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Boiling-point elevation
Boiling-point elevation
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Boiling-point elevation is the phenomenon whereby the boiling point of a liquid (a solvent) will be higher when another compound is added, meaning that a solution has a higher boiling point than a pure solvent. This happens whenever a non-volatile solute, such as a salt, is added to a pure solvent, such as water. The boiling point can be measured accurately using an ebullioscope.

Explanation

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The change in chemical potential of a solvent when a solute is added explains why boiling point elevation takes place.

The boiling point elevation is a colligative property, which means that boiling point elevation is dependent on the number of dissolved particles but not their identity. [1]It is an effect of the dilution of the solvent in the presence of a solute. It is a phenomenon that happens for all solutes in all solutions, even in ideal solutions, and does not depend on any specific solute–solvent interactions. The boiling point elevation happens both when the solute is an electrolyte, such as various salts, and a nonelectrolyte. In thermodynamic terms, the origin of the boiling point elevation is entropic and can be explained in terms of the vapor pressure or chemical potential of the solvent. In both cases, the explanation depends on the fact that many solutes are only present in the liquid phase and do not enter into the gas phase (except at extremely high temperatures).

The vapor pressure affects the solute shown by Raoult's Law while the free energy change and chemical potential are shown by Gibbs free energy. Most solutes remain in the liquid phase and do not enter the gas phase, except at very high temperatures.

In terms of vapor pressure, a liquid boils when its vapor pressure equals the surrounding pressure. A nonvolatile solute lowers the solvent’s vapor pressure, meaning a higher temperature is needed for the vapor pressure to equalize the surrounding pressure, causing the boiling point to elevate.

Original vapor pressure
A nonvolatile solute lowers the solvent’s vapor pressure

In terms of chemical potential, at the boiling point, the liquid and gas phases have the same chemical potential. Adding a nonvolatile solute lowers the solvent’s chemical potential in the liquid phase, but the gas phase remains unaffected. This shifts the equilibrium between phases to a higher temperature, elevating the boiling point.

Relationship between Freezing-point Depression

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Freezing-point depression is analogous to boiling point elevation, though the magnitude of freezing-point depression is higher for the same solvent and solute concentration. These phenomena extend the liquid range of a solvent in the presence of a solute.

[edit]

The extent of boiling-point elevation can be calculated by applying Clausius–Clapeyron relation and Raoult's law together with the assumption of the non-volatility of the solute. The result is that in dilute ideal solutions, the extent of boiling-point elevation is directly proportional to the molal concentration (amount of substance per mass) of the solution according to the equation:[2]

ΔTb = Kb · bc

where the boiling point elevation, is defined as Tb (solution) − Tb (pure solvent).

  • Kb, the ebullioscopic constant, which is dependent on the properties of the solvent. It can be calculated as Kb = RTb2M/ΔHv, where R is the gas constant, and Tb is the boiling temperature of the pure solvent [in K], M is the molar mass of the solvent, and ΔHv is the heat of vaporization per mole of the solvent.
  • bc is the colligative molality, calculated by taking dissociation into account since the boiling point elevation is a colligative property, dependent on the number of particles in solution. This is most easily done by using the van 't Hoff factor i as bc = bsolute · i, where bsolute is the molality of the solution.[3] The factor i accounts for the number of individual particles (typically ions) formed by a compound in solution. Examples:
    • i = 1 for sugar in water
    • i = 1.9 for sodium chloride in water, due to the near full dissociation of NaCl into Na+ and Cl− (often simplified as 2)
    • i = 2.3 for calcium chloride in water, due to nearly full dissociation of CaCl2 into Ca2+ and 2Cl− (often simplified as 3)

Non integer i factors result from ion pairs in solution, which lower the effective number of particles in the solution.

Equation after including the van 't Hoff factor

ΔTb = Kb · bsolute · i

The above formula reduces precision at high concentrations, due to nonideality of the solution. If the solute is volatile, one of the key assumptions used in deriving the formula is not true because the equation derived is for solutions of non-volatile solutes in a volatile solvent. In the case of volatile solutes, the equation can represent a mixture of volatile compounds more accurately, and the effect of the solute on the boiling point must be determined from the phase diagram of the mixture. In such cases, the mixture can sometimes have a lower boiling point than either of the pure components; a mixture with a minimum boiling point is a type of azeotrope.

