The colligative properties of solutions consist of freezing point depression, boiling point elevation, vapor pressure lowering and osmotic pressure.
Mole fraction : molarity, molality, % w/w and % w/v
Overview of colligative properties :
Vapor pressure lowering : CaCl_{2}
Freezing point depression : examples : Glucose : Urea : Ethanol : NaCl : CaCl_{2}
Boiling point elevation : Glucose
Osmotic pressure
Osmotic potential
Self-generation of osmotic pressure at interfaces
Reverse osmosis
"Solutes simply dilute the solvent;..The colligative properties follow directly from this..."
Andrews 1976 [2403]
Colligative properties depend on the number of particles (for example, molecules or ions) in a solution and not on the molecular weight or size of the of the particles. They apply to all solvents, and they do depend on the solvent. These properties apply to ideal solutions ^{o} with real solutions varying to different extents that can give rise to useful information concerning the solute-solute and solute-solvent interactions. The colligative properties of solutions ^{d} ideally depend on changes in the entropy of the solution on dissolving the solute, which is determined by the concentration of the solute molecules or ions but does not depend on their structure. The rationale for these colligative properties is the increase in entropy on mixing solutes with the water. This extra entropy provides extra energy available in the solution which has to be overcome when ice or water vapor (neither of which contains a non-volatile solute) ^{b} is formed. It is available to help break the bonds of the ice causing melting at a lower temperature and to reduce the entropy loss when water condenses,^{ m} so encouraging the condensation and reducing the vapor pressure. Put another way, the solute stabilizes the water in the solution relative to pure water. The colligative properties are thermodynamic properties determined under equilibrium conditions and do not give information concerning the rates of any process.
The energy change occurring over a small change (dx) in the mole fraction of water (x_{w}) is RTdx/x_{w}. Therefore the total energy change going from pure water (x_{w}= 1) to this new value (x_{w}) is
= RTLn(x_{w}/1) = RTLn(x_{w})
The entropy change reduces the chemical potential (μ_{w}, J·mol^{-1})^{ a} by -RTLn(x_{w}) (that is, the negative energy term RTLn(x_{w}) is added to the chemical potential to get the potential of the solution;
= + RTLn(x_{w})
where R is the molar gas constant, T is the temperature (K) x_{w} is the (dimensionless) mole fraction of the ‘free’ (bulk) water (0 < x_{w} < 1 )^{ h}.
Note that x_{w} may be replaced by the water activity (a_{w}) in this expression. Dissolving a solute in liquid water thus makes the liquid water more stable (lower energy) whereas the ice and vapor phases remain unchanged as no solute is present in ice crystals and no non-volatile solute is present in the vapor. Thus, the entire decrease in the chemical potential of the water when it makes an ideal solution on dissolving solute is due to the dilution of the solvent by the solute. This dilution ideally does not affect the enthalpy(H), simply increasing the randomness of the solution. Deviations from ideal behavior ^{i} provide useful insights into the interactions of molecules and ions with water (the enthalpy term). The colligative properties of materials encompass all liquids but on this web page, we concentrate on water and aqueous solutions.
Chemical potentials of gaseous, liquid and frozen water and solution versus temperature
The above figure is based on an ideal solution where x_{w} = 0.9 with x_{S} = 0.1 (that is, 6.167 molal). The freezing point depression is at -11.47 °C (that is, at 6.167 ˣ 1.8597 K), the boiling point elevation is at 103.16 °C (that is, at 6.167 ˣ 0.5129 K), and the vapor pressure is 0.9 ˣ that of pure water. The chemical potential changes by an amount equal to RTLn(0.9) (that is, 8.314 ˣ 273.15 ˣ -0.10536 = -239.28 J ˣ mol^{-1} at 0 °C). See below for an explanation of terms and the derivation of the equations. The small change of the chemical potential of ice with temperature gives rise to a larger freezing point depression than the boiling point elevation. Thus, solutions have higher boiling points and lower freezing points than pure water. [Back to Top ]
Wherever water is present in solution it may be considered as being either 'bound' or 'free', although there will be transitional water between these states. When considering the colligative properties, 'water' is considered bound to any solute when it has a very low entropy compared with pure liquid water. This occurs, for example, when salts form strongly bound hydration shells [1494], with the hydration shells of some ions (for example, Mg^{2+}) giving clear Raman spectra in solution [1503]. The average number of water molecules (h) bound to the solute ion more strongly (by at least 55.6 kJ ˣ mol^{-1}) ^{n} than are bound to other water molecules, and effective in removing water molecules from freezing point depression determinations, are shown for some ions below [2654]. h (now called the thermodynamic hydration number) is a number that allows the remaining ions to behave ideally; it is not a coordination number. Vapor pressure, boiling point elevation and osmotic pressure data give similar thermodynamic hydration numbers, allowing for temperature effects [2654]. This concept of 'bound' and 'free' water was described in 1917 [2750].
