Colligative properties of water

Chemical potentials of gaseous, liquid, 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) (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 ]

(1)

In the presence of a solute,

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.

(2)

As above, in a solution,

(3)

Therefore,

(4)

and

(5)

(6)

Hence,

(7)

Freezing point depression

When a solute dissolves 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, opposite) until the eutectic point (c) is reached. Further cooling solidifies the remaining solution with the mixed solid cooling to point (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 the solute present, the chemical potential of the water reduces by an amount -RTLn(x_w).^c

There will be a new melting point where

  =              

 =  + 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

As ΔG = ΔH - TΔS, at constant pressure,         

and,                  

(This is the Gibbs-Helmholtz equation)        

Therefore, differentiating Ln(x_w) = -ΔG_fus/RT we get,

Solving the right-hand side and replacing Ln(x_w) by Ln(1-x_S) and ignoring the small temperature dependence of ΔH_fus, ^g

Ignoring the small temperature dependence of ΔH_fus 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

The freezing point lowering ΔT is (T_m - T), a positive value. Suppose T is close to T_m  then T ˣ T_m = T_m². This introduces a small error (c), see below).

x_S is the mole fraction of the solute = moles solute/(moles water + moles solute). At low x_S, x_S may be approximated by moles solute/moles water = molality ˣ molar mass of water = m_SM_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,

ΔT = m_SK_f

This is the commonly used cryoscopic equation,

where K_f (the cryoscopic constant) for water is                     (K ˣ kg ˣ mol⁻¹)

K_f for water is 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 effect of this assumption is shown right where the data for the enthalpy of the fusion of ice is from [906] and [76] for the upper central line and Dorsey and Angell et al. [1098] for the lower central line. However, in practice, the changes occurring in solution [1098] will be somewhat less as the solutes destroy much of the aqueous structuring responsible for the sizeable specific heat increases of the liquid phase on cooling. However, 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_SK_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_SK_f), is greater than using the mole fraction equation (ΔT = x_SK_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 water is removed from the 'free' (freezable) water pool with the addition of the solute, the addition of incremental amounts of solute has a progressively greater lowering effect on the freezing point. Freezing point depression may thus 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', which depend on the method of determination [2654]. The thermodynamic hydration numbers for strongly bound water depend on temperature and pressure and will reduce at high concentrations where ion-pairing may be significant or insufficient water is 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 solute concentration. As the water freezes to become ice, the concentration of the remaining solution increases. 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 equation for the straight-line relationship (see right) is

(1/T - 1/T_m) = 0.001382 ˣ -Ln(1-x_S)

(R² = 0.99997) giving a value for ΔH_fus of 6016 J ˣ mol⁻¹. [Back to Top ]

Urea

The lower blue line (on the right) follows the 'ideal' colligative equation, ΔT = m_SK_f, for urea (H₂NCONH₂). The red line tracks the experimental freezing point curve [70], deviating from the relationship obtained directly from the unapproximated colligative equations (black line) at higher molality where no bound water of hydration is found ^j. Allowance has been made for the dimerization of the urea (that is, urea + urea (urea)₂) with an equilibrium constant of 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 ]

Ethanol

The upper blue line (on the right) follows the 'ideal' colligative equation, ΔT = m_SK_f., for ethanol (CH₃CH₂OH). 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 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 ]

NaCl

The molality is that of the Na⁺ plus Cl⁻ ions. The blue line (upper at higher molality) follows the 'ideal' colligative equation, ΔT = m_SK_f. The red line (upper at lower molality) tracks the experimental freezing point curve [70]. It deviates slightly from the relationship obtained directly from the complete unapproximated colligative equations above (black line), 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.2H₂O 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 ]

CaCl₂

The molality is that of the Ca²⁺ plus Cl⁻ ions. The upper blue line follows the 'ideal' colligative equation, ΔT = m_SK_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 CaCl₂ 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 CaCl₂ aqueous solution occurs at -50 °C at ionic molality of 12.7 (4.24 m CaCl₂, 32 % w/w) (variously reported at -54.23 °C at 3.83 m CaCl₂ 29.85% w/w [1121]). At higher concentrations, up to 8.8 m CaCl₂(49 % w/w, at +28.9 °C) solid CaCl₂.6H₂O precipitates on cooling rather than ice forming. At increasingly higher solution concentrations, there are solubility curves for CaCl₂.4H₂O, CaCl₂.2H₂O, and CaCl₂.H₂O.

The equation for the straight line relationship is

(1/T - 1/T_m) = 0.001450 ˣ -Ln(1-x_S)

(R² = 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 ]

Boiling point elevation

The equations for boiling point elevation are derived similarly to those of freezing point depression above. The theoretical expression for the ebullioscopic constant (K_b) is

                            (K ˣ kg ˣ mol⁻¹)                 (22)

where M_w is the molar mass of water (kg ˣ mol⁻¹), R is the gas constant (kg ˣ m²ˣ 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 m² 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 remaining solution's concentration increases which causes the boiling point of the solution to increase until the solution reaches saturation when the boiling point remains constant. The salt precipitates out of the remaining solution until there is no solution left.

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Glucose

The lower blue line (see right) follows the 'ideal' colligative equation, ΔT = m_SK_b, for glucose (C₆H₁₂O₆). The red line tracks the experimental boiling point curve. The colligative equations may be best fitted by making allowance for the lower 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 ]

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^e The term Ln(1 − x) may be expanded in terms of a Maclaurin series

Ln(1 − x) = -(x + x²/2 + x³/3 + x⁴/4 + x⁵/5 + x⁶/6 + x⁷/7 + .........)

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 mole fraction determines the freezing point depression [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 constant enthalpy of fusion of ice: glucose 2.8 H₂O/mol bound using the data up to 1.8 molal; urea -0.2 H₂O/mol bound using the data up to 7.84 molal; NaCl 4.0 H₂O/mol bound using the data up to the eutectic concentration (reported 10.4 molal ions concentration); CaCl₂ 11.2 H₂O/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. He proposed his linear relationship between the mole fraction and the vapor pressure in 1887; ( F.-M. Raoult, Loi générale des tensions de vapeur des dissolvants, Comptes rendus hebdomadaires des séances de l'Académie des sciences, 104 (1887) 1430-1433).

By assuming equilibria between the free water in the solution and its hydrated forms and that some water may be hidden from the bulk by binding the solute(s) [4102], Raoult’s law has been proven valid over the full range of concentrations [4103]. [Back]

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