Ebullioscopic constants

[edit]

Values of the ebullioscopic constants Kb for selected solvents:[4]

Compound Boiling point in °C Ebullioscopic constant Kb in units of [(°C·kg)/mol] or [°C/molal]
Acetic acid 118.1 3.07
Benzene 80.1 2.53
Carbon disulfide 46.2 2.37
Carbon tetrachloride 76.8 4.95
Naphthalene 217.9 5.8
Phenol 181.75 3.04
Water 100 0.512

Uses

[edit]

Together with the formula above, the boiling-point elevation can be used to measure the degree of dissociation or the molar mass of the solute. This kind of measurement is called ebullioscopy (Latin-Greek "boiling-viewing"). However, superheating is a factor that can affect the precision of the measurement and would be challenging to avoid because of the decrease in molecular mobility. Therefore, ΔTb would be hard to measure precisely even though superheating can be partially overcome by the invention of the Beckmann thermometer. In reality, cryoscopy is used more often because the freezing point is often easier to measure with precision.

See also

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References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Boiling-point elevation

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from Grokipedia
Boiling point elevation is a colligative property of solutions in which the of a increases when a nonvolatile solute is dissolved in it, due to the solute particles reducing the solvent's and thereby requiring a higher to reach the boiling condition where vapor pressure equals . 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 in dilute systems. 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 of the solute (moles of solute per of ) 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 and 2.53 °C/m for ). 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. 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. Boiling point elevation finds practical applications in determining the of unknown solutes through ebullioscopic measurements, as well as in like formulations (e.g., in raises the to prevent overheating in engines) and food preparation, where adding salt to slightly elevates its during cooking. For instance, a 6.98 m solution in has a of about 104 °C, demonstrating the effect's utility in thermal management. As one of the four primary —alongside vapor pressure lowering, , and —boiling point elevation underscores the thermodynamic principles governing non-ideal solution behavior.

Fundamentals

Definition and Mechanism

Boiling-point elevation refers to the increase in the of a when a non-volatile solute is dissolved in it, defined as the difference (ΔT_b) between the of the solution and that of the pure . This elevation is directly proportional to the molal concentration of the solute particles in the solution. At the molecular level, the addition of a non-volatile solute reduces the of the above the solution compared to the pure , a phenomenon known as vapor pressure lowering. Since occurs when the of the liquid equals the surrounding , the lowered vapor pressure means the solution must be heated to a higher to achieve this equilibrium and begin . This mechanism arises because solute particles occupy surface sites that would otherwise be available for molecules to evaporate, thereby decreasing the rate of evaporation and requiring elevated temperatures for . A common illustration of this effect is observed when table salt () is added to ; the resulting saltwater solution has a higher than pure , meaning it takes longer to reach under the same conditions, such as cooking pasta in salted . 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 stemming from his formulation of between 1887 and 1888.

Colligative Property Characteristics

Colligative properties are physical characteristics of solutions that depend solely on the number of solute particles dissolved in the , rather than on the chemical identity or nature of those particles. This includes phenomena such as lowering, , , and , which are most accurately observed in dilute, ideal solutions where solute-solvent interactions mimic those in the pure . exemplifies this, as the addition of solute particles reduces the 's , requiring a higher to reach for . A primary characteristic of boiling point elevation as a colligative property is its direct proportionality to the of the solution, which measures the concentration in moles of solute per of , ensuring consistency across different solvents. This proportionality holds for non-volatile, non- 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, yields an i close to 2 in dilute aqueous solutions, though often slightly less due to ion pairing. 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. Limitations of this colligative behavior become evident with volatile solutes, which add their own partial , 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.