Ion | h | Ion | h | Ion | h | Ion | h |
F^{-} | 6.2 | Li^{+} | 6.6 | Mg^{2+} | 14 | HCO_{2-} | 3.5 |
Cl^{-} | 0 | Na^{+} | 3.9 | Ca^{2+} | 12 | HCO_{3-} | 6.3 |
Br^{-} | 0 | K^{+} | 1.7 | Sr^{2+} | 12 | SO_{42-} | 8.9 |
I^{-} | 0 | Rb^{+} | 1.8 | Ba^{2+} | 10.5 | CO_{32-} | 11.8 |
OH^{-} | 6.0 | Cs^{+} | 0.6 | Fe^{3+} | 18 | PO_{43-} | 20.6 |
H^{+} | 6.7 | NH_{4}^{+} | 1.8 | Al^{3+} | 22 | citrate^{3-} | 25.0 |
The mole fraction of 'free' water (x_{w} ) in an m (molal) salt solution is given by,
x_{w} = (m_{w} - m ˣ H_{d})/(m_{w} - m ˣ H_{d} - m ˣ i_{e})
where m_{w} is the molality of pure water, m is the molality of the salt, H_{d} is the hydrodesmic number (the dynamic average number of strongly bound water to the ions), and i_{e} is number of particles given by the extent of dissociation and ion pair formation [3505]. For example, it has been found that values of H_{d} = 3.7 and i_{e} = 1.77 (23% ion pair, 77% dissociated) satisfy the (concentration independent) law of mass action for the dissociation of NaCl. On the left is a plot of the experimental p/p_{0} of NaCl solutions vs calculated molar fractions of free water solvent x_{w} from 0.10 m up to saturation at 6.144 m with a constant number of particles produced per mole i_{e} = 1.77 and a constant hydrodesmic number H_{d} = 3.7. p/p_{0} is the ratio of the vapor pressure above the solution, p, to the vapor pressure of pure water, p_{0}, see Raoult's Law below.
Bound water should be considered part of the solute and not part of the dissolving 'free' water. The effective molality of a hydrated solute should not include the bound water within the kilogram of dissolving water [250]. In contrast to the molality, bound water makes no difference to the numerical value of the molarity.
As the 'bound' water contributes little to the dielectric relaxation processes at low frequencies, the dielectric drops in solutions and may be used to estimate the 'bound' water. The reciprocal relative permittivity (dielectric constant) (1/ε) increases linearly as the water activity drops over wide ranges of salt solutions [1220]. However, as the proportionality is dependent on the salt composition, the hydrated salt must contribute significantly to the dielectric effect.
The partial vapor pressure of water is the pressure of the water vapor with which it is in equilibrium as part of the gaseous phase it is in contact. The saturation vapor pressure is the equilibrium value of this vapor pressure which, in a given mixture of gases such as air, is the same regardless of the mixture’s composition and depends only on temperature and on the energy required for vaporization. It is commonly found in textbooks and elsewhere on the Web that the vapor pressure is lowered due to the presence of other molecules 'blocking' the surface or that it depends on the volume fraction of materials. These explanations are entirely false and highly misleading as they lead to the erroneous conclusion that the vapor pressure lowering may be dependent on molecular size rather than energetic entropic effects. Also, the surface of the water should not be considered as identical to the bulk aqueous liquid phase. The correct explanation for the vapor pressure lowering is that the presence of solute increases the entropy of the solution (a randomly disbursed mixture having greater entropy than a single material); such rise in entropy increasing the energy required for removing solvent molecules from the liquid phase to the vapor phase.
