Colligative Properties And Determination Of Molar Mass

Colligative Properties And Determination Of Molar Mass

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Colligative Properties: Colligative properties are properties of solutions that depend only on the number of solute particles dissolved in a given amount of solvent, and not on the chemical identity of the solute or solvent. In ideal solutions, these properties are directly proportional to the mole fraction of the solute.

The four major colligative properties are:

1. Relative lowering of vapor pressure
2. Elevation of boiling point
3. Depression of freezing point
4. Osmotic pressure

These properties are particularly useful for determining the molar mass of non-volatile solutes.

Relative Lowering Of Vapour Pressure

Definition: When a non-volatile solute is dissolved in a solvent, the vapor pressure of the solvent above the solution is lower than that of the pure solvent at the same temperature. The lowering of vapor pressure ($\Delta P$) is the difference between the vapor pressure of the pure solvent ($P^0$) and the vapor pressure of the solution ($P_s$): $\Delta P = P^0 - P_s$. The relative lowering of vapor pressure is the ratio of the lowering of vapor pressure to the vapor pressure of the pure solvent: $\frac{\Delta P}{P^0} = \frac{P^0 - P_s}{P^0}$.

Raoult's Law: For a solution containing a non-volatile solute, the vapor pressure of the solution ($P_s$) is equal to the mole fraction of the solvent ($\chi_{solvent}$) multiplied by the vapor pressure of the pure solvent ($P^0$).

$$P_s = \chi_{solvent} \times P^0$$

The lowering of vapor pressure is then:

$$\Delta P = P^0 - P_s = P^0 - (\chi_{solvent} \times P^0) = P^0 (1 - \chi_{solvent})$$

Since the sum of mole fractions is 1 ($\chi_{solvent} + \chi_{solute} = 1$), we have $1 - \chi_{solvent} = \chi_{solute}$.

Therefore, the relative lowering of vapor pressure is equal to the mole fraction of the solute:

$$\frac{\Delta P}{P^0} = \frac{P^0 - P_s}{P^0} = \chi_{solute}$$

Determining Molar Mass:

The mole fraction of the solute ($\chi_{solute}$) is defined as:

$$\chi_{solute} = \frac{n_{solute}}{n_{solvent} + n_{solute}}$$

Where $n_{solute}$ is the number of moles of solute and $n_{solvent}$ is the number of moles of solvent.

If the solution is dilute (i.e., the amount of solute is much smaller than the amount of solvent), we can approximate $n_{solvent} + n_{solute} \approx n_{solvent}$.

$$\chi_{solute} \approx \frac{n_{solute}}{n_{solvent}}$$

Substituting this into Raoult's Law:

$$\frac{P^0 - P_s}{P^0} \approx \frac{n_{solute}}{n_{solvent}}$$

Let $w_1$ be the mass of the solute and $M_1$ be its molar mass. Let $w_2$ be the mass of the solvent and $M_2$ be its molar mass.

$n_{solute} = \frac{w_1}{M_1}$ and $n_{solvent} = \frac{w_2}{M_2}$

So, the equation becomes:

$$\frac{P^0 - P_s}{P^0} = \frac{w_1/M_1}{w_2/M_2}$$

Rearranging to find the molar mass of the solute ($M_1$):

$$M_1 = \frac{w_1}{w_2} \times \frac{M_2}{\frac{P^0 - P_s}{P^0}}$$

This equation allows us to determine the molar mass of a non-volatile solute if we know its mass, the mass of the solvent, the molar mass of the solvent, and the vapor pressures of the pure solvent and the solution.

Elevation Of Boiling Point

Definition: The boiling point elevation ($\Delta T_b$) is the increase in the boiling point of a solvent when a non-volatile solute is added. The boiling point of the solution is higher than that of the pure solvent.

Relationship: The elevation of the boiling point is directly proportional to the molal concentration (molality) of the solute particles.

$$\Delta T_b = K_b \times m$$

Where:

• $\Delta T_b$ is the boiling point elevation (in °C or K).
• $K_b$ is the ebullioscopic constant (or molal boiling point elevation constant) of the solvent (e.g., for water, $K_b = 0.52$ °C kg/mol).
• $m$ is the molality of the solution (moles of solute per kilogram of solvent).