Theoretical Aspects

Boiling Point Elevation Formula

The boiling point elevation, denoted as ΔTb\Delta T_b, represents the increase in the boiling temperature of a solution compared to the pure solvent and is given by the equation ΔTb=i⋅Kb⋅m\Delta T_b = i \cdot K_b \cdot m where ii is the van't Hoff factor, KbK_b is the ebullioscopic constant of the solvent, and mm is the molality of the solute. In this formula, mm measures the concentration of the solute in terms of moles of solute particles per of , ensuring the property depends on the number of particles rather than their identity. The KbK_b is a solvent-specific property that indicates the boiling point increase per unit molality for a non-dissociating solute. The van't Hoff factor ii accounts for the number of particles a solute dissociates into in solution; for non-electrolytes like glucose, i=1i = 1, while for electrolytes like (NaCl), which dissociates into two ions, i=2i = 2 under ideal conditions./08%3A_Solutions/8.04%3A_Colligative_Properties-_Boiling_Point_Elevation_and_Freezing_Point_Depression) This formula applies under the assumptions of ideal dilute solutions, where solute-solvent interactions are negligible, the solute is non-volatile (contributing no ), and occurs at standard ./16%3A_The_Chemical_Activity_of_the_Components_of_a_Solution/16.10%3A_Colligative_Properties_-_Boiling-point_Elevation) For example, in a 1 molal of glucose (where i=1i = 1 and Kb≈0.512∘C/mK_b \approx 0.512^\circ \text{C}/\text{m} for ), the elevation is ΔTb=1⋅0.512⋅1=0.512∘C\Delta T_b = 1 \cdot 0.512 \cdot 1 = 0.512^\circ \text{C}, raising the from 100∘C100^\circ \text{C} to approximately 100.512∘C100.512^\circ \text{C}.

Derivation from Raoult's Law

Raoult's law states that the partial vapor pressure of a over an ideal dilute solution is equal to the of the multiplied by the of the pure at the same temperature: P=XsolventPsolvent∘P = X_{\text{solvent}} P^\circ_{\text{solvent}}, where XsolventX_{\text{solvent}} is the of the and Psolvent∘P^\circ_{\text{solvent}} is the of the pure . This law applies to solutions with non-volatile solutes, as the solute does not contribute to the total . Boiling occurs when the total vapor pressure of the solution equals the external atmospheric pressure PatmP_{\text{atm}}. For the pure solvent, the normal boiling point Tb∘T_b^\circ is defined as the temperature at which Psolvent∘(Tb∘)=PatmP^\circ_{\text{solvent}}(T_b^\circ) = P_{\text{atm}}. In the solution, the boiling point TbT_b is the temperature where XsolventPsolvent∘(Tb)=PatmX_{\text{solvent}} P^\circ_{\text{solvent}}(T_b) = P_{\text{atm}}, so Psolvent∘(Tb)=Patm/XsolventP^\circ_{\text{solvent}}(T_b) = P_{\text{atm}} / X_{\text{solvent}}. Since Xsolvent=1−XsoluteX_{\text{solvent}} = 1 - X_{\text{solute}} and the solution is dilute (Xsolute≪1X_{\text{solute}} \ll 1), this approximates to Xsolvent≈1−XsoluteX_{\text{solvent}} \approx 1 - X_{\text{solute}}, yielding Psolvent∘(Tb)≈Patm(1+Xsolute)P^\circ_{\text{solvent}}(T_b) \approx P_{\text{atm}} (1 + X_{\text{solute}}). To relate the temperature change ΔTb=Tb−Tb∘\Delta T_b = T_b - T_b^\circ to the solute concentration, the Clausius-Clapeyron is used, which describes the temperature dependence of the for the pure : dln⁡Psolvent∘dT=ΔHvapRT2,\frac{d \ln P^\circ_{\text{solvent}}}{dT} = \frac{\Delta H_{\text{vap}}}{R T^2}, where ΔHvap\Delta H_{\text{vap}} is the (assumed constant), RR is the , and TT is the . Integrating over a small temperature interval gives ln⁡(Psolvent∘(Tb)Psolvent∘(Tb∘))≈ΔHvapRTb2ΔTb.\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, Psolvent∘(Tb)/Psolvent∘(Tb∘)≈1+XsoluteP^\circ_{\text{solvent}}(T_b) / P^\circ_{\text{solvent}}(T_b^\circ) \approx 1 + X_{\text{solute}}, and using ln⁡(1+Xsolute)≈Xsolute\ln(1 + X_{\text{solute}}) \approx X_{\text{solute}} for small XsoluteX_{\text{solute}}, yields ΔTb≈R(Tb∘)2ΔHvapXsolute.\Delta T_b \approx \frac{R (T_b^\circ)^2}{\Delta H_{\text{vap}}} X_{\text{solute}}. The mole fraction of the solute XsoluteX_{\text{solute}} is related to the molality mm (moles of solute per kg of solvent) by Xsolute≈m⋅(Msolvent/1000)X_{\text{solute}} \approx m \cdot (M_{\text{solvent}} / 1000) for dilute aqueous solutions, where MsolventM_{\text{solvent}} is the molar mass of the solvent in g/mol. For electrolytes that dissociate into ii particles (van't Hoff factor), this becomes Xsolute≈im⋅(Msolvent/1000)X_{\text{solute}} \approx i m \cdot (M_{\text{solvent}} / 1000). Substituting gives ΔTb≈R(Tb∘)2Msolvent1000ΔHvap⋅im=Kb⋅im,\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 Kb=R(Tb∘)2Msolvent/(1000ΔHvap)K_b = R (T_b^\circ)^2 M_{\text{solvent}} / (1000 \Delta H_{\text{vap}}) is the molal boiling-point elevation constant. This derivation relies on key approximations: the solution is dilute such that Xsolute≪1X_{\text{solute}} \ll 1 and higher-order terms can be neglected, and ΔHvap\Delta H_{\text{vap}} is temperature-independent over the small ΔTb\Delta T_b. These assumptions hold well for ideal or near-ideal dilute solutions.