(1)
In the presence of a solute,
P_{solute} = x_{w} ˣ P_{pure liquid }= (1 - x_{S}) ˣ P_{pure liquid}
where P_{solute} is the vapor pressure over a solution containing the solute (here assumed non-volatile) and P_{pure liquid }is the pressure of the vapor in the gas phase above the pure liquid. This equation is also known as Raoult's law ^{p} and may also be applied (ideally) to mixtures of volatile liquids.
The above figure is based on an ideal solution where x_{w} = 0.9 with x_{S} = 0.1 (as in the top figure), the vapor pressure is 0.9 ˣ that of pure water.
Raoult's law may be simply derived,
(2)
As above, in a solution,
= + RTLn(x_{w})
The vapor phase is in equilibrium with the liquid phase.
(3)
Therefore,
=
(4)
and
=
Assuming ideal gas behavior (a good approximation at these low pressures),
(5)
(6)
Hence,
+ RTLn(x_{w}) =
(7)
x_{w}= P_{solute}/P_{pure liquid } or P_{solute} = x_{w} ˣ P_{pure liquid}
This is Raoult's law. As with other colligative properties, Raoult's law is also applicable to polymers, except at high enough concentrations as to cause significant molecular overlap [1101]. [Back to Top ]
When a solute is dissolved in water, the melting point of the resultant solution may be described using a phase diagram, as shown right. The bold red lines indicate the phase changes. A solution at (a) on cooling will not freeze at 0 °C but will meet the melting point line at a lower freezing point (b). If no supercooling takes place ^{k} some ice will form resulting in a more concentrated solution, freezing at an even lower temperature. This will continue (following the blue dotted line until the eutectic point (c) is reached. Further cooling will result in solidification of the remaining solution with the mixed solid cooling to (d). The eutectic point is at the lowest temperature that the solution can exist at equilibrium and is at the point where the ice line meets the solubility line of the solid solute (or one of its solid hydrates), shown on the right.
Although the eutectic should be independent of the original concentration, ESR studies with spin probes show that interactions between solutes may cause variation at initially low concentrations [1540]. Morse and Frazer first investigated the relationships involving the freezing point of solutions.
In an equilibrium mixture of pure water and pure ice at the melting point (273.15 K) the chemical potential of the ice () must equal the chemical potential of the pure water (), that is, = . With solute present, the chemical potential of the water is reduced by an amount -RTLn(x_{w}).^{c}
There will be a new melting point where
=
and,
= + RTLn(x_{w}) (8)
= + RTLn(x_{w}) (9)
where x_{w} is the mole fraction of the water, R is the gas constant (= 8.314472 J mol^{-1} K^{-1}), and T is the new melting point of ice in contact with the solution. Note that x_{w} may be replaced by a_{w} (the water activity) in this equation. The chemical potential of the ice is not affected by the solute which dissolves only in the liquid water and is generally completely excluded from the ice lattice. The process of melting is also called fusion. The enthalpy change during this process ΔH_{fus} is H^{liquid} - H^{solid}. ^{f} Thus,
RTLn(x_{w}) = - = -ΔG_{fus}
Ln(x_{w}) = -ΔG_{fus}/RT
(11)
As ΔG = ΔH - TΔS, at constant pressure,
(12)
and,
(This is the Gibbs-Helmholtz equation)
Therefore, differentiating Ln(x_{w}) = -ΔG_{fus}/RT we get,
(13)
(14)
Therefore,
(15)
(16)
Solving the right-hand side and replacing Ln(x_{w}) by Ln(1-x_{S}) and ignoring the small temperature dependence of ΔH_{fu}_{s}, ^{g}
[250] (17)
Ignoring the small temperature dependence of ΔH_{fu}_{s} introduces a small error (a), see below. The corrected value for ΔH_{fus} may later be obtained from a plot of Ln(1-x_{S}) versus (1/T_{m}-1/T).
