Colligative Properties of Solutions

Colligative Properties of Dilute Solutions

Colligative properties are physicalproperties of solutions that depend on the number of solute particles in a givenamount of solvent, rather than on the identity of the solute particles themselves. They are observed in dilute solutions, typically where the solute concentration is less than about 1 mol/L.

Examples of colligative properties include vapor pressure lowering, boiling point elevation, freezing point depression, andosmotic pressure.

Understanding these properties is crucial in various fields, especially in medicine and daily life, such as preventing ice formation on roads or ensuring correct concentrations for intravenous injections.

1. Vapor Pressure

Matter exists in three phases: solid, liquid, and gas. Vaporization is the process where a liquid changes to a gaseous state. When this occurs below the boiling point, it's called evaporation. Condensation is the reverse process, where a gas changes into a liquid.

1.1 Definition of Vapor Pressure

At a given temperature, vapor pressure (`$p$`) is the pressure exerted by a vapor that is in dynamic equilibrium with its liquid phase. This means the rate of evaporation equals the rate of condensation.

• The standard unit for vapor pressure is Pascals (Pa) or kilopascals (kPa).
• Other units include atmospheres (atm), torr, and mmHg.
• Unit Conversions:
  1. $1 \, \text{atm} = 101.3 \, \text{kPa} = 1.013 \times 10^5 \, \text{Pa}$
  2. $1 \, \text{atm} = 760 \, \text{torr} = 760 \, \text{mmHg}$
• Vapor pressure increases with increasing temperature.
• Volatile liquids have high vapor pressure because they easily vaporize (e.g., ethanol, ether, acetone).
• Nonvolatile liquids have low vapor pressure because they are harder to vaporize.

2. Vapor Pressure Lowering

When a nonvolatile solute (e.g., sugar) is dissolved in a puresolvent (e.g., water), the vapor pressure of the resulting solution (`$p$`) is lower than that of the pure solvent (`$p^*$`) at the same temperature. This phenomenon is called vapor pressure lowering.

2.1 Explanation

The nonvolatile solute particles occupy aportion of the solvent's surface, reducing the surface area available for solvent molecules to escape into the vapor phase. This decreases the escaping tendency of solvent molecules, leading to a lower vapor pressure.

$p < p^*$

2.2 Raoult'sLaw

Raoult's Law describes the relationship between the vapor pressure of a solution and the mole fraction of the solvent:

$p = p^* \cdot x_A$

Where:

• `$p$` is the vapor pressure of the solution.
• `$p^*$` is the vapor pressure of the pure solvent.
• `$x_A$` is the mole fraction of the solvent. Since a solute is present, `$x_A < 1$`.

The vapor pressure lowering (`$\Delta p$`) is the difference between the vapor pressure of the pure solvent and the solution:

$\Delta p = p^* - p$

Substituting Raoult's Law into the equation for vapor pressure lowering:

$\Delta p = p^* - (p^* \cdot x_A)$

$\Delta p = p^* (1 - x_A)$

Since `$x_A + x_B = 1$` (where `$x_B$` is the mole fraction of the solute), we can write `$1 - x_A = x_B$`.

$\Delta p = p^* \cdot x_B$

Therefore, vapor pressure lowering is directly proportional to the mole fraction of the solute.

2.3 Application for Dilute Solutions

For very dilute solutions, where the number of solvent moles (`$n_A$`) is much greater than thenumber of solute moles (`$n_B$`), the mole fraction of the solute can be approximated:

$x_B \approx \frac{n_B}{n_A}$

If `$n_A$` is expressed in massand molar mass of A, and `$n_B$` from molality (`$m$`), then:

$x_B = m \times 10^{-3} \times \text{Molar mass of A}$

So, $\Deltap = p^* \times m \times 10^{-3} \times \text{Molar mass of A}$. This can be simplified to $\Delta p = K \cdot m$, where `$K = p^* \times 10^{-3} \times \text{Molar mass of A}$` is aconstant for a given solvent at a specific temperature.

Note: Raoult's Law applies only to nonvolatile, nonionizing dilute solutions.

3. Boiling Point Elevation

The boiling point ofa liquid is the temperature at which its vapor pressure equals the external pressure.

• When the vapor pressure equals the external pressure, the liquid begins to boil.
• The normal boiling point is the boiling point at an external pressure of 1.00 atm (101.3 kPa). For pure water, this is $100^\circ\text{C}$.
• Boiling point changes with external pressure.

3.1 Definition of Boiling Point Elevation

The boiling point elevation (`$\Delta T_b$`) is the phenomenon where the boiling point of a solution is higher than that of the pure solvent at the same external pressure.

This occurs because the presence of a nonvolatile solute lowers the vapor pressure of the solution. To reach the external pressure (and thus boil), the solution must beheated to a higher temperature than the pure solvent.

3.2 Calculation of Boiling Point Elevation

The boiling point elevation is directly proportional to the molality (`$m$`) of the solute:

$\Delta T_b = K_b \cdot m$

Where:

• `$\Delta T_b$` is the boiling point elevation of the solution (in K or $^\circ\text{C}$).
• `$m$` is the molality of the solute (moles of solute per kg of solvent, in mol·kg$^{-1}$).
• `$K_b$` is the molal boiling point elevation constant (or ebullioscopic constant).
  1. `$K_b$` is a characteristic physical property of thesolvent, independent of the nature of the solute.
  2. It can be measured experimentally.
  3. Its unit is K·kg·mol$^{-1}$ (or $^\circ\text{C}$·kg·mol$^{-1}$).

Table: `$K_b$` of some common solvents

4. Freezing Point Depression

The freezing point is the temperature at which a solid and liquid phase can coexist in equilibrium. The melting point of a solid and the freezing point of its liquid are identical. At this temperature, solid and liquid coexist in equilibrium.

4.1 Definition of Freezing Point Depression

The freezing point depression (`$\Delta T_f$`) is the phenomenon where the freezing point of a solution is lower than that of the pure solvent.

The presence of soluteparticles interferes with the formation of the ordered solid structure of the solvent, requiring a lower temperature to initiate freezing.

4.2 Calculation of Freezing Point Depression

The freezing point depression is directly proportional to the molality (`$m$`) of the solute:

$\Delta T_f = K_f \cdot m$

Where:

• `$\Delta T_f$` is the freezing point depression of the solution (in K or $^\circ\text{C}$).
• `$m$` is the molality of the solute (in mol·kg$^{-1}$).
• `$K_f$` is the molal freezing point depression constant (or cryoscopic constant).
  1. `$K_f$` is a characteristic physical property of the solvent, independent of the nature of the solute.
  2. It can be measured experimentally.
  3. Its unit is K·kg·mol$^{-1}$ (or $^\circ\text{C}$·kg·mol$^{-1}$).

Table: `$K_f$` of some common solvents

4.3 Example Calculation

Problem: A substance extractedfrom urine (90 mg) is dissolved in 12 g of distilled water. The freezing point of the solution is 0.233 K lower than pure water ($\Delta T_f = 0.233 \, \text{K}$). Calculate the relative molecular mass of the substance.

Given: `$K_f$` for water = 1.86 K·kg·mol$^{-1}$.

