Boiling-point elevation
Boiling point elevation is a colligative property of solutions in which the boiling point of a solvent increases when a nonvolatile solute is dissolved in it, due to the solute particles reducing the solvent's vapor pressure and thereby requiring a higher temperature to reach the boiling condition where vapor pressure equals atmospheric pressure.[1] This effect is independent of the solute's chemical identity and depends solely on the number of solute particles relative to the solvent, making it proportional to the solution's molality in dilute systems.[2] The magnitude of boiling point elevation, denoted as ΔT_b, is quantitatively described by the formula ΔT_b = K_b × m, where m is the molality of the solute (moles of solute per kilogram of solvent) and K_b is the molal boiling point elevation constant, a solvent-specific value that reflects the solvent's inherent properties (e.g., 0.512 °C/m for water and 2.53 °C/m for benzene).[2] For electrolytes that dissociate into ions, the formula is adjusted by the van't Hoff factor i to account for the effective number of particles: ΔT_b = i × K_b × m, as formalized by Jacobus Henricus van't Hoff in his 1884 work on solution thermodynamics.[1] This adjustment is crucial for accurate predictions, such as in a 1.00 m aqueous NaCl solution, where i ≈ 2 leads to a boiling point of approximately 101.02 °C.[3] Boiling point elevation finds practical applications in determining the molar mass of unknown solutes through ebullioscopic measurements, as well as in industrial processes like antifreeze formulations (e.g., ethylene glycol in water raises the boiling point to prevent overheating in engines) and food preparation, where adding salt to water slightly elevates its boiling point during cooking.[2] For instance, a 6.98 m ethylene glycol solution in water has a boiling point of about 104 °C, demonstrating the effect's utility in thermal management.[3] As one of the four primary colligative properties—alongside vapor pressure lowering, freezing point depression, and osmotic pressure—boiling point elevation underscores the thermodynamic principles governing non-ideal solution behavior.[1]Fundamentals
Definition and Mechanism
Boiling-point elevation refers to the increase in the boiling point of a solvent when a non-volatile solute is dissolved in it, defined as the difference (ΔT_b) between the boiling point of the solution and that of the pure solvent. This elevation is directly proportional to the molal concentration of the solute particles in the solution.[2] At the molecular level, the addition of a non-volatile solute reduces the vapor pressure of the solvent above the solution compared to the pure solvent, a phenomenon known as vapor pressure lowering. Since boiling occurs when the vapor pressure of the liquid equals the surrounding atmospheric pressure, the lowered vapor pressure means the solution must be heated to a higher temperature to achieve this equilibrium and begin boiling. This mechanism arises because solute particles occupy surface sites that would otherwise be available for solvent molecules to evaporate, thereby decreasing the rate of solvent evaporation and requiring elevated temperatures for boiling.[4] A common illustration of this effect is observed when table salt (sodium chloride) is added to water; the resulting saltwater solution has a higher boiling point than pure water, meaning it takes longer to reach boiling under the same conditions, such as cooking pasta in salted water.[5] This phenomenon was first systematically observed and studied by French chemist François-Marie Raoult in the late 19th century, as part of his investigations into colligative properties stemming from his formulation of Raoult's law between 1887 and 1888.[1]Colligative Property Characteristics
Colligative properties are physical characteristics of solutions that depend solely on the number of solute particles dissolved in the solvent, rather than on the chemical identity or nature of those particles. This includes phenomena such as vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure, which are most accurately observed in dilute, ideal solutions where solute-solvent interactions mimic those in the pure solvent. Boiling point elevation exemplifies this, as the addition of solute particles reduces the solvent's vapor pressure, requiring a higher temperature to reach atmospheric pressure for boiling.[6][1] A primary characteristic of boiling point elevation as a colligative property is its direct proportionality to the molality of the solution, which measures the concentration in moles of solute per kilogram of solvent, ensuring consistency across different solvents. This proportionality holds for non-volatile, non-electrolyte solutes, where each solute particle contributes equally to the effect without dissociation. For electrolyte solutes, however, the observed elevation deviates due to ionic dissociation, quantified by the van't Hoff factor (i), which represents the effective number of particles produced per formula unit of solute; for instance, sodium chloride yields an i close to 2 in dilute aqueous solutions, though often slightly less due to ion pairing.