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: ΔT=K⋅m⋅i\Delta T = K \cdot m \cdot i, where KK is the respective constant, mm is the molality, and ii is the van't Hoff factor accounting for dissociation. The mechanisms differ fundamentally due to the distinct phase transitions involved. Boiling-point elevation occurs because the solute lowers the solvent's , requiring a higher to achieve equilibrium with the external during the endothermic process. In contrast, arises from the need to equalize the between the solid and phases in the presence of solute, shifting the exothermic fusion equilibrium to a lower . The ebullioscopic constant (KbK_b) and cryoscopic constant (KfK_f) differ in value because the (ΔHvap\Delta H_\text{vap}) is significantly larger than the (ΔHfus\Delta H_\text{fus}), resulting in Kb<KfK_b < K_f for most solvents like water. 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 (TbT_b), while freezing-point depression lowers the normal freezing temperature (TfT_f), effectively widening the liquid range of the solution. 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.

Ebullioscopic Constants

The ebullioscopic constant, denoted KbK_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 ΔTb=Kb⋅m\Delta T_b = K_b \cdot m, where mm is the molality, and has units of °C kg/mol (or °C/m)./14:_Properties_of_Solutions/14.02:_Colligative_Properties) The value of KbK_b for a given solvent is determined theoretically from its thermodynamic properties using the relation Kb=RTb2Msolvent1000ΔHvap,K_b = \frac{R T_b^2 M_\text{solvent}}{1000 \Delta H_\text{vap}}, where RR is the gas constant (8.314 J/mol·K), TbT_b is the normal boiling point of the pure solvent in Kelvin, MsolventM_\text{solvent} is the molar mass of the solvent in g/mol, and ΔHvap\Delta H_\text{vap} is the molar enthalpy of vaporization at TbT_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. Experimentally, KbK_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 Kb=ΔTb/mK_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 TbT_b. Values of KbK_b vary significantly among solvents due to differences in their boiling points and enthalpies of vaporization; higher TbT_b and lower ΔHvap\Delta H_\text{vap} generally yield larger KbK_b. Representative values for common solvents, based on experimental measurements, are provided in the table below.
SolventKbK_b (°C kg/mol)
Water0.512
Ethanol1.22
Benzene2.53
Acetic acid3.07
Acetone1.71
These values are typically reported at standard atmospheric pressure (1 atm) and near the normal boiling point. Although KbK_b is often treated as constant for practical calculations in dilute solutions, it exhibits dependence on temperature and pressure because TbT_b and ΔHvap\Delta H_\text{vap} vary with these conditions; for instance, increasing pressure raises TbT_b, thereby increasing KbK_b via the Tb2T_b^2 term. Corrections for non-ideal solutions involve incorporating activity coefficients or virial coefficients to account for solute-solvent interactions that deviate from Raoult's law, particularly at higher concentrations.

Practical Applications

Analytical Techniques

Ebullioscopy is an analytical technique that utilizes boiling-point elevation to determine the molecular weight of an unknown non-volatile solute by measuring the temperature increase ΔT_b in a known solvent. The principle relies on the colligative nature of boiling-point elevation, where ΔT_b is proportional to the molality m of the solute, as given by the formula ΔT_b = K_b m (with K_b as the ebullioscopic constant of the solvent). Once m is calculated from the measured ΔT_b, the molar mass M of the solute is obtained using M = (w_solute / w_solvent) × (1000 / m), where w_solute and w_solvent are the masses in grams of the solute and solvent, respectively (the factor of 1000 converts solvent mass to kilograms). This method, introduced by Ernst Beckmann in 1890, provides a direct way to quantify solute concentration based on vapor pressure lowering in the solution. The procedure typically employs an ebullioscope apparatus equipped with a sensitive thermometer, such as the Beckmann thermometer, which has a narrow range (e.g., 6°C span) and can measure temperature differences to 0.001°C. The process begins by calibrating the boiling point of the pure solvent under controlled conditions, often at reduced pressure to minimize superheating, using a setup with a boiling chamber, condenser, and manometer for precise pressure adjustment. A known mass of the solute is then dissolved in a weighed amount of solvent (typically 10–20 g), and the solution is boiled in the apparatus; the steady-state boiling temperature is recorded once equilibrium is reached, usually after 5–15 minutes of boiling facilitated by a Cottrell pump to ensure uniform heating around the thermometer bulb. The ΔT_b is the difference between the solution's boiling point and that of the pure solvent, often measured differentially in a dual-chamber design for greater accuracy. This technique offers several advantages for laboratory analysis, including its simplicity and the minimal sample size required (often 0.1–1 g of solute), making it suitable for non-volatile organic compounds where high precision is needed without complex equipment. It is particularly accurate for solutes that do not dissociate or associate in solution, yielding results comparable to other colligative methods for low-molecular-weight substances. However, ebullioscopy has notable limitations, such as susceptibility to errors from volatile solutes that contribute to the vapor phase and alter the expected ΔT_b, as well as from superheating or transient hot spots during boiling that can skew temperature readings. Impurities in the solvent or solute can also introduce inaccuracies by affecting the baseline boiling point, and the method is less effective for high-molecular-weight polymers due to the small ΔT_b produced. Modern alternatives, such as vapor pressure osmometry, have largely supplanted ebullioscopy for routine analyses owing to greater sensitivity and ease of operation at ambient temperatures.

Industrial and Everyday Uses

In everyday cooking, adding salt to water for boiling pasta exemplifies boiling-point elevation on a small scale. Typical recipes recommend 1–2 tablespoons of salt per gallon of water, resulting in a concentration of approximately 5–10 g/L NaCl, which raises the boiling point by about 0.1–0.2°C. This slight increase allows the water to reach a marginally higher temperature, potentially aiding in more even cooking of the pasta without significantly altering the process time, though the primary purpose of salting is flavor enhancement. In industrial applications, such as automotive engine coolants, boiling-point elevation is leveraged through the addition of solutes like ethylene glycol or propylene glycol to water. A common 50/50 mixture of ethylene glycol and water elevates the boiling point from 100°C to approximately 106–107°C at atmospheric pressure, enabling the coolant to operate at higher temperatures without vaporizing and losing efficiency. This prevents overheating in engines, where the system is often pressurized to further raise the effective boiling point to around 120–130°C, improving heat transfer and system reliability. Boiling-point elevation plays a role in distillation processes across industries, where differences in boiling points, influenced by component concentrations, facilitate separation. In petroleum refining, crude oil is subjected to fractional distillation, separating hydrocarbons into fractions like gasoline (boiling range 40–200°C) and diesel (200–350°C) based on their inherent boiling points. Similarly, in the food industry, concentrating sugar syrups for products like jams or candies exploits boiling-point elevation; as water evaporates, sugar concentration rises, increasing the boiling point from 100°C to over 110°C at 70–80% solids, allowing precise control of viscosity and texture without scorching. In engineering and environmental contexts, boiling-point elevation influences boiler operations and desalination. Boiler water treatment often involves additives like phosphates or polymers to manage total dissolved solids (TDS), which can elevate the boiling point by 0.5–2°C at high concentrations, helping maintain liquid phase under pressure and reducing cavitation risks in feed pumps where localized pressure drops might otherwise induce boiling. In desalination, particularly multi-effect distillation systems, boiling-point elevation in concentrated brines (up to 3–5°C at salinities >100 g/kg) reduces by diminishing the temperature driving force for , leading to up to 62% overestimation of energy performance if neglected in high-recovery designs (>70%), with economic impacts including millions in miscalculated costs.

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