Solution:

1. Use the freezing point depression formula: $\Delta T_f = K_f \cdot m$
2. Rearrange to find molality (`$m$`): `$m = \frac{\Delta T_f}{K_f}$`
3. Calculate molality: `$m = \frac{0.233 \, \text{K}}{1.86 \, \text{K} \cdot \text{kg} \cdot \text{mol}^{-1}} = 0.1252 \, \text{mol} \cdot \text{kg}^{-1}$`
4. Molality is moles of solute per kg of solvent: `$m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}}$`
5. Calculate moles of solute: `$\text{moles of solute} = m \times \text{mass of solvent (kg)}$`
6. Convert mass of water to kg: `$12 \, \text{g} =0.012 \, \text{kg}$`
7. `$\text{moles of solute} = 0.1252 \, \text{mol} \cdot \text{kg}^{-1} \times 0.012 \, \text{kg} = 0.0015024 \, \text{mol}$`
8. Calculate molar mass: `$\text{Molar mass} = \frac{\text{mass of solute}}{\text{moles of solute}}$`
9. Convert mass of solute to g: `$90 \, \text{mg} = 0.090 \, \text{g}$`
10. `$\text{Molar mass} = \frac{0.090 \, \text{g}}{0.0015024 \, \text{mol}} \approx 59.9 \, \text{g} \cdot \text{mol}^{-1}$`

This molar mass is close to that of urea (`$\text{CO}(\text{NH}_2)_2$`), which is approximately 60.06 g/mol.

5. Colligative Properties for Electrolytic Solutions (van't Hoff factor)

The equations for colligative properties (`$\Delta p = K \cdot m$`, `$\Delta T_b = K_b \cdot m$`, `$\Delta T_f = K_f \cdot m$`) apply to nonvolatile, nonelectrolytic dilute solutions.

For electrolytic solutions (e.g., NaCl, CaCl$_2$), the solute dissociates into multiple ions, increasing the number of particles in the solution. These equations must bemodified using the van't Hoff factor (`$i$`).

5.1 van't Hoff Factor (`$i$`)

The van't Hoff factor (`$i$`) represents the number of particles a solute dissociates into when dissolved in a solvent. Itaccounts for the actual number of solute particles in the solution.

• For non-electrolytes (e.g., glucose, sugar), `$i = 1$`.
  • Example: Glucose solution (0.1 mol·kg$^{-1}$) has 0.1 moles of particles.
• For strong electrolytes, `$i$` is approximately equal to the number of ions formed per formula unit.
  • Example: NaCl dissociates into Na$^+$ and Cl$^-$, so `$i = 2$`.
    • 0.1 mol·kg$^{-1}$ NaCl solution has `$2 \times 0.1 = 0.2$` moles of particles.
    • Therefore, `$\Delta T_f (\text{NaCl}) = 2 \cdot K_f \cdot m$`as compared to `$\Delta T_f (\text{Glucose}) = K_f \cdot m$`.
  • Example: CaCl$_2$ dissociates into Ca$^{2+}$ and 2 Cl$^-$, so `$i = 3$`.

5.2 Modified Colligative Property Equations for Electrolytes

• Vapor Pressure Lowering: `$\Delta p = i \cdot K \cdot m$`
• Boiling Point Elevation: `$\DeltaT_b = i \cdot K_b \cdot m$`
• Freezing Point Depression: `$\Delta T_f = i \cdot K_f \cdot m$`

5.3 Ranking Freezing Points of Solutions

To rank solutions by freezing point, we needto compare their effective molality (`$i \cdot m$`), as a lower freezing point corresponds to a higher effective molality.

Problem: List the following aqueous solutions in order of their expected freezing points (from lowest to highest): 0.050 m CaCl$_2$; 0.15 m NaCl; 0.10 m HCl; 0.10 m C$_{12}$H$_{22}$O$_{11}$ (sucrose).

Solution: Calculate `$i \cdot m$` for each solution:

• 0.050 m CaCl$_2$: CaCl$_2$ dissociates into 1 Ca$^{2+}$ and 2 Cl$^-$, so `$i = 3$`.
  • Effective molality: `$3 \times 0.050 \, \text{m} =0.150 \, \text{m}$`
• 0.15 m NaCl: NaCl dissociates into 1 Na$^+$ and 1 Cl$^-$, so `$i = 2$`.
  • Effective molality:`$2 \times 0.15 \, \text{m} = 0.30 \, \text{m}$`
• 0.10 m HCl: HCl dissociates into 1 H$^+$ and 1 Cl$^-$, so`$i = 2$`.
  • Effective molality: `$2 \times 0.10 \, \text{m} = 0.20 \, \text{m}$`
• 0.10 m C$_{12}$H$_{22}$O$_{11}$ (sucrose): Sucrose is a non-electrolyte, so `$i = 1$`.
  • Effective molality: `$1 \times 0.10 \, \text{m} = 0.10 \, \text{m}$`

Effective molalities in increasing order:

• 0.10 m C$_{12}$H$_{22}$O$_{11}$ (0.10 m)
• 0.050 m CaCl$_2$ (0.150 m)
• 0.10 m HCl (0.20 m)
• 0.15 m NaCl (0.30 m)

Since a higher effective molality leads to a greaterfreezing point depression (lower freezing point), the order of freezing points from highest to lowest will be the reverse of the effective molality order:

Order of Freezing Points (highest to lowest):

1. 0.10 m C$_{12}$H$_{22}$O$_{11}$ (highest freezing point)
2. 0.050 m CaCl$_2$
3. 0.10 m HCl
4. 0.15 m NaCl (lowest freezing point)

Summary of Colligative Properties

Colligative properties (vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure) are dependent on the number of solute particles, not their chemical nature.

• These properties can be determined experimentally.
• They can be used to determine the molar mass of a solute.
• In medicine, freezing point depression is often used to characterize biological samples.
• The equations apply to nonvolatile, nonelectrolytic dilute solutions. For electrolytic solutions, the van'tHoff factor (`$i$`) must be included to account for ion dissociation.

The importance of colligative properties is evident in daily phenomena, such as using salt to melt ice, antifreeze in car radiators, and understanding cellular responses in different solutions.

Colligative Properties of Dilute Solutions

Colligative properties are properties ofsolutions that depend on the number of solute particles in a given amountof solvent, not on the kind or identity of the solute particles. These properties are particularly observed in dilute solutions, typically with solute concentrations less than about $1 \text{ mol/L}$.

Examples of everyday phenomena related to colligative properties include:

• Adding salt toroads in winter to prevent ice formation.
• Adding antifreeze (like ethylene glycol) to car radiators to prevent water from freezing.
• Red blood cells bursting when placed in pure water due due to osmosis.
• The need for precisely concentrated salt solutions (e.g., intravenous injections) in medicine.

1. Vapor Pressure and Vapor Pressure Lowering

Vapor pressure is the pressure exerted bya vapor in dynamic equilibrium with its liquid phase at a given temperature. The symbol for vapor pressure is $p$, and its common units are Pa, kPa, atm, torr, or mmHg.

• $1 \text{ atm} = 101.3 \text{ kPa}= 1.013 \times 10^5 \text{ Pa}$
• $1 \text{ atm} = 760 \text{ torr} = 760 \text{ mmHg}$

A volatile liquid (e.g., ethanol, ether, acetone) has a high vapor pressure, meaning it vaporizes easily. A nonvolatile liquid has a low vapor pressure and vaporizes with difficulty.

1.1 Vapor Pressure Lowering

When a nonvolatile solute (e.g., sugar) isdissolved in a pure solvent (e.g., water), the vapor pressure of the resulting solution ($p$) will be less than that of the pure solvent ($p^*$). This phenomenon is called vapor pressure lowering.