If x_{S} << 1 then Ln(1-x_{S}) = - x_{S} (This introduces a small error (b), see below)^{e}
(18)
and,
The freezing point lowering ΔT is (T_{m} - T), a positive value. If T is close to T_{m} then T ˣ T_{m} = T_{m}^{2}. This introduces a small error (c), see below).
(19)^{h}
Therefore,
As x_{S} is the mole fraction of the solute = moles solute/(moles water + moles solute), at low x_{S} it may be approximated by moles solute/moles water = molality ˣ molar mass of water = m_{S}M_{w} (where m_{S} is the molality = moles per kg water, and M_{w} is the molar mass of water = 0.018015268 kg mol^{-1}).
With this approximation,
(20)
ΔT = m_{S}K_{f}
(21)
This is the commonly used cryoscopic equation,
where K_{f} (the cryoscopic constant) for water is
where K_{f} for water is and calculated to be 1.8597 K ˣ kg ˣ mol^{-1} using R = 8.314472 J ˣ mol^{-1} ˣ K^{-1}, T_{m} = 273.15 K, and ΔH_{fus}= 6009.5 J ˣ mol^{-1}. K_{f} falls to 1.849, 1.830, 1.826, and 1.736 at -10 °C, -20 °C, -30 °C and -70 °C, respectively [2864].
The equation ΔT = m_{S}K_{f} (shown opposite) thus makes several assumptions. The assumption (a) concerning the constancy of ΔH_{fus} introduces some error. For example, it is well known that the specific heat of pure water increases considerably on supercooling whereas that of ice decreases. This leads to a drop in ΔH_{fus} with the degree of supercooling [1748]. |
The effect is shown opposite where the data for the enthalpy of fusion of ice used is from [906] and [76] for the upper central line and from Dorsey and Angell et al. [1098] for the lower central line. In practice, however, the changes occurring in solution [1098] will be somewhat less as the solutes will destroy much of the aqueous structuring responsible for the sizeable specific heat increases of the liquid phase on cooling. However, there is also heat lost or gained from the solution on concentration because of the solute concentrations. The overall latent heats of fusion of ethanol and NaCl solutions are also shown [1475].
The assumptions (b, error due to the Ln(1-x) approximation ^{e }shown by the green 'Ln-corr' curve, and c, due to the temperature approximation shown by the blue 'T-corr' curve) operate in different directions, and mostly compensate for each other to only produce small net errors in ΔT (shown as the dotted red line opposite, indicating the actual 'ideal' behavior). Thus, there is only a net 2% error, when using these two approximations, in 10-molal solutions for the ideal behavior utilizing the equation ΔT = m_{S}K_{f}..
The error introduced by using the molality (m_{S}) rather than the mole fraction (x_{S}) may be significant in solutions greater than a few molal and is shown in the diagram on the right. At higher molality, the expected freezing point depression using the molality equation (ΔT = m_{S}K_{f}) is greater than using the mole fraction equation (ΔT = x_{S}K_{f}/M). This effect is partially compensated by the impact of the Ln(1-x) approximation explained above.
In reality, deviations from 'ideal' behavior occur through solute-solute and solute-solvent interactions so introducing a partial dependence on the identity of the solute, (for example, see [1012]): (a) solutes may interact (salts forming ion-pairs or larger clusters), lowering their effective concentration, (b) solutes may bind variable amounts of water [1100], depending on both their identity and concentration and the identity and concentration of other solutes, so removing water from the 'free' water pool that freezes [445]. As such water is removed from the 'free' (freezable) water pool with the addition of the solute, addition of incremental amounts of solute has a progressively greater lowering effect on the freezing point. Freezing point depression may thus be used to determine the 'hydration' number of ions and such values are greater than those obtained by boiling point elevation, indicating the greater hydration of ions at low temperatures (for example, see [1064]). It is important to note that the 'hydration numbers for strongly bound water' are not the same as the often cited 'hydration number of ions', or 'coordination numbers', that depend on the method of determination [2654]. The thermodynamic hydration numbers for strongly bound water depends on temperature and pressure and will reduce at high concentrations where ion-pairing may be significant, or there may be insufficient water available. Colligative properties apply to all solutes, including polymers, except at high enough concentrations as to cause substantial molecular overlap [1101].
