Question

Write the equations relating boiling-point elevation and freezing-point depression to the concentration of the solution. Define all the terms, and give their units.

Short answer

Boiling-point elevation: $\mathrm{\Delta }{T}_b=i\cdot K_b\cdot m$; Freezing-point depression: $\mathrm{\Delta }{T}_f=i\cdot K_f\cdot m$.

Step-by-step solution

Understanding Boiling-point Elevation

Boiling-point elevation is a colligative property, which means it depends on the number of solute particles in a solution, not on the type of solute. The equation for boiling-point elevation is given by:$$\mathrm{\Delta }{T}_b=i\cdot K_b\cdot m$$where:- $\mathrm{\Delta }{T}_b$ is the boiling-point elevation (°C)- $i$ is the van 't Hoff factor (unitless), representing the number of particles the solute breaks into- $K_b$ is the ebullioscopic constant of the solvent (°C·kg/mol)- $m$ is the molality of the solution (mol/kg)

Understanding Freezing-point Depression

Freezing-point depression is another colligative property. It can be described using the following equation:$$\mathrm{\Delta }{T}_f=i\cdot K_f\cdot m$$where:- $\mathrm{\Delta }{T}_f$ is the freezing-point depression (°C)- $i$ remains the van 't Hoff factor (unitless)- $K_f$ is the cryoscopic constant of the solvent (°C·kg/mol)- $m$ remains the molality of the solution (mol/kg)

Defining the Terms and Units

For both boiling-point elevation and freezing-point depression, the relevant terms and their units are as follows:- **$\mathrm{\Delta }{T}_b$ and $\mathrm{\Delta }{T}_f$:** Change in boiling/freezing point, both measured in degrees Celsius (°C).- **$i$:** van 't Hoff factor, unitless, representing the number of particles into which a solute disassociates.- **$K_b$ and $K_f$:** Ebullioscopic and cryoscopic constants for the solvent, respectively, measured in °C·kg/mol.- **$m$:** Molality of the solution, measured in moles of solute per kilogram of solvent (mol/kg).

Key concepts

Boiling-point Elevation

Boiling-point elevation is one of the fascinating colligative properties of solutions. It is specifically important in determining how adding a solute to a solvent can raise the boiling point of that solvent. This effect occurs because solute particles interfere with the vaporization of the solvent. Energy needs to be provided to overcome this interference, leading to a higher boiling point. For example, when salt is added to water, the resultant solution boils at a temperature higher than pure water.

The equation used to calculate this rise is: $$\mathrm{\Delta }{T}_b=i\cdot K_b\cdot m$$Where:

• $\mathrm{\Delta }{T}_b$ is the change in boiling point (in °C).
• $i$ is the van 't Hoff factor (unitless), which represents the number of particles into which the solute dissociates.
• $K_b$ is the ebullioscopic constant (in °C·kg/mol), a property unique to each solvent.
• $m$ denotes the molality (in mol/kg), which measures the concentration of the solute in the solution.

Boiling-point elevation is crucial in fields like cooking and industrial chemistry.

Freezing-point Depression

Freezing-point depression is another intriguing colligative property that explains why solutions freeze at lower temperatures than pure solvents. By adding a solute to a solvent, the purity is disrupted and the orderly crystalline structure is harder to achieve. This means the solution must be cooled to a lower temperature to achieve solidification. This characteristic is utilized in various real-world applications, such as using salt to melt ice on roads.

The mathematical representation of this phenomenon is: $$\mathrm{\Delta }{T}_f=i\cdot K_f\cdot m$$Where:

• $\mathrm{\Delta }{T}_f$ stands for the change in freezing point (in °C).
• $i$ is still the van 't Hoff factor (unitless).
• $K_f$ is the cryoscopic constant (in °C·kg/mol), a value specific to each solvent.
• $m$ symbolizes the molality of the solution (in mol/kg).

This principle is not only fascinating on a theoretical level but is also highly practical.

van 't Hoff Factor

The van 't Hoff factor, denoted as $i$, plays a vital role in colligative properties. It quantifies the effect of solute particles on properties like boiling-point elevation and freezing-point depression. The van 't Hoff factor represents the number of particles a solute releases into the solution.

For instance:

• If a solute like sodium chloride (NaCl) dissociates fully, it results in two particles: sodium (Na${}^{+}$) and chloride (Cl${}^{-}$), making $i=2$.
• For non-electrolytes, such as glucose, the factor $i=1$, since it does not disassociate in solution.

The factor assumes ideal conditions, but deviations can occur in real-world scenarios, depending on interactions between the solute and solvent particles.
The van 't Hoff factor is crucial for accurately predicting how a solute will alter these properties in solutions.

Molality

Molality is a measure of the concentration of a solution based on the amount of solute per kilogram of solvent. It is represented by the symbol $m$ and is expressed in moles per kilogram (mol/kg). Unlike molarity, which depends on the total volume of the solution, molality is determined by the mass of the solvent only.

Here’s why molality is favored in certain calculations:

• It remains unchanged with temperature because it only depends on the mass, making it more reliable for precise thermodynamic property estimations like boiling-point elevation and freezing-point depression.
• Its use in colligative property equations helps predict how solutes will affect the physical properties of solutions.

Understanding molality is critical for scientists and engineers who design processes that depend on precise solution concentrations.

Ebullioscopic Constant

The ebullioscopic constant, symbolized as $K_b$, is a property intrinsic to a solvent that indicates its sensitivity to boiling-point elevation when a solute is added. Measured in °C·kg/mol, $K_b$ is a crucial factor in calculating how much the boiling point of a solvent will increase for a given molality of solution.

This constant is useful in practical applications such as:

• Determining the boiling-point variation when creating antifreeze solutions.
• Engineering processes where precise boiling points are paramount.

Every solvent has its unique $K_b$, equivalent to how much its boiling point is elevated by a 1 molal solution. Effectively utilizing the ebullioscopic constant allows for accurate predictions and control within industrial and scientific contexts.

Cryoscopic Constant

The cryoscopic constant, denoted by $K_f$, is a unique characteristic of a solvent, indicating its tendency to have its freezing point decreased when a solute is dissolved. This constant, measured in °C·kg/mol, helps calculate the extent of freezing-point depression given a solute's molality.

Important uses of the cryoscopic constant include:

• Predicting the freezing points of various solutions, which is vital in preparing safe antifreeze mixtures.
• Understanding and managing natural water solutions, preventing freezing in climatic regions.

Each solvent has its own $K_f$, indicative of how much the freezing point descends per molal concentration of solute. Mastery of this concept is essential for those engaged in fieldwork and designing solutions to environmental challenges.