At the same temperature, nonvolatile solute particles occupy part ofthe surface of the solvent, affecting the escaping tendency of the solvent molecules. Thus, the vapor pressure of the solution is reduced.

According to Raoult's Law, for a nonvolatile, non-ionizing dilute solution:

• The vapor pressureof the solution is proportional to the mole fraction of the solvent: $p = p^* x_A$
• The term $x_A$ represents the mole fraction of the solvent, which is less than 1, hence $p < p^*$.

The vapor pressure lowering ($\Delta p$) is defined as the difference between the vapor pressure of the pure solvent and that of the solution:

$\Delta p = p^* - p$

Substituting Raoult's Law: $\Delta p = p^* -(p^* x_A) = p^*(1 - x_A)$.

Since $x_A + x_B = 1$, where $x_B$ is the mole fraction of the solute, then $1 - x_A = x_B$.

Therefore, $\Delta p = p^* x_B$

In dilute solutions where the number of moles of solvent ($n_A$) is much greater than the number of moles of solute ($n_B$) ($n_A \gg n_B$), the molefraction of solute $x_B$ can be approximated as:

$x_B \approx \frac{n_B}{n_A} = \frac{m \cdot M_A}{1000}$

Where $m$ is the molality of the solute, and $M_A$ is the molar mass of the solvent. This leads to $\Delta p = p^* \frac{m \cdot M_A}{1000} = k \cdot m$, where $k = p^* \frac{M_A}{1000}$ is aconstant specific to the solvent.

2. Boiling Point Elevation

2.1 Boiling Point Definition

The boiling point of a liquid is the temperature at which its vapor pressure equals the external pressure. When the external pressure is $101.3 \text{ kPa}$ ($1 \text{ atm}$), this is known as the normal boiling point. The boiling point changes with external pressure.

2.2 Boiling Point Elevation Phenomenon

The boiling point of a solution is higher than that of the pure solvent.This phenomenon is called boiling point elevation ($\Delta T_b$).

The boiling point elevation is directly proportional to the molality ($m$) of the solute:

$\Delta T_b = K_b \cdot m$

Where:

• $\Delta T_b$: Boiling point elevation of the solution.
• $m$: Molality of the solute (moles of solute per kg of solvent).
• $K_b$: The molal boiling point elevation constant (ebullioscopic constant).

Characteristics of $K_b$:

• It is a characteristic physical property of the solvent, independent of the nature of the solute.
• It can be determined experimentally.
• Its unit is $\text{K} \cdot \text{kg} \cdot \text{mol}^{-1}$ or \text{^\circ C} \cdot \text{kg} \cdot \text{mol}^{-1}.

Example $K_b$ values forsome solvents:

3. Freezing Point Depression

3.1 Freezing Point Definition

The freezing point of a substance is the temperature at which its solid and liquid phases can coexist in equilibrium. The melting pointof a solid and the freezing point of its liquid are identical. For water, the freezing point is 0 \text{^\circ C}.

3.2 Freezing Point Depression Phenomenon

The freezing point of a solution is lower than that of thepure solvent. This phenomenon is called freezing point depression ($\Delta T_f$).

The freezing point depression is directly proportional to the molality ($m$) of the solute:

$\Delta T_f = K_f \cdot m$

Where:

• $\Delta T_f$: Freezing point depression of the solution.
• $m$: Molality of the solute.
• $K_f$: The molal freezing point depression constant (cryoscopic constant).

Characteristics of $K_f$:

• It is a characteristic physical property of the solvent, independent of the nature of the solute.
• It can be measured experimentally.
• Its unit is $\text{K} \cdot \text{kg} \cdot \text{mol}^{-1}$ or \text{^\circ C} \cdot \text{kg} \cdot \text{mol}^{-1}.

Example $K_f$ values for some solvents:

3.3 Example Calculation

Problem: A substance extracted from urine (90 mg) is dissolved in 12 g of distilled water. The freezing point of the solution is $0.233 \text{ K}$ lower than that of pure water. Calculate therelative molecular mass of the substance. ($K_f$ for water is $1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}$)

Solution:

1. Given:$\Delta T_f = 0.233 \text{ K}$, mass of solute = $0.090 \text{ g}$, mass of solvent = $0.012 \text{ kg}$, $K_f = 1.86 \text{ K} \cdot\text{kg} \cdot \text{mol}^{-1}$.
2. Use the freezing point depression formula: $\Delta T_f = K_f \cdot m$.
3. First, calculate molality ($m$): $m = \frac{\Delta T_f}{K_f} =\frac{0.233 \text{ K}}{1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}} \approx 0.125 \text{ mol} \cdot \text{kg}^{-1}$.
4. Molality is also defined as $m = \frac{\text{moles of solute}}{\text{kg of solvent}}$.
5. Moles of solute = $m \cdot \text{kg of solvent} = 0.125 \text{ mol} \cdot \text{kg}^{-1} \cdot 0.012 \text{ kg} = 0.0015 \text{ mol}$.
6. Molar mass of solute = $\frac{\text{mass of solute}}{\text{moles of solute}} = \frac{0.090 \text{ g}}{0.0015 \text{ mol}} = 60 \text{ g} \cdot \text{mol}^{-1}$.

The relative molecular mass of the substance is approximately $60$. This corresponds to urea ($\text{CO}(\text{NH}_2)_2$).

4. Colligative Properties for Electrolytic Solutions (Van't Hoff Factor)

The colligative property equations discussed so far (vapor pressure lowering, boiling point elevation, freezing point depression) apply to nonvolatile, nonelectrolytic dilute solutions.

For electrolytic solutions, the solute dissociates into multiple ions, increasing the effective number of particles in the solution. To account for this, the van't Hoff factor ($i$) is introduced.

The van't Hoff factor ($i$) represents the number of particles(ions or molecules) that 1 mole of solute yields in solution.

• For glucose solution, where glucose is a nonelectrolyte, $i = 1$.
• For $\text{NaCl}$ solution, $\text{NaCl}$ dissociates into $\text{Na}^+$ and $\text{Cl}^-$ ions, so $i = 2$.
• For $\text{CaCl}_2$ solution, $\text{CaCl}_2$ dissociates into $\text{Ca}^{2+}$ and $2\text{Cl}^-$ ions, so $i = 3$.

The colligative property equations are modified for electrolytic solutions:

• $\Delta p = i \cdot K \cdot m$
• $\Delta T_b = i \cdot K_b \cdot m$
• $\DeltaT_f = i \cdot K_f \cdot m$

These modified equations are used for nonvolatile electrolytic solutions.

4.1 Example: Comparing Electrolytic and Nonelectrolytic Solutions

If glucose solution and $\text{NaCl}$ solution havethe same molal concentration (e.g., $0.1 \text{ mol} \cdot \text{kg}^{-1}$), they will not have the same $\Delta T_f$.

• For $0.1 \text{ mol} \cdot \text{kg}^{-1}$ glucose solution: $\Delta T_f (\text{Glucose}) = K_f \cdot m$ (since $i=1$).
• For $0.1 \text{ mol} \cdot \text{kg}^{-1}$ $\text{NaCl}$ solution: $\Delta T_f(\text{NaCl}) = 2 \cdot K_f \cdot m$ (since $\text{NaCl}$ dissociates into 2 ions, $i=2$).
• Therefore, $\Delta T_f (\text{NaCl}) = 2 \cdot \Delta T_f (\text{Glucose})$.