Determining Molar Mass:

Molality ($m$) is defined as:

$$m = \frac{n_{solute}}{w_{solvent} \text{ (in kg)}}$$

Since $n_{solute} = \frac{w_1}{M_1}$, we can write:

$$m = \frac{w_1/M_1}{w_2 \text{ (in kg)}}$$

Substituting this into the boiling point elevation equation:

$$\Delta T_b = K_b \times \frac{w_1}{M_1 \times w_2 \text{ (in kg)}}$$

Rearranging to find the molar mass of the solute ($M_1$):

$$M_1 = \frac{K_b \times w_1}{w_2 \text{ (in kg)} \times \Delta T_b}$$

This equation allows us to determine the molar mass of a non-volatile solute by measuring the boiling point elevation of its solution.

Depression Of Freezing Point

Definition: The freezing point depression ($\Delta T_f$) is the decrease in the freezing point of a solvent when a non-volatile solute is added. The freezing point of the solution is lower than that of the pure solvent.

Relationship: The depression of the freezing point is directly proportional to the molal concentration (molality) of the solute particles.

$$\Delta T_f = K_f \times m$$

Where:

• $\Delta T_f$ is the freezing point depression (in °C or K).
• $K_f$ is the cryoscopic constant (or molal freezing point depression constant) of the solvent (e.g., for water, $K_f = 1.86$ °C kg/mol).
• $m$ is the molality of the solution.

Determining Molar Mass:

Similar to boiling point elevation, using the definition of molality ($m = \frac{w_1/M_1}{w_2 \text{ (in kg)}}$):

$$\Delta T_f = K_f \times \frac{w_1}{M_1 \times w_2 \text{ (in kg)}}$$

Rearranging to find the molar mass of the solute ($M_1$):

$$M_1 = \frac{K_f \times w_1}{w_2 \text{ (in kg)} \times \Delta T_f}$$

This equation allows us to determine the molar mass of a non-volatile solute by measuring the freezing point depression of its solution.

Osmosis And Osmotic Pressure

Osmosis: Osmosis is the spontaneous net movement of solvent molecules through a semipermeable membrane from a region of higher solvent concentration (lower solute concentration) to a region of lower solvent concentration (higher solute concentration). A semipermeable membrane allows the passage of solvent molecules but not solute particles.

Osmotic Pressure ($\pi$): Osmotic pressure is the minimum pressure that needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It can be thought of as the pressure difference required to stop osmosis.

Van't Hoff Equation: For dilute solutions, the osmotic pressure is given by the Van't Hoff equation, which is analogous to the ideal gas law:

$$\pi = i \times M \times R \times T$$

Where:

• $\pi$ is the osmotic pressure (in atm or Pa).
• $i$ is the Van't Hoff factor, which represents the number of particles the solute dissociates into in solution (for non-electrolytes like glucose, $i=1$; for electrolytes, $i$ is the number of ions formed per formula unit, e.g., for NaCl, $i=2$).
• $M$ is the molar concentration (molarity) of the solution (mol/L).
• $R$ is the ideal gas constant (0.0821 L atm/mol K or 8.314 J/mol K).
• $T$ is the absolute temperature (in Kelvin).

Determining Molar Mass:

Molarity ($M$) is defined as:

$$M = \frac{n_{solute}}{V_{solution} \text{ (in L)}}$$

Since $n_{solute} = \frac{w_1}{M_1}$, we can write:

$$M = \frac{w_1/M_1}{V_{solution} \text{ (in L)}}$$

Substituting this into the Van't Hoff equation:

$$\pi = i \times \frac{w_1}{M_1 \times V_{solution} \text{ (in L)}} \times R \times T$$

Rearranging to find the molar mass of the solute ($M_1$):

$$M_1 = \frac{i \times w_1 \times R \times T}{V_{solution} \text{ (in L)} \times \pi}$$

Osmotic pressure is often used to determine the molar mass of macromolecules (like polymers and proteins) because it can be measured accurately at relatively low concentrations, and the solute particles are often too large to be easily handled by other methods.

Example: 3.0 g of a non-electrolyte solute was dissolved in 500 mL of water. The osmotic pressure of the solution was found to be 0.30 atm at 27°C. Calculate the molar mass of the solute.

