure components are presented as p-T diagrams (see Potentials and properties), phase diagram for a mixture, are usually based on a T-x diagram (temperature versus concentration, at ambient pressure), where one may see at a glance the main thermodynamic characteristics of the system: solubility, melting and boiling points, and their temperature dependence.
There are functions that may be deduced from the phase diagram, as the enthalpy of solution (related to solubility dependence on temperature), but some others are usually plotted aside, like density variation with
Thermodynamics of solutions
10