The freezing point of a solution is dependent on its concentration. As the water freezes to become ice, the concentration of the remaining solution increases and this causes the freezing point of the solution to decrease until the solution reaches saturation when the freezing point remains constant, and the salt precipitates out of the remaining solution until there is no liquid left. Due to freezing point depression and boiling point elevation, the liquid range of all solvents increases in the presence of solutes. [Back to Top ]
The upper blue line follows the 'ideal' colligative equation, ΔT = m_{S}K_{f}. for glucose (C_{6}H_{12}O_{6}). The red line tracks the experimental freezing point curve [70]. It hardly deviates from the relationship obtained directly from the unapproximated colligative equations above (black line), where an allowance has been made for bound water of hydration (2.8 water molecules bound to each glucose molecule) ^{j} and the changes in enthalpy of fusion with temperature (here the fitted ΔH_{fus} is 6016 J mol^{-1}).
The eutectic for glucose solutions is at 2.5 m and -5 °C. At higher concentrations, solid glucose α-monohydrate precipitates on cooling rather than ice forming.
The equation for the straight line relationship is
(1/T - 1/T_{m}) = 0.001382 ˣ -Ln(1-x_{S})
(R^{2} = 0.99997) giving a value for ΔH_{fus} of 6016 J ˣ mol^{-1}. [Back to Top ]
The lower blue line follows the 'ideal' colligative equation, ΔT = m_{S}K_{f}. for urea (H_{2}NCONH_{2}). The red line tracks the experimental freezing point curve [70], deviating from the relationship obtained directly from the unapproximated colligative equations above (black line) at higher molality where no bound water of hydration is found^{ j} but allowance has been made for the dimerization of the urea (that is, urea + urea (urea)_{2}) with equilibrium constant 0.0007. Further deviation occurs at higher molality due to the multiple associations of urea molecules [364]. The fitted enthalpy of fusion (ΔH_{fus}) is 6022 J ˣ mol^{-1} here.
Note that mid-infrared pump-probe spectroscopy studies indicate that one molecule of water may be more tightly bound to each molecule of urea [1130]. This may occur together with the above equilibria. [Back to Top ]
The upper blue line follows the 'ideal' colligative equation, ΔT = m_{S}K_{f}. for ethanol (CH_{3}CH_{2}OH). The red line follows the experimental freezing point curve [70], hardly deviating from the relationship obtained directly from the unapproximated colligative equations above (black line) where allowance has been made for bound water of hydration (1.8 water molecules bound to each ethanol molecule) and the changes in enthalpy of fusion with temperature (here the fitted ΔH_{fus} is 6092 J ˣ mol^{-1}). There is considerable deviation at higher molalities, however, as ethanol forms hydrogen-bonded chains and solid clathrate I and II hydrates (at around -63 °C) rather than ice [1102], and with solutions eventually behaving as a solution of water in ethanol rather than ethanol in water. A more extensive freezing point graph is given elsewhere.
There are recent interesting papers on the increased hydrogen-bonding in alcoholic drinks [1119] and the formation of azeotropes [1746] and 'peculiar points' (possible formation of molecular complexes) [2024]. [Back to Top ]
The molality is that of the Na^{+} plus Cl^{-} ions. The blue line (upper at higher molality) follows the 'ideal' colligative equation, ΔT = m_{S}K_{f}. The red line (upper at lower molality) tracks the experimental freezing point curve [70]. It deviates slightly from the relationship (black line) obtained directly from the complete unapproximated colligative equations above, but where an allowance has been made for bound water of hydration (2.3 water molecules bound to each NaCl unit)^{ j} (see also the activity coefficient of salts). The changes in enthalpy of fusion with temperature are here fitted where ΔH_{fus} is 6085 J ˣ mol^{-1}. The actual freezing point depression line (red) shows some deviations away from the theoretical line (black), particularly at higher molality, due to ion-pair (probably separated by bound water) formation.
A minimum in the degree of dissociation of NaCl has been found to be about 0.78 at an ionic molality of about 3 [1108]. Such minima occur with most strong electrolytes and may be explained as the increased ionic concentration interferes with the individual water separated ion-pair formation.
The eutectic point for NaCl aqueous solution occurs at -21.1 °C at ionic molality of 10.34 m (5.17 m NaCl, 23.2 % w/w). At higher concentrations up to 6.0 m NaCl (26.1 % w/w at -0.01 °C) solid NaCl.2H_{2}O precipitates on cooling rather than ice forming and at higher temperatures, there is the solubility line for unhydrated NaCl showing a slight increase in solubility with increasing temperature. [Back to Top ]
The molality is that of the Ca^{2+} plus Cl^{-} ions. The upper blue line follows the 'ideal' colligative equation, ΔT = m_{S}K_{f}. The red line tracks the experimental freezing point curve [70], deviating slightly from the relationship obtained directly from the unapproximated colligative equations above where allowance has been made for bound water of hydration (9.1 water molecules bound to each CaCl_{2} unit)^{ j} and the changes in enthalpy of fusion with temperature (here the fitted ΔH_{fus} is 5736 J ˣ mol^{-1}). The actual freezing point depression line (red) shows some deviations away from the theoretical line (black), particularly at lower molality, due to ion-pair (probably separated by bound water) formation, as with NaCl above.
The eutectic point for CaCl_{2} aqueous solution occurs at -50 °C at ionic molality of 12.7 (4.24 m CaCl_{2}, 32 % w/w) (variously reported at -54.23 °C at 3.83 m CaCl_{2} 29.85% w/w [1121]). At higher concentrations, up to 8.8 m CaCl_{2}(49 % w/w, at +28.9 °C) solid CaCl_{2}.6H_{2}O precipitates on cooling rather than ice forming. At increasingly higher solution concentrations there are solubility curves for CaCl_{2}.4H_{2}O, CaCl_{2}.2H_{2}O, and CaCl_{2}.H_{2}O.
The equation for the straight line relationship is
(1/T - 1/T_{m}) = 0.001450 ˣ -Ln(1-x_{S})
(R^{2} = 0.9992) giving a value for ΔH_{fus} of 5736 J mol^{-1}. The deviations at low molality due to solvent separated ion-pairs mentioned above can be seen at low -Ln(1-x) values, where the experimental data line (red) dips below the theoretical straight line (black). Work similar to this has been published [1064]. [Back to Top ]
The equations for boiling point elevation is derived in a way similar to those of freezing point depression above. The theoretical expression for the ebullioscopic constant (K_{b}) is
(22)
where M_{w} is the molar mass of water (kg ˣ mol^{-1}), R is the gas constant (kg ˣ m^{2}ˣ s^{-2} ˣ mol^{-1} ˣ K^{-1}), T_{b } is the boiling point of water (K), and ΔH_{vap} is the enthalpy of vaporization of water (kg ˣ m^{2} ˣ s^{-2} ˣ mol^{-1}). ^{f} This results in K_{b} = 0.5129 K ˣ kg ˣ mol^{-1}. As molecules tend to bind less water at higher temperatures, boiling point elevation is less affected than freezing point depression by any such reduction in 'free' water [1064].
The boiling point of a solution is dependent on its concentration. As the water boils off, the concentration of the remaining solution increases and this causes the boiling point of the solution to increase until the solution reaches saturation when the boiling point remains constant and the salt precipitates out of the remaining solution until there is no solution left.
[Back to Top ]
The lower blue line follows the 'ideal' colligative equation, ΔT = m_{S}K_{b}. for glucose (C_{6}H_{12}O_{6}). The red line tracks the experimental boiling point curve. The colligative equations may be best fitted making allowance for the smaller bound water of hydration (1.1 water molecules bound to each glucose molecule) than is found in the freezing point depression above, and the changes in enthalpy of vaporization with temperature (here the fitted ΔH_{vap} is 40.4 kJ ˣ mol^{-1}). [Back to Top ]
[Back to Top ]
^{a} The chemical potential (μ) is a term first used by Willard Gibbs and is the same as the molar Gibbs energy of formation, ΔG_{f}, for a pure substance. For materials in a mixture
where the substance changing is A with n_{A} representing the number of A molecules, and n_{B} represents the number of molecules of all other materials present. The molar chemical potential of water in an electrolyte solution (molality m; activity a),H_{2}O_{(T, P, m)}, can be written as
H_{2}O_{(T, P, m)}= H_{2}O_{(T, P, 0)} + RT ln {a_{H2O(T, P, m)}}
[Back]
^{b} In contrast to liquid water, ice (ice Ih) is a very poor solvent for almost all solutes. Even gases are ejected from the solution on freezing the solution. [Back]
^{c} The energy change occurring over a small change (dx) in the mole fraction of water (x_{w}) is RTdx/x_{w}. Therefore the total energy change going from pure water (x_{w}= 1) to this new value (x_{w}) is = RTLn(x_{w}/1)= RTLn(x_{w}). [Back]
^{d} A colligative property of gasses is the molar volume (22.711 L at 100 kPa and 273.15 K). The colligative properties of liquids apply to all solvents, and osmotic pressure can also apply to gases [2100]. [Back]
^{e} The term Ln(1 - x) may be expanded in terms of a Maclaurin series
Ln(1 - x) = -(x + x^{2}/2 + x^{3}/3 + x^{4}/4 + x^{5}/5 + x^{6}/6 + x^{7}/7 + .........)
This series quickly converges for values of x close to zero. [Back]
^{f} The (latent) heat of fusion (ΔH_{fus}) is the amount of thermal energy required to convert the solid form into the liquid form, at constant temperature and pressure. The (latent) heat of vaporization (ΔH_{vap}) is the amount of thermal energy required to convert the liquid form into the gaseous form, at constant temperature and pressure. [Back]
^{g} Experimentally, the (latent) heat of fusion of the solution (ΔH) appears to vary linearly with the mole fraction of the solute (x_{s}) (that is, ΔH = (1 - kx_{s}) ˣ ΔH_{fus} where k is a constant for that particular solute and ΔH_{fus} is the (latent) heat of fusion of pure water, see [1123]) and thus with melting point temperature. Although presented differently, this relationship gives identical equations to the above analysis where k equals the number of moles of hydrating water per mole solute molecules or ions. The similarity is unsurprising given that the bound water does not freeze and therefore does not contribute to the total heat of fusion. [Back]
^{h} Raoult recognized in 1882 that the freezing point depression was determined by the mole fraction [1286]. [Back]
^{i} The term 'ideal' indicates adherence to a particular equation; it does not indicate any more fundamental truth. [Back]
^{j}. Reference [250] gives hydration values calculated with unchanging enthalpy of fusion of ice: glucose 2.8 H_{2}O/mol bound using the data up to 1.8 molal; urea -0.2 H_{2}O/mol bound using the data up to 7.84 molal; NaCl 4.0 H_{2}O/mol bound using the data up to the eutectic concentration (reported 10.4 molal ions concentration); CaCl_{2} 11.2 H_{2}O/mol bound using the data up to 8.54 molal ions concentration. [Back]
^{k}^{} Some solutes, such as hydrophilic polymers like polyvinyl alcohol, increase the chance of supercooling [1498]. [Back]
^{m}^{} The efficiency of water condensation (for example, dew formation) is determined by the geometry of the surface. Asymmetric millimetric bumps covered in a slippery lubricant (such as the surface of cactus spines) give the highest efficiency [2555]. [Back]
^{n}^{} The high binding energy required (≈ 55.6 kJ ˣ mol^{-1}, [2654]) is because a tightly bound water molecule will form two new hydrogen bonds (≈ 2 ˣ 24 ˣ kJ ˣ mol^{-1}) on release. Therefore to prevent competitive release, the water molecule(s) must be held by a stronger link. [Back]
^{o}^{} A mixture is ideal when its physical properties change linearly with changing molecular concentrations. [Back]
^{p} François-Marie Raoult (1830 - 1901) was a French chemist who researched into the properties and behavior of solutions, including the vapor pressure lowering and the depression of freezing points. [Back]
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This page was established in 2006 and last updated by Martin Chaplin on 29 March, 2019