4.2 Ordering Freezing Points of Solutions

To list aqueous solutions in order of their expected freezing points, we must consider both molality ($m$) and the van't Hoff factor ($i$) to determine the effective particle concentration ($i \cdot m$). A larger$i \cdot m$ value results in a greater freezing point depression, meaning a lower freezing point.

Problem: List the following aqueous solutions in order of their expected freezing points (from lowest to highest): $0.050 \text{ m } \text{CaCl}_2$; $0.15 \text{ m } \text{NaCl}$; $0.10 \text{ m } \text{HCl}$; $0.10 \text{ m } \text{C}_{12}\text{H}_{22}\text{O}_{11}$ (sucrose).

Solution:

1. $0.050 \text{ m } \text{CaCl}_2$: $\text{CaCl}_2 \to \text{Ca}^{2+} + 2\text{Cl}^-$, so$i=3$. Effective concentration $= 3 \cdot 0.050 \text{ m} = 0.150 \text{ m}$.
2. $0.15 \text{ m } \text{NaCl}$: $\text{NaCl} \to \text{Na}^+ + \text{Cl}^-$, so $i=2$. Effective concentration $= 2 \cdot 0.15 \text{ m} = 0.300 \text{ m}$.
3. $0.10 \text{ m } \text{HCl}$: $\text{HCl} \to \text{H}^+ + \text{Cl}^-$, so $i=2$. Effective concentration $= 2 \cdot 0.10 \text{ m} = 0.200 \text{ m}$.
4. $0.10 \text{ m } \text{C}_{12}\text{H}_{22}\text{O}_{11}$: Sucrose is a nonelectrolyte, so $i=1$. Effective concentration $= 1 \cdot 0.10 \text{ m} = 0.100 \text{ m}$.

Ordering by effective concentration (from highest to lowest, corresponding to lowest to highest freezing point):

• $0.15 \text{ m } \text{NaCl}$ ($0.300 \text{ m}$effective) - Lowest freezing point.
• $0.10 \text{ m } \text{HCl}$ ($0.200 \text{ m}$ effective).
• $0.050 \text{ m } \text{CaCl}_2$ ($0.150 \text{ m}$ effective).
• $0.10 \text{ m } \text{C}_{12}\text{H}_{22}\text{O}_{11}$ ($0.100 \text{ m}$ effective) - Highest freezing point.

Summary of Colligative Properties

The colligative properties (vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure) are dependent on the number of solute particles, not their identity. They can be determined experimentally and are useful for determining the molar mass of solutes.

In medicine, these principles are applied, for example, using $\Delta T_f = K_f \cdot m$ to determine the molar mass ($M_B$) of substances in biological samples or to ensure correct concentrations for intravenous injections.

Colligative Properties of Dilute Solutions

Colligative properties are physicalproperties of solutions that depend solely on the number of solute particles in a givenamount of solvent, and not on the nature or identity of the solute particles. These properties are particularly observed in dilute solutions.

The main colligative properties include:

• Vapor pressure lowering
• Boiling point elevation
• Freezing point depression
• Osmotic pressure

1. Vapor Pressure Lowering

Vapor pressure is the pressure exerted by a vapor in dynamic equilibrium with its liquid phase at a given temperature. Whena nonvolatile solute is dissolved in a solvent, the vapor pressure of the resulting solution is lower than that of the pure solvent.

1.1 Understanding Vapor Pressure

• Matter exists in three states: solid, liquid, and gas. For H₂O, these are ice, water, and water vapor.
• Evaporation is the process of a liquid changing to a gaseous state below its boiling point.
• Condensation is the process of a gas changing to aliquid.
• The symbol for vapor pressure is $p$, and common units are Pa or kPa. Other units include atm, torr, and mmHg.
• Relationship among units: $1 \text{ atm} = 101.3 \text{ kPa} = 1.013 \times 10^5 \text{ Pa} = 760 \text{ torr} = 760 \text{ mmHg}$.
• Vapor pressure increases with increasing temperature.
• Volatile liquids: Easily vaporized, havinghigh vapor pressure (e.g., ethanol, ether, acetone).
• Nonvolatile liquids: Hard to vaporize, having low vapor pressure.

When a nonvolatile solute (B) like sugar is added to a pure solvent (A) like water, the solute particles occupy aportion of the solvent's surface. This reduces the number of solvent molecules escaping into the vapor phase, thus lowering the vapor pressure of the solution.

1.2 Raoult's Law

According to Raoult's Law, the vapor pressure of a solution ($p$) containinga nonvolatile solute is directly proportional to the mole fraction of the solvent ($x_A$).

$p = p^* \cdot x_A$

Where:

• $p$ is the vapor pressure of the solution.
• $p^*$ is the vapor pressure of the pure solvent.
• $x_A$ is the mole fraction of the solvent.

Since $x_A < 1$ for a solution, it follows that $p < p^*$.

1.3 Vapor Pressure Lowering ($\Delta p$)

The vapor pressure lowering ($\Delta p$) is the difference between the vapor pressure of the pure solvent and the vapor pressure of the solution.

$\Delta p = p^* - p$

Substituting Raoult's Law into this equation:

$\Delta p = p^* - (p^* \cdot x_A)$
$\Delta p = p^* (1 - x_A)$
Since $x_A + x_B = 1$, then $1 - x_A = x_B$.
$\Delta p = p^* \cdot x_B$

Where $x_B$ is the mole fraction of the solute.

Notes:

• This equation applies to nonvolatile, nonionizing dilute solutions.
• In dilute solutions, where the number of moles of solvent ($n_A$) is much greater than the number of moles of solute ($n_B$), $x_B$ can be approximated.
• For very dilute solutions, $x_B = \frac{n_B}{n_A + n_B} \approx \frac{n_B}{n_A}$.
• If $n_A$ is expressed in mass and $m$ is molality: $x_B= m \times 10^{-3} \times \text{molar mass of A}$.
• This approximation leads to $\Delta p = p^* \times m \times 10^{-3} \times \text{molar mass of A} = k \cdot m$, where $k$ is aconstant.

2. Boiling Point Elevation

2.1 Boiling Point Definition

The boiling point of a liquid is the temperature at which its vapor pressure equals the external pressure.

• When vapor pressure equals external pressure, theliquid begins to boil.
• The normal boiling point is the boiling point at a pressure of 1.00 atm (101.3 kPa). For pure water, this is 100°C.
• The boiling point changes with external pressure.

2.2 Boiling Point Elevation ($\Delta T_b$)

The boiling point elevation is the phenomenon where the boiling point of a solution is higher than that of the pure solvent, due to the presence of a nonvolatile solute. A lowered vapor pressure means a higher temperature is requiredto reach the external pressure, hence an elevated boiling point.

$\Delta T_b = K_b \cdot m$

Where:

• $\Delta T_b$ is the boiling point elevation.
• $m$ is the molality of the solute (moles of solute per kg of solvent).
• $K_b$ is the molal boiling point elevation constant (also called the ebullioscopic constant).

Characteristics of $K_b$:

• It is a characteristic physical property of the solvent, independent of the nature of the solute.
• Can be measured experimentally.
• Unit: K·kg·mol^-1 (or °C·kg·mol^-1).

Example $K_b$ values for some solvents:

3. Freezing Point Depression

3.1 Freezing Point Definition

The freezing point is the temperature at which a solid and liquid phase of a substancecan coexist in equilibrium.

• The melting point of a solid and the freezing point of its liquid are identical.
• At the freezing point, solid and liquid coexist in equilibrium.