Answer:

Abnormal Molar Masses

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Abnormal Molar Masses: Colligative properties are based on the assumption that the solute exists as discrete particles (molecules or ions) and does not undergo any association (combining to form larger molecules) or dissociation (breaking apart into ions) in the solution. When a solute undergoes association or dissociation, the actual number of solute particles in the solution deviates from the number of moles of solute added. This leads to observed colligative property values that differ from those predicted, and consequently, the calculated molar mass using the standard formulas appears "abnormal" (either higher or lower than the true molar mass).

Causes of Abnormal Molar Masses:

1. Association of Solute Molecules: Some solutes, particularly organic molecules, can associate in solution to form larger molecules. This reduces the total number of particles in the solution compared to the number of moles of solute added.

Example: Carboxylic acids like benzoic acid ($C_6H_5COOH$) can form hydrogen-bonded dimers in non-polar solvents.

2 $C_6H_5COOH \rightarrow (C_6H_5COOH)_2$ (dimer)

If $n$ moles of solute associate to form $n/2$ moles of dimer, the number of particles is halved. This leads to a smaller observed colligative property (e.g., less boiling point elevation, less freezing point depression, higher osmotic pressure if calculated naively from molar mass) and a calculated molar mass that appears twice its actual value.

2. Dissociation of Solute Molecules: Electrolytes (acids, bases, salts) dissociate in polar solvents (like water) into ions. This increases the total number of particles in the solution compared to the number of moles of solute added.

Example: Sodium chloride (NaCl) dissociates in water into $Na^+$ and $Cl^-$ ions.

$NaCl(s) \xrightarrow{H_2O} Na^+(aq) + Cl^-(aq)$

If 1 mole of NaCl dissociates completely, it yields 2 moles of ions ($Na^+$ and $Cl^-$). The number of particles is doubled. This leads to observed colligative properties that are approximately twice what would be expected for a non-electrolyte, and the calculated molar mass would be approximately half its true value.

Van't Hoff Factor (i): To account for association or dissociation, the Van't Hoff factor ($i$) is introduced into the colligative property equations. The factor $i$ is the ratio of the observed colligative property to the colligative property calculated on the assumption of no association or dissociation.

For colligative properties related to moles:

• Relative Lowering of Vapor Pressure: $\frac{P^0 - P_s}{P^0} = i \times \chi_{solute}$
• Boiling Point Elevation: $\Delta T_b = i \times K_b \times m$
• Freezing Point Depression: $\Delta T_f = i \times K_f \times m$
• Osmotic Pressure: $\pi = i \times M \times R \times T$

Calculating Molar Mass with Van't Hoff Factor:

The equations for determining molar mass ($M_1$) are modified by including the Van't Hoff factor:

• From Boiling Point Elevation: $M_1 = \frac{i \times K_b \times w_1}{w_2 \text{ (in kg)} \times \Delta T_b}$
• From Freezing Point Depression: $M_1 = \frac{i \times K_f \times w_1}{w_2 \text{ (in kg)} \times \Delta T_f}$
• From Osmotic Pressure: $M_1 = \frac{i \times w_1 \times R \times T}{V_{solution} \text{ (in L)} \times \pi}$

Determining the Van't Hoff Factor:

For dissociation:

If a solute dissociates into $n$ ions per formula unit, and the degree of dissociation is $\alpha$ (fraction of molecules that dissociate), then:

$$i = 1 + \alpha(n - 1)$$

For association:

If $n$ molecules associate to form one larger molecule (e.g., $n$ monomers form one polymer), and the degree of association is $\alpha$ (fraction of molecules that associate), then:

$$i = 1 + \alpha(\frac{1}{n} - 1)$$

In practice, $i$ is often determined experimentally by measuring a colligative property and then calculating $i$ using the modified equations, or the degree of dissociation/association ($\alpha$) is determined from $i$. This allows for a more accurate determination of the true molar mass.

Example: A 0.1 molal solution of potassium chloride (KCl) in water freezes at -0.372°C. Calculate the molar mass of KCl and its Van't Hoff factor ($i$). (Kf for water = 1.86 °C kg/mol)

Answer: