Question

List four properties of a solution that depend on the total concentration but not the type of particle or particles present as solute. Write the mathematical expression that describes how each of these properties depends on concentration.

Short answer

The four colligative properties that depend on the total concentration but not on the type of particles present as solute are: 1. Boiling Point Elevation: \(ΔT_b = K_b · m\) 2. Freezing Point Depression: \(ΔT_f = K_f · m\) 3. Osmotic Pressure: \(π = c · R · T\) 4. Vapor Pressure Lowering: \(ΔP = P^*_A - P\) (Raoult's law: \(P = X_A · P^*_A\))

Step-by-step solution

1. Boiling Point Elevation

Boiling Point Elevation is a colligative property that states that adding a non-volatile solute to a pure solvent will raise its boiling point. This effect depends on the concentration of solute particles. The relation between boiling point elevation and the concentration of solute is given by: \(ΔT_b = K_b · m\) Where: - \(ΔT_b\) is the boiling point elevation, - \(K_b\) is the ebullioscopic constant of the solvent, - \(m\) is the molality of the solute.

2. Freezing Point Depression

Freezing Point Depression is another colligative property that occurs when a non-volatile solute is added to a pure solvent, lowering its freezing point. The effect is related to the concentration of solute particles. The mathematical expression for freezing point depression is: \(ΔT_f = K_f · m\) Where: - \(ΔT_f\) is the freezing point depression, - \(K_f\) is the cryoscopic constant of the solvent, - \(m\) is the molality of the solute.

3. Osmotic Pressure

Osmotic pressure is the pressure required to prevent the flow of solvent across a semi-permeable membrane that separates solutions of different concentrations. It is a colligative property that depends on the total concentration of solute particles in the solution. The mathematical expression for osmotic pressure is given by: \(π = c · R · T\) Where: - \(π\) is the osmotic pressure, - \(c\) is the molar concentration of the solute, - \(R\) is the universal gas constant, - \(T\) is the absolute temperature.

4. Vapor Pressure Lowering

Vapor Pressure Lowering occurs when a non-volatile solute is added to a pure solvent, decreasing its vapor pressure. This effect is related to the concentration of solute particles. The mathematical expression for vapor pressure lowering is given by Raoult's law: \(P = X_A · P^*_A\) Where: - \(P\) is the vapor pressure of the solution, - \(X_A\) is the mole fraction of the solvent in the solution, - \(P^*_A\) is the vapor pressure of the pure solvent. Remember that Vapor Pressure Lowering is represented by: \(ΔP = P^*_A - P\)

Key concepts

Boiling Point Elevation

Understanding boiling point elevation is crucial when studying solutions and their behaviors. It's a colligative property, which means it depends on the number of solute particles in a solvent, not their identity. When a non-volatile solute is dissolved in a solvent, it disrupts the natural boiling process. This results in a need for higher temperature to provide the energy necessary to allow solvent molecules to escape into the gas phase, thus the boiling point increases.
In everyday language, you can imagine it as if the solute particles are 'holding onto' the solvent particles, making it harder for them to vaporize. The equation that describes this phenomenon is simple and elegant:
\[\[\begin{align*} \DeltaT_b &= K_b \times m ewline \DeltaT_b &= \text{boiling point elevation} ewline K_b &= \text{ebullioscopic constant (specific to each solvent)} ewline m &= \text{molality of the solute (moles of solute per kg of solvent)} ewline \end{align*}\]\]For instance, if you're cooking at high altitudes, your pastas will take longer to cook because water boils at a higher temperature due to lower atmospheric pressure, not due to boiling point elevation—but the principle by which the temperature changes is quite similar!

Freezing Point Depression

The concept of freezing point depression is complementary to boiling point elevation but in the opposite direction, temperature-wise. Adding a non-volatile solute to a solvent decreases the solvent's freezing point. This is because the solute particles disrupt the formation of the orderly solid structure, thereby requiring a lower temperature to achieve the same organizational structure necessary for the solid phase.
In simpler terms, the solute gets in the way of the solvent molecules that are trying to form ice. The relationship governing this effect is captured by the following expression:
\[\[\begin{align*} \DeltaT_f &= K_f \times m ewline \DeltaT_f &= \text{freezing point depression} ewline K_f &= \text{cryoscopic constant (unique to each solvent)} ewline m &= \text{molality of the solute} ewline \end{align*}\]\]This is the reason why salt is strewn on icy roads in winter; the salt lowers the freezing point of water, causing ice to melt even when the ambient temperature is below water's normal freezing point.

Osmotic Pressure

Osmotic pressure might sound complex, but it's a concept we can see in action in biological systems, such as when plants absorb water from the soil. It occurs due to the natural tendency of a solvent to move through a semi-permeable membrane from an area of low solute concentration to one of high concentration to achieve equilibrium. The osmotic pressure is the force that must be applied to prevent this flow.
It's a bit like a crowd pushing against a gate; the osmotic pressure is the force needed to keep the gate closed. The mathematical relationship for osmotic pressure is given by:
\[\[\begin{align*} \pi &= c \times R \times T ewline \pi &= \text{osmotic pressure} ewline c &= \text{molar concentration of the solute (moles per liter of solution)} ewline R &= \text{universal gas constant} ewline T &= \text{absolute temperature (in Kelvin)} ewline \end{align*}\]\]Osmotic pressure is vital for processes like desalination of seawater and even in medical treatments involving dialysis.

Vapor Pressure Lowering

When approaching the topic of vapor pressure lowering, it's useful to think about what happens when you add salt to water. The vapor pressure of the solution is lower than that of the pure solvent because the salt particles occupy space at the surface of the liquid, hindering evaporation of the solvent. A lower vapor pressure means that fewer solvent molecules can escape into the gas phase at a given temperature.
The formula expressing this relationship stems from Raoult's law, which assumes ideal behavior in solutions:
\[\[\begin{align*} \Delta P &= P^*_A - P ewline P &= X_A \times P^*_A ewline \Delta P &= \text{vapor pressure lowering} ewline P^*_A &= \text{vapor pressure of the pure solvent} ewline P &= \text{vapor pressure of the solution} ewline X_A &= \text{mole fraction of the solvent in the solution} ewline \end{align*}\]\]This property is significant in industries that rely on evaporation techniques, such as in the production of concentrated fruit juices or in the paint industry, where solvents are designed to evaporate at different rates depending on the application.