3.2 Freezing Point Depression ($\Delta T_f$)

Thefreezing point depression is the phenomenon where the freezing point of a solution is lower than that of the pure solvent. The presence of solute particles interferes with the formation of the ordered solid structure of the solvent, requiring a lower temperature for freezing to occur.

$\Delta T_f =K_f \cdot m$

Where:

• $\Delta T_f$ is the freezing point depression.
• $m$ is the molality of the solute.
• $K_f$ is the molal freezing point depression constant (also called the cryoscopic constant).

Characteristics of $K_f$:

• It is a characteristic physical property of the solvent, independent of the nature of the solute.
• Can be measured experimentally.
• Unit: K·kg·mol^-1 (or °C·kg·mol^-1).

Example $K_f$ values for some solvents: (Refer to the table in section 2.2)

3.3 Applications and Examples

• Road de-icing: Salt (e.g., NaCl) is spread on roads in winter to lower the freezing point of water, preventing ice formation.
• Antifreeze: Substances like ethylene glycol are added to automobile radiators to preventwater from freezing in cold temperatures.
• Molar mass determination: Colligative properties can be used to determine the molar mass of an unknown solute.

Example Calculation:
A substance is extracted from urine. 90 mg of this substance is dissolved in 12 g of distilled water. The freezing point of the solution is lower than that of pure water by 0.233 K. Calculate the relative molecular mass of the substance.

• Given: $\Delta T_f = 0.233 \text{ K}$, massof solute = 0.090 g, mass of solvent = 0.012 kg.
• For water, $K_f = 1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}$.
• Weknow $\Delta T_f = K_f \cdot m$.
• $m = \frac{\Delta T_f}{K_f} = \frac{0.233 \text{ K}}{1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}} = 0.125 \text{ mol} \cdot \text{kg}^{-1}$.
• Molality ($m$) is moles of solute / kg of solvent.
• Moles of solute = $m \times \text{mass of solvent (kg)} = 0.125 \text{ mol} \cdot \text{kg}^{-1} \times 0.012 \text{ kg} = 0.0015 \text{ mol}$.
• Molar mass = mass of solute / moles of solute = $\frac{0.090 \text{ g}}{0.0015 \text{ mol}} = 60 \text{ g} \cdot \text{mol}^{-1}$.
• The substance is urea (CO(NH₂)₂), which has a molar mass of approximately 60 g/mol.

4. Van't Hoff Factor ($i$) for Electrolytes

The equations for colligative properties (vapor pressure lowering, boiling point elevation, freezing point depression) discussed so far apply to nonvolatile, nonelectrolytic dilute solutions. Forelectrolytic solutions, the solute dissociates into multiple ions in the solvent, increasing the number of particles.

• For glucose solution vs. NaCl solution with the same concentration (e.g., 0.1 mol·kg^-1):
  • Glucose does not dissociate,so 0.1 mol of glucose yields 0.1 mol of particles. $\Delta T_f (\text{Glucose}) = K_f \cdot m$.
  • NaCl dissociates into Na⁺ and Cl^- ions, so 0.1 mol of NaCl yields 0.2 mol of particles. $\Delta T_f (\text{NaCl}) = 2 \cdot K_f \cdot m$.
• The van't Hoff factor (i) accounts for the number of particles produced per formula unit of solute.
• For NaCl, $i \approx 2$ (1 Na⁺ + 1 Cl^-).
• For CaCl₂, $i \approx 3$ (1 Ca²⁺ + 2 Cl^-).

The modifiedequations for electrolytic solutions are:

$\Delta p = i \cdot K' \cdot m$ (where $K'$ is a general constant related to $p^*$)
$\Delta T_b = i \cdot K_b \cdotm$
$\Delta T_f = i \cdot K_f \cdot m$

These equations apply to electrolytic solutions, and the colligative properties depend on the effective number of particles (moles $\times i$).

4.1 Ordering Solutions by Freezing Point

To order solutions by their expected freezing points, calculate the effective molality ($i \cdot m$) for each solution. A higher effective molality means a greater freezing point depression, hence a lower freezing point.

Example: List the following aqueous solutions in order of their expected freezing points (from lowest to highest): 0.050 m CaCl₂; 0.15 m NaCl; 0.10 m HCl; 0.10 m C₁₂H₂₂O₁₁ (sucrose).

• 0.050 m CaCl₂: CaCl₂ dissociates into 3 ions ($i=3$). Effective molality = $3 \times 0.050 \text{ m} = 0.150 \text{ m}$.
• 0.15 m NaCl: NaCl dissociates into 2 ions ($i=2$). Effective molality = $2 \times 0.15 \text{ m} = 0.30 \text{ m}$.
• 0.10 m HCl: HCl dissociates into 2 ions ($i=2$). Effective molality = $2 \times 0.10 \text{ m} = 0.20 \text{ m}$.
• 0.10 m C₁₂H₂₂O₁₁: Sucrose is a nonelectrolyte ($i=1$). Effective molality = $1 \times 0.10 \text{ m} = 0.10 \text{ m}$.

Ordering by effective molality (highest to lowest,which corresponds to lowest freezing point to highest freezing point):

1. 0.15 m NaCl (0.30 m effective) - Lowest freezing point.
2. 0.10 m HCl (0.20 m effective).
3. 0.050 m CaCl₂ (0.150 m effective).
4. 0.10 m C₁₂H₂₂O₁₁ (0.10 m effective) - Highest freezing point.

Finalorder (lowest freezing point first): 0.15 m NaCl, 0.10 m HCl, 0.050 m CaCl₂, 0.10 m C₁₂H₂₂O₁₁.

Summary and KeyTakeaways

• Colligative properties depend on the number of solute particles, not their identity.
• The four main colligative properties are vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.
• Vapor Pressure Lowering: $\Delta p = p^* \cdot x_B$. The presence of a nonvolatile solute lowers the vapor pressure of a solvent.
• Boiling Point Elevation: $\Delta T_b = K_b \cdot m$. A lower vapor pressure leads to a higher boiling point for the solution.
• Freezing Point Depression: $\Delta T_f = K_f \cdot m$. Solute particles disrupt crystal formation, leading to a lower freezing point.
• The constants $K_b$ and $K_f$ are characteristic of the solvent.
• For electrolyticsolutions, the van't Hoff factor ($i$) is used to account for particle dissociation: $\Delta X = i \cdot K \cdot m$.
• Colligative properties have practical applications in determining molar masses and in various daily life and medical contexts (e.g., antifreeze, de-icing, intravenous solutions).

Colligative Properties of Dilute Solutions

Colligative properties are physicalproperties of solutions that depend solely on the number of solute particles in a givenamount of solvent, not on the type or identity of the solute. These properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

These phenomena are particularly important in medicine and daily life, affecting everything from intravenous injections to antifreeze in car radiators and the bursting of red blood cells in purewater.

What are Dilute Solutions?

Dilute solutions are those in which the concentration of the solute is relatively low, typically less than 1 mol/L. Many biological fluids, such as blood, gastric juice, saliva, tears, and urine, are dilute solutions.

Solutions can be broadly categorized into:

• Nonelectrolytic solutions: Solutes like sugar (C₁₂H₂₂O₁₁) dissolve without dissociating into ions.

• Electrolytic solutions: Solutes like NaCl or CaCl₂ dissociate into ions when dissolved in a solvent. For example, 1 mole of NaCl yields 2 moles of particles (Na and Cl), and 1 mole of CaCl₂ yields 3 moles of particles (Ca and 2Cl).

1. Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in dynamic equilibrium withits liquid phase at a given temperature. This phenomenon is critical to understanding how solutes alter the properties of a solvent.