[6][1] The magnitude of boiling point elevation is influenced by the solvent's inherent properties, the volatility of the solute, and the ideality of the solution. Non-volatile solutes are assumed, as they do not contribute significantly to the total vapor pressure, allowing the effect to stem primarily from solvent molecules. In ideal solutions, solute particles dilute the solvent uniformly without altering molecular interactions, but deviations arise when solute-solvent attractions differ markedly from solvent-solvent ones.[7][1] Limitations of this colligative behavior become evident with volatile solutes, which add their own partial vapor pressure, complicating the pure solvent's contribution and invalidating simple proportionality. In concentrated solutions, non-ideal effects dominate, where activity coefficients—measures of effective concentration accounting for intermolecular forces—deviate from unity, leading to unpredictable elevations beyond dilute regimes. These constraints highlight that colligative models are approximations best suited to low solute concentrations.[6][1]Theoretical Aspects
Boiling Point Elevation Formula
The boiling point elevation, denoted as \Delta T_b, represents the increase in the boiling temperature of a solution compared to the pure solvent and is given by the equation \Delta T_b = i \cdot K_b \cdot m where i is the van't Hoff factor, K_b is the ebullioscopic constant of the solvent, and m is the molality of the solute.[8][9] In this formula, molality m measures the concentration of the solute in terms of moles of solute particles per kilogram of solvent, ensuring the property depends on the number of particles rather than their identity.[1] The ebullioscopic constant K_b is a solvent-specific property that indicates the boiling point increase per unit molality for a non-dissociating solute.[2] The van't Hoff factor i accounts for the number of particles a solute dissociates into in solution; for non-electrolytes like glucose, i = 1, while for electrolytes like sodium chloride (NaCl), which dissociates into two ions, i = 2 under ideal conditions./08%3A_Solutions/8.04%3A_Colligative_Properties-_Boiling_Point_Elevation_and_Freezing_Point_Depression)[9] This formula applies under the assumptions of ideal dilute solutions, where solute-solvent interactions are negligible, the solute is non-volatile (contributing no vapor pressure), and boiling occurs at standard atmospheric pressure./16%3A_The_Chemical_Activity_of_the_Components_of_a_Solution/16.10%3A_Colligative_Properties_-_Boiling-point_Elevation)[4] For example, in a 1 molal aqueous solution of glucose (where i = 1 and K_b \approx 0.512^\circ \text{C}/\text{m} for water), the boiling point elevation is \Delta T_b = 1 \cdot 0.512 \cdot 1 = 0.512^\circ \text{C}, raising the boiling point from $100^\circ \text{C} to approximately $100.512^\circ \text{C}.[10][1]Derivation from Raoult's Law
Raoult's law states that the partial vapor pressure of a solvent over an ideal dilute solution is equal to the mole fraction of the solvent multiplied by the vapor pressure of the pure solvent at the same temperature: P = X_{\text{solvent}} P^\circ_{\text{solvent}}, where X_{\text{solvent}} is the mole fraction of the solvent and P^\circ_{\text{solvent}} is the vapor pressure of the pure solvent.[11] This law applies to solutions with non-volatile solutes, as the solute does not contribute to the total vapor pressure.[12] Boiling occurs when the total vapor pressure of the solution equals the external atmospheric pressure P_{\text{atm}}. For the pure solvent, the normal boiling point T_b^\circ is defined as the temperature at which P^\circ_{\text{solvent}}(T_b^\circ) = P_{\text{atm}}. In the solution, the boiling point T_b is the temperature where X_{\text{solvent}} P^\circ_{\text{solvent}}(T_b) = P_{\text{atm}}, so P^\circ_{\text{solvent}}(T_b) = P_{\text{atm}} / X_{\text{solvent}}.[11] Since X_{\text{solvent}} = 1 - X_{\text{solute}} and the solution is dilute (X_{\text{solute}} \ll 1), this approximates to X_{\text{solvent}} \approx 1 - X_{\text{solute}}, yielding P^\circ_{\text{solvent}}(T_b) \approx P_{\text{atm}} (1 + X_{\text{solute}}).[13] To relate the temperature change \Delta T_b = T_b - T_b^\circ to the solute concentration, the Clausius-Clapeyron equation is used, which describes the temperature dependence of the vapor pressure for the pure solvent: \frac{d \ln P^\circ_{\text{solvent}}}{dT} = \frac{\Delta H_{\text{vap}}}{R T^2}, where \Delta H_{\text{vap}} is the enthalpy of vaporization (assumed constant), R is the gas constant, and T is the temperature. Integrating over a small temperature interval gives \ln \left( \frac{P^\circ_{\text{solvent}}(T_b)}{P^\circ_{\text{solvent}}(T_b^\circ)} \right) \approx \frac{\Delta H_{\text{vap}}}{R T_b^2} \Delta T_b. Substituting the approximation for the vapor pressures, P^\circ_{\text{solvent}}(T_b) / P^\circ_{\text{solvent}}(T_b^\circ) \approx 1 + X_{\text{solute}}, and using \ln(1 + X_{\text{solute}}) \approx X_{\text{solute}} for small X_{\text{solute}}, yields \Delta T_b \approx \frac{R (T_b^\circ)^2}{\Delta H_{\text{vap}}} X_{\text{solute}}. [11][13] The mole fraction of the solute X_{\text{solute}} is related to the molality m (moles of solute per kg of solvent) by X_{\text{solute}} \approx m \cdot (M_{\text{solvent}} / 1000) for dilute aqueous solutions, where M_{\text{solvent}} is the molar mass of the solvent in g/mol. For electrolytes that dissociate into i particles (van't Hoff factor), this becomes X_{\text{solute}} \approx i m \cdot (M_{\text{solvent}} / 1000). Substituting gives \Delta T_b \approx \frac{R (T_b^\circ)^2 M_{\text{solvent}}}{1000 \Delta H_{\text{vap}}} \cdot i m = K_b \cdot i m, where K_b = R (T_b^\circ)^2 M_{\text{solvent}} / (1000 \Delta H_{\text{vap}}) is the molal boiling-point elevation constant.