1.1 Evaporation and Condensation

• Evaporation (Vaporization): The process by which a liquid changes to a gaseous state. It occurs below the boiling point.

• Condensation: The process by which a gas changes to a liquid state.

In a closed system, at equilibrium, the rate of evaporation equals the rate of condensation, and the pressure exerted by the gas phase is the vapor pressure.

1.2 Characteristicsof Vapor Pressure

• Symbol and Units:

  The symbol for vapor pressure is .

  Common units are Pa (Pascal) or kPa (kilopascal).

  Other units include atm, torr, and mmHg:

  • 1 atm = 101.3 kPa = 1.013 × 10 Pa

  • 1 atm = 760 torr = 760 mmHg

• Temperature Dependence: Vapor pressure increases with increasing temperature.

• Substance Dependence: Different liquids have different vapor pressures at the same temperature.

• Volatility:

  • Volatile liquids: Easily vaporized, high vapor pressure (e.g., ethanol, ether, acetone).

  • Nonvolatile liquids: Hard to vaporize, low vapor pressure.

2. Vapor Pressure Lowering ()

When a nonvolatile solute is dissolved in a solvent, the vaporpressure of the resulting solution () is lower than that of the pure solvent () at the same temperature. This is because the solute particles occupy part of the solvent's surface, reducing the escaping tendency of solvent molecules into the gas phase.

This phenomenon is called vaporpressure lowering ().

According to Raoult's Law, for an ideal solution:

Where:

• is the vapor pressure of the solution.

• is the vapor pressure of the pure solvent.

• is the mole fraction of the solvent.

Since , it follows that .

2.1 Calculating Vapor Pressure Lowering

The vapor pressure lowering () is defined as:

Substituting Raoult's Law ():

Since (where is the mole fraction of the solute):

The mole fraction of solute () in a dilute solution () can be approximated as:

Where is the molality of the solute, is molar mass of the solute in g/mol and ismolar mass of the solvent and is the mass of solvent in g.

For dilute solutions, vapor pressure lowering can also be expressed in terms of molality:

Where is a constant related to , , and molar mass of A. For dilute solutions: .

Note: Raoult'slaw applies to nonvolatile, nonionizing dilute solutions.

3. Boiling Point Elevation ()

The boiling point of a liquid is the temperature at which its vapor pressure equals the external pressure. When a nonvolatile solute is added to asolvent, the vapor pressure of the solution is lowered. To reach the boiling point, a higher temperature is required for the solution to achieve the external pressure, resulting in an elevated boiling point.

3.1 Boiling Point Definition

• The temperature at which a liquid's vapor pressure equals the external pressure.

• At an external pressure of 101.3 kPa (1 atm), pure water boils at 100°C. This is the normal boiling point.

• Boiling points change with external pressure variations.

3.2 Boiling Point Elevation

The boiling point elevation () is the difference between the boiling point of the solution () and the boiling point of the pure solvent ().

For dilute, nonvolatile, nonelectrolytic solutions, the boiling point elevation is directly proportional to the molality () of the solute:

Where:

• : The boiling point elevation of the solution (in K or °C).

• : The molality of the solute (in mol/kg).

• : The molal boiling point elevation constant (ebullioscopic constant).

Characteristics of :

• It is a characteristic physical property of the solvent, independent of the nature of thesolute.

• It can be measured experimentally.

• Its unit is K·kg·mol (or °C·kg·mol).

Example values for some solvents:

4. Freezing Point Depression ()

The freezing point of a substance is the temperature at which its solid and liquid phases coexist inequilibrium. Similar to boiling point elevation, adding a nonvolatile solute to a solvent lowers the freezing point of the mixture.

4.1 Freezing Point Definition

• The temperature at which a solid and liquid phase can coexist at equilibrium.

• The melting point of a solid and the freezing pointof its liquid are identical.

• For pure water, the freezing point is 0°C.

4.2 Freezing Point Depression

The freezing point depression () is the difference between thefreezing point of the pure solvent () and the freezing point of the solution ().

For dilute, nonvolatile, nonelectrolytic solutions, the freezing point depression is directly proportional to the molality () of the solute:

Where:

• : The freezing point depression of the solution (in K or °C).

• : The molality of the solute (in mol/kg).

• : The molal freezing point depression constant (cryoscopic constant).

Characteristics of :

• It is a characteristic physical property of the solvent, independent of the nature of the solute.

• It can be measured experimentally.

• Its unit is K·kg·mol (or °C·kg·mol).

Example values for some solvents:

4.3 Application: Determining Molar Mass

Colligative properties can be used to determine the molar mass of an unknown nonvolatile solute.

Example: Extract one substance from urine. Dissolve 90 mg (0.090 g) of this extraction into12 g (0.012 kg) of distilled water. The freezing point of the solution is lower than that of pure water by 0.233 K ( K). Calculate the molar mass of the substance. ( for water= 1.86 K·kg·mol)

We know .

So, .

Molality () = .

Moles of solute = .

Molar mass = .

The molar mass of the substance is approximately 60 g/mol, which corresponds to urea (CO(NH)).

5. Van't Hoff Factor () for Electrolytes

The equationsfor colligative properties (vapor pressure lowering, boiling point elevation, and freezing point depression) discussed so far apply to nonvolatile, nonelectrolytic dilute solutions. For electrolytic solutions, the solute dissociates into ions, increasing the effective number of particles in the solution.

5.1 Electrolytic Solutions

If glucose solution and NaCl solution have the same molal concentration (e.g., 0.1 mol·kg), they will not have the same .

• Glucose solution: Glucose is a nonelectrolyte, so 1 mole of glucose yields 1 mole of particles. ()

• NaCl solution: NaCl is an electrolyte, dissociating into Na and Cl ions. So, 1 mole of NaCl yields approximately 2 moles ofparticles. ()

To account for the dissociation of electrolytes, the van't Hoff factor () is introduced.

represents the number of particles (ions or molecules) that 1 mole of a solute produces in solution.

• For nonelectrolytes (e.g., glucose), .

• For strong electrolytes:

  • NaCl: (Na, Cl)

  • CaCl: (Ca, 2Cl)

5.2 Modified Colligative Property Equations for Electrolytes

When dealing with electrolytic solutions, the colligative property equations are modified by multiplying by the van't Hoff factor :

These equations apply to nonvolatile electrolytic solutions.

5.3 Ordering Freezing Points of Solutions

To order solutions by their expectedfreezing points, we calculate for each solution:

Example: List the following aqueous solutions in order of their expected freezing points (from lowest to highest): 0.050 m CaCl; 0.15 m NaCl;0.10 m HCl; 0.10 m CHO (sucrose).

• Sucrose (CHO): Nonelectrolyte, .

• HCl: Strong electrolyte ().

• CaCl: Strong electrolyte ().

• NaCl: Strong electrolyte ().

The greater the value, the greater the freezing point depression (), and thus the lower the freezing point.

Ordering from lowest freezing point (highest ) to highest freezing point (lowest ):

1. 0.15 m NaCl () Lowest freezing point

2. 0.10 m HCl ()

3. 0.050 m CaCl ()

4. 0.10 m CHO () Highest freezing point

Key Takeaways

• Colligative properties depend on the number of solute particles, not their identity.

• Adding a nonvolatile solute lowers the vapor pressure, elevates the boiling point, and depresses the freezing point of a solvent.

• Raoult's lawquantifies vapor pressure lowering for ideal, dilute solutions.

• The extent of boiling point elevation () and freezing point depression () is directly proportional to the molality () of the solute, with constants and being characteristic of the solvent.

• For electrolytic solutions, the van't Hoff factor () accounts for the dissociation of solutes into multiple particles, modifying the colligative property equations.

• These properties are crucial for understanding biological systems and practical applications like antifreeze.

Colligative Properties of Dilute Solutions

Colligative properties are physicalproperties of solutions that depend solely on the number of solute particles dissolved in a solvent, not ontheir identity. These properties are particularly relevant for dilute solutions, typically defined as solutions where the concentration of solute is less than about 1 mol/L.

Examples of everyday phenomena related to colligative properties include:

• Adding salt to roads in winter to prevent iceformation.
• Adding antifreeze to car radiators to prevent water from freezing.
• The bursting of red blood cells when placed in pure water due to osmotic pressure.
• Medical preparations like intravenous injections requiring preciseconcentration of salt solutions.

1. Vapor Pressure and Vapor Pressure Lowering

Matter can exist in three states: solid, liquid, and gas. Vaporization is the process bywhich a liquid changes to a gaseous state. When this occurs below the boiling point, it's called evaporation. The reverse process, changing a gas to a liquid, is called condensation.

1.1. Vapor Pressure

Definition: At a given temperature, the pressure exerted by a vapor in dynamic equilibrium with its liquid is called the vapor pressure.

• Symbol: $p$
• Units: Pascals (Pa) or kilopascals (kPa). Other units include atmospheres (atm), torr, and mmHg.

  Unit Relationships:
  $1 \text{ atm} = 101.3 \text{ kPa} = 1.013 \times 10^5 \text{ Pa}$
  $1 \text{ atm} = 760 \text{ torr} = 760 \text{ mmHg}$

• Vapor pressure increases with increasing temperature.
• Volatile liquids: Liquids that are easily vaporized andhave high vapor pressure (e.g., ethanol, ether, acetone).
• Nonvolatile liquids: Liquids that are hard to vaporize and have low vapor pressure.

1.2. Vapor Pressure Lowering

When a nonvolatile solute (like sugar) is dissolvedin a pure solvent (like pure water), the vapor pressure of the resulting solution is lower than that of the pure solvent at the same temperature.

• The nonvolatile solute particles occupy part of the surface of the solvent, affecting the escaping tendency of solvent molecules.
• Therefore, $p < p^*$, where $p$ is the vapor pressure of the solution and $p^*$ is the vapor pressure of the pure solvent. This phenomenon is called vapor pressure lowering.

1.3. Raoult's Law

Raoult's Law describes the relationship between the vapor pressure of a solution and the mole fraction of its components.

For a solution containing a nonvolatile solute, the vapor pressure of the solution ($p$) is directly proportional to the mole fraction of the solvent ($x_A$).

$p = p^* \times x_A$

Where:

• $p$ is the vapor pressure of the solution.
• $p^*$ is the vapor pressure of the pure solvent.
• $x_A$ isthe mole fraction of the solvent ($x_A = n_A / (n_A + n_B)$).

The vapor pressure lowering ($\Delta p$) is given by:

$\Delta p = p^* - p$

Substituting Raoult's Law into the equation for vapor pressure lowering:

$\Delta p = p^* - (p^* x_A)$
Since $x_A + x_B = 1$, then $x_A = 1 - x_B$.
$\Delta p = p^* - p^*(1 - x_B) \implies \Delta p = p^* - p^* + p^* x_B$
$\Delta p = p^* x_B$

Where:

• $\Delta p$is the vapor pressure lowering.
• $x_B$ is the mole fraction of the solute.

Note: Raoult's Law generally applies to nonvolatile, nonionizing dilute solutions.

For dilute solutions where$n_A \gg n_B$ (mole of solvent is much greater than mole of solute), the mole fraction of solute $x_B$ can be related to molality ($m$). We can derive a simplified form:

$x_B = m \times 10^{-3} \times \text{Molar Mass of } A$

So, $\Delta p = p^* \times m \times 10^{-3} \times \text{Molar Mass of } A$. This can be simplified to $\Delta p = k \times m$, where $k = p^* \times 10^{-3} \times \text{Molar Mass of } A$ is a constant.

2. Boiling Point Elevation

2.1. Boiling Point

Definition: The boiling point of a liquid is the temperature atwhich its vapor pressure equals the external pressure.

• When the vapor pressure of a liquid equals the external pressure, the liquid begins to boil.
• The boiling point changes with external pressure.
• Normal boiling point: The boiling point at an external pressure of 1.00 atm (101.3 kPa). For pure water, the normal boiling point is $100^\circ\text{C}$.

Vapor Pressure of Water at Different Temperatures:

2.2. Boiling Point Elevation

Definition: Theboiling point elevation is the phenomenon where the boiling point of a solution is higher than that of its pure solvent.

• This occurs because the presence of a nonvolatile solute lowers the vapor pressure of the solution, requiring a higher temperature to reach the external pressure for boiling.

The boiling point elevation ($\Delta T_b$) is directly proportional to the molality ($m$) of the solute:

$\Delta T_b = K_b \times m$

Where:

• $\Delta T_b$is the boiling point elevation.
• $m$ is the molality of the solute (moles of solute per kilogram of solvent).
• $K_b$ is the molal boiling point elevation constant (also called the ebullioscopic constant).

Characteristics of $K_b$:

• It is a characteristic physical property of the solvent, independent of the nature of the solute.
• It can be measured experimentally.
• Its unit is $\text{K} \cdot \text{kg} \cdot \text{mol}^{-1}$ or \text{^\circ C} \cdot \text{kg} \cdot \text{mol}^{-1}.

$K_b$ Values for Some Solvents:

3. Freezing Point Depression

3.1. Freezing Point

Definition: The freezing point isthe temperature at which a solid and liquid phase can coexist at equilibrium. The melting point of a solid and the freezing point of its liquid are identical.

For example, for water, $\text{H}_2\text{O(s)} \iff \text{H}_2\text{O(l)}$ at $0^\circ\text{C}$ at standard pressure. The freezing point is also the temperature at which the vapor pressure of the solid and liquid are equal.

Vapor Pressure of Water (Solid and Liquid) at Different Temperatures:

3.2. Freezing Point Depression

Definition: The freezing point depression is the phenomenon where thefreezing point of a solution is lower than that of its pure solvent.

• This occurs because the presence of a solute lowers the vapor pressure of the liquid solvent, making it necessary to lower the temperature further to reach the freezing point where the solid and liquid phases can coexist.

The freezing point depression ($\Delta T_f$) is directly proportional to the molality ($m$) of the solute:

$\Delta T_f = K_f \times m$

Where:

• $\Delta T_f$is the freezing point depression.
• $m$ is the molality of the solute.
• $K_f$ is the molal freezing point depression constant (also called the cryoscopic constant).

Characteristics of $K_f$:

• It is a characteristic physical property of the solvent, independent of the nature of the solute.
• It can be measured experimentally.
• Its unit is $\text{K} \cdot \text{kg} \cdot \text{mol}^{-1}$ or \text{^\circ C} \cdot \text{kg} \cdot \text{mol}^{-1}.

Example: Calculation of Molar Mass Using Freezing Point Depression

Extract onesubstance from urine. Dissolve 90 mg this extraction into 12 g distilled water. The freezing point of the solution is lower than that of pure water by 0.233 K. Calculate the relative molecular mass of the substance. ($K_f$ for water = $1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}$)

Given: $\Delta T_f = 0.233 \text{ K}$, mass of solute = 0.090 g, mass of solvent = 0.012 kg, $K_f = 1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}$.

From $\Delta T_f = K_f \times m$, we have $m =\frac{\Delta T_f}{K_f}$.
$m = \frac{0.233 \text{ K}}{1.86 \text{ K} \cdot \text{kg} \cdot \text{mol}^{-1}} = 0.125 \text{ mol} \cdot \text{kg}^{-1}$
Molality ($m$) is moles of solute / kg of solvent.
Moles of solute = $m \times \text{mass of solvent (kg)}$
Moles of solute = $0.125 \text{ mol} \cdot \text{kg}^{-1} \times 0.012 \text{ kg} = 0.0015 \text{ mol}$
Molar mass = $\frac{\text{mass of solute}}{\text{moles of solute}} = \frac{0.090 \text{ g}}{0.0015 \text{ mol}} = 60 \text{ g} \cdot \text{mol}^{-1}$
The substance is urea ($\text{CO(NH}_2)_2$), which has a molar mass of $60.06 \text{ g} \cdot \text{mol}^{-1}$.

4. Osmotic Pressure

This section is mentioned in the outline but details are not provided in the source material. However, it is a key colligative property.

5. Colligative Properties for Electrolytic Solutions (van't Hoff Factor)

The equations for colligative properties ($ \Delta p $, $ \Delta T_b $, $ \Delta T_f $) apply to <mark>nonvolatile, nonelectrolytic dilute solutions</mark>.</p> <p>However, for electrolytic solutions (which dissociate into ions in solution), the number of particles in solution is greater than the number of moles of solute added. For example, 1 mole of NaCl yields 2 moles of particles ($\text{Na}^+$ and $\text{Cl}^-$), and 1 mole of $\text{CaCl}_2$ yields 3 moles of particles ($\text{Ca}^{2+}$ and $2\text{Cl}^-$).

To account for this, the van't Hoff factor ($i$) is introduced, which represents the number of particles obtainedper formula unit of solute.

For non-electrolytic solutions: $i = 1$
For electrolytic solutions: $i > 1$

The modified equations for colligative properties in electrolytic solutions are:

• Vapor pressure lowering: $\Delta p = i \cdot K \cdot m$ (where $K$ is a constant related to $p^*$)
• Boiling point elevation: $\Delta T_b = i \cdot K_b \cdot m$
• Freezing point depression: $\Delta T_f = i \cdot K_f \cdot m$

Example: Glucose solution vs. NaCl solution

If a $0.1 \text{ mol} \cdot \text{kg}^{-1}$ glucose solution and a $0.1 \text{ mol} \cdot \text{kg}^{-1}$ NaCl solution are compared, will they have the same $\Delta T_f$?

For glucose ($\text{C}_{12}\text{H}_{22}\text{O}_{11}$), it is a non-electrolyte, so $i = 1$.
$\Delta T_f (\text{Glucose}) = K_f \cdot m$

For NaCl, it is an electrolyte thatdissociates into $\text{Na}^+$ and $\text{Cl}^-$, so $i \approx 2$ (assuming complete dissociation).
$\Delta T_f (\text{NaCl}) = 2 \cdot K_f \cdot m$

Therefore, the NaCl solution will have a freezing point depression approximately twice that of the glucose solution, meaning it will freeze at a lower temperature.

Ordering Solutions by Freezing Point:

List the following aqueous solutions in order of their expected freezing points (from lowest to highest):
$0.050 \text{ m CaCl}_2$; $0.15 \text{ m NaCl}$; $0.10 \text{ m HCl}$; $0.10 \text{ m C}_{12}\text{H}_{22}\text{O}_{11}$ (sucrose).

To determine the freezing point, we need to calculate $i \cdot m$ for each solution. The greater this value, the lower the freezing point (greater $\Delta T_f$).

• $0.050 \text{ m CaCl}_2$: $\text{CaCl}_2 \to \text{Ca}^{2+} + 2\text{Cl}^-$, so $i = 3$. $i \cdot m = 3 \times 0.050 = 0.150$
• $0.15 \text{ m NaCl}$: $\text{NaCl} \to \text{Na}^+ + \text{Cl}^-$, so $i = 2$. $i \cdot m = 2 \times 0.15 = 0.300$
• $0.10 \text{ m HCl}$: $\text{HCl} \to \text{H}^+ + \text{Cl}^-$, so $i = 2$. $i \cdot m = 2 \times 0.10 = 0.200$
• $0.10 \text{ m C}_{12}\text{H}_{22}\text{O}_{11}$: Sucrose is a non-electrolyte, so $i = 1$. $i \cdot m = 1 \times 0.10 =0.100$

Ordering by $i \cdot m$ from largest to smallest (which corresponds to lowest to highest freezing point):
$0.15 \text{ m NaCl } (i \cdot m = 0.300) <0.10 \text{ m HCl } (i \cdot m = 0.200) < 0.050 \text{ m CaCl}_2 (i \cdot m = 0.150) < 0.10 \text{ m C}_{12}\text{H}_{22}\text{O}_{11} (i \cdot m = 0.100)$

Expected freezing points (lowest to highest):
$0.15 \text{ m NaCl} < 0.10 \text{m HCl} < 0.050 \text{ m CaCl}_2 < 0.10 \text{ m C}_{12}\text{H}_{22}\text{O}_{11}$

Summary of Colligative Properties

The four main colligative properties are:

• Vapor pressure lowering ($\Delta p$)
• Boiling point elevation ($\Delta T_b$)
• Freezing point depression ($\Delta T_f$)
• Osmotic pressure ($\Pi$) (Not detailed in provided text)

These properties are significant in various applications, particularly in medicine and industrial processes. They allow for the determination of the molar mass of an unknown solute and are crucial for understanding biological systems, as evident in thepreparation of biological samples and intravenous solutions.

Colligative Properties of Dilute Solutions

Colligative properties are physicalproperties of solutions that depend directly on the number of solute particles present, rather than onthe nature or identity of the solute particles themselves. These properties are observed in dilute solutions, typically with solute concentrations less than about 1 mol/L.

The main colligative properties include:

• Vapor Pressure Lowering ($\\Delta p$)
• Boiling Point Elevation ($\\Delta T_b$)
• Freezing Point Depression ($\\Delta T_f$)
• Osmotic Pressure (not covered in this note)

1. Vapor Pressure Lowering

Vapor pressure is the pressure exerted by a vapor in dynamic equilibrium with its liquid phase at a given temperature. When a nonvolatile solute (like sugar) is dissolved in a puresolvent (like water), it occupies part of the solvent's surface, hindering the solvent molecules from escaping into the vapor phase. This leads to a reduction in the solution's vapor pressure compared to the pure solvent.

Definition and Raoult's Law

The vapor pressure of a solution ($p$) will be less than that of the pure solvent ($p^*$). This phenomenon is called vapor pressure lowering.

For a nonvolatile solute and a non-ionizing dilute solution, Raoult's Law states:

$p = p^* \times x_A$

Where:

• $p$: vapor pressure of the solution
• $p^*$: vapor pressure of the pure solvent
• $x_A$: mole fraction of the solvent

Since $x_A < 1$,it follows that $p < p^*$.

Vapor Pressure Lowering ($\\Delta p$)

The vapor pressure lowering, $$\\Delta p$$ , is defined as the difference between the vapor pressure of the pure solvent and the solution:

$\\Delta p = p^*- p$