[12][11] This derivation relies on key approximations: the solution is dilute such that X_{\text{solute}} \ll 1 and higher-order terms can be neglected, and \Delta H_{\text{vap}} is temperature-independent over the small \Delta T_b. These assumptions hold well for ideal or near-ideal dilute solutions.[13][11]Comparative Properties
Relation to Freezing Point Depression
Boiling-point elevation and freezing-point depression are both colligative properties that stem from the interactions between solute particles and solvent molecules, which disrupt the phase equilibria of the pure solvent. These effects depend on the number of solute particles rather than their nature, leading to changes in the solvent's vapor pressure and chemical potential. In both cases, the magnitude of the temperature shift is proportional to the molality of the solution and follows a similar mathematical form: \Delta T = K \cdot m \cdot i, where K is the respective constant, m is the molality, and i is the van't Hoff factor accounting for dissociation.[1][14] The mechanisms differ fundamentally due to the distinct phase transitions involved. Boiling-point elevation occurs because the solute lowers the solvent's vapor pressure, requiring a higher temperature to achieve equilibrium with the external pressure during the endothermic vaporization process. In contrast, freezing-point depression arises from the need to equalize the chemical potential between the solid and liquid phases in the presence of solute, shifting the exothermic fusion equilibrium to a lower temperature. The ebullioscopic constant (K_b) and cryoscopic constant (K_f) differ in value because the enthalpy of vaporization (\Delta H_\text{vap}) is significantly larger than the enthalpy of fusion (\Delta H_\text{fus}), resulting in K_b < K_f for most solvents like water.[1][15][16] Thermodynamically, both properties are analogous, derived from the Gibbs phase rule, which governs the degrees of freedom in phase equilibria, and concepts of fugacity, which equate the escaping tendencies of components across phases. Boiling-point elevation increases the normal boiling temperature (T_b), while freezing-point depression lowers the normal freezing temperature (T_f), effectively widening the liquid range of the solution.[17][18] In practice, these properties are exploited together in cryoscopy and ebullioscopy to determine the molecular weights of solutes by measuring temperature shifts and extrapolating to infinite dilution for accuracy, with ebullioscopy often preferred for nonvolatile compounds due to fewer complications from solid-phase interactions.[19]Ebullioscopic Constants
The ebullioscopic constant, denoted K_b, is a characteristic property of a solvent that relates the molality of a non-volatile solute to the resulting elevation in the solvent's boiling point. It appears in the boiling point elevation equation as \Delta T_b = K_b \cdot m, where m is the molality, and has units of °C kg/mol (or °C/m)./14:_Properties_of_Solutions/14.02:_Colligative_Properties) The value of K_b for a given solvent is determined theoretically from its thermodynamic properties using the relation K_b = \frac{R T_b^2 M_\text{solvent}}{1000 \Delta H_\text{vap}}, where R is the gas constant (8.314 J/mol·K), T_b is the normal boiling point of the pure solvent in Kelvin, M_\text{solvent} is the molar mass of the solvent in g/mol, and \Delta H_\text{vap} is the molar enthalpy of vaporization at T_b. The factor of 1000 accounts for the definition of molality in moles per kilogram of solvent. This formula assumes ideal dilute solution behavior and derives from the Clausius-Clapeyron equation applied to the solvent's vapor pressure lowering.[20] Experimentally, K_b is determined by preparing solutions of known molality with a non-volatile solute and measuring the precise boiling point elevation using ebullioscopic methods, such as the Landsberger-Walker apparatus or differential boiling point apparatus, under controlled constant pressure. The constant is then calculated as K_b = \Delta T_b / m from data extrapolated to infinite dilution to minimize solute-solute interactions. Accuracy depends on factors like solvent purity (impurities can alter vapor pressure), precise temperature control (typically ±0.01 °C), and maintaining atmospheric pressure to avoid shifts in T_b.[12] Values of K_b vary significantly among solvents due to differences in their boiling points and enthalpies of vaporization; higher T_b and lower \Delta H_\text{vap} generally yield larger K_b. Representative values for common solvents, based on experimental measurements, are provided in the table below.| Solvent | K_b (°C kg/mol) |
|---|---|
| Water | 0.512 |
| Ethanol | 1.22 |
| Benzene | 2.53 |
| Acetic acid | 3.07 |
| Acetone | 1.71 |