Colligative Properties

From MyMCAT

1 Introduction
2 Molality
3 Vapor Pressure Lowering
4 Boiling Point Elevation
5 Freezing Point Depression
6 Osmotic Pressure

[edit] Introduction

Colligative properties are properties of solutions that depend on the number of particles in a given volume of solvent and not on the mass of the particles. Colligative properties include: lowering of vapor pressure; elevation of boiling point; depression of freezing point; and osmotic pressure. These topics often come up as discrete questions (questions independent of a passage) or as pseudo-discrete questions (questions that only require simple information from a passage, such as a concentration of a specific solution.)

[edit] Molality

As explained above, the following properties described do not depend on the volume of the solution, but instead on the actual number free particles. Thus concentration, or molarity, which is measured in moles/L is not useful as we are not concerned with the volume of solution!

Molality (mol/kg, molal, or simply m) denotes the number of moles of a given substance per kilogram of solvent (not solution). For instance: adding 1.0 moles of dissolved particles to 2.0 kilograms of solvent constitutes a solution with a molality of 0.5 mol/kg. Such a solution may be described as "0.5 molal".

Note that molality is sometimes represented by the symbol (m), while molarity by the symbol (M). The two symbols should not be confused! The units of M (Molarity) are mol/L while those of m (molality) are mol/kg.

$M \mbox{ (molarity) } = \frac{\mbox{ moles of solute }}{\mbox{ volume of solution }}$ $m \mbox{ (molality) } = \frac{\mbox{ moles of solute }}{\mbox{ kilograms of solvent }}$

Be sure to understand the difference between the two terms. Molarity is a ratio of moles solute to total volume, whereas molality is a ratio of moles solute to kilograms solvent. (Notice how molality does not involve volumes at all, instead it compares solute to solvent, which is what colligative properties involve.)

[edit] Vapor Pressure Lowering

Consider a closed container filled with pure water. Initially particles of liquid water will escape (evaporate) in the container, after some time however, some of these particles will return to the liquid phase below. This process will proceed until an equilibrium is established and there are as many molecules returning to the liquid as there are escaping. At this point the vapor is said to be saturated, and the pressure of that vapor is called the vapor pressure.

Raoult's law states that for a solution which is a mixture, the vapor pressure of the solution is equal to the sum of the vapor pressures for their pure components multiplied by how much of the solution is taken up by that component (ie the mole ratio).

$\ P_{solution}= (P_{1})_{pure} x_1 + (P_{2})_{pure} x_2 + \cdots$

and the individual vapor pressure for each component is

$\ P_{i}=(P_{i})_{pure} x_i$

where

$(P_i)_{pure}\,$ is the vapor pressure of the pure component

$x_i\,$ is the mole fraction of the component in solution (moles i / total # moles in solution)

Consequently, as the number of components in a solution increases, the individual vapor pressures decrease, since the mole fraction of each component decreases with each additional component. If a pure solute (ie a salt) which has a vapor pressure of zero (they do NOT evaporate) is dissolved in a solvent, the vapor pressure of the final solution will be lower than that of the pure solvent.

One can see then that as more of a solute is added to water, its mole fraction increases while the mole fraction of the water decreases. As a result, the partial pressure of the water component decreases and because the solute doesn't have any of its own partial pressure to contribute to the total pressure. Thus, overall, the vapor pressure for the mixture decreases...

The following diagram depicts just how Raoult's formula changes,

The left shows a pure solution, because the mole ratio is 1.0, the solution's vapor pressure is just the pressure of the pure solvent (since it is not a solution), the center shows a solution of two different molecules, the vapor pressure then is the combination of the pure vapor pressures of each molecule multiplied by their mole fraction. The far right solution however contains a dissolved ion which does not have a vapor pressure since it does not evaporate and thus the final solution's vapor pressure is reduced since the mole fraction in the multiplication is now less than 1.

[edit] Boiling Point Elevation

By definition, boiling occurs when a solution's vapor pressure reaches the pressure of the atmosphere around it. As a glass of water increases in temperature, its vapor pressure increases slightly because more particles are being shifted to gas, once the water reaches 100 degrees celcius however, the vapor pressure of the water will equal the atmospheric pressure and additional energy added to the system will rapidly start convertings particles in the liquid phase to the gas phase.

Note however, that the boiling point for a liquid can change depending on the surrounding pressure. The boiling point of a liquid is lowered if the pressure of the surrounding gases is decreased. For example, water will boil at a lower temperature at the top of a mountain, where the atmospheric pressure on the water is less, than it will at sea level, where the pressure is greater. In the laboratory, liquids can be made to boil at temperatures far below their normal boiling points by heating them in vacuum flasks under greatly reduced pressure.

Now consider what would happen if we tried to boil a flask of pure water and a flask of sea water. From our previous examination of vapor pressure, it should be clear that adding solutes to a solution lowers the solution's vapor pressure, thus the sea water would have a lower vapor pressure than the pure water. Keeping this idea in mind, now let us return to the definition of boiling. Boiling occurs when the vapor pressure of a solution matches the pressure of the atmosphere around it. At 100 degrees, pure water reaches its boiling point because its vapor pressure becomes equal to the atmospheric pressure, but what about the sea water? At 100 degrees, the sea water's vapor pressure must be less than the pure water's vapor pressure (because there are solutes in it), but if this is true, then its pressure is actually lower than the atmospheric pressure. As a result, the sea water will NOT boil at 100 degrees. For the sea water to boil, we would have to raise the temperature even more, in fact, enough to bring its lowered vapor pressure back up to atmospheric pressure.

The formula for boiling point is as follows,

$\Delta\mbox{T}_{b} = \mbox{K}_{b}\mbox{m}_{solute}i$

where K_b is the ebullioscopic constant, a constant unique to the liquid (the pure solvent), m is the molality, and i is the van 't Hoff factor, which accounts for the number of individual particles (typically ions) formed by a compound in solution (for instance, NaCl produces two molecules for everyone one NaCl molecule, thus its i factor is 2).

[edit] Freezing Point Depression

Freezing-point depression describes the phenomenon that the freezing point of a liquid (the solvent) is depressed when another compound is added, meaning that a solution has a lower freezing point than a pure solvent. The most common example of this is the addition of salt to wet, cold roads to prevent them from becoming icy when the temperature drops below below 0°C.

Why this occurs is not the simplest explanation. Let us consider what happens as something melts (the opposite of freezing). A frozen solid has very low entropy, its molecules are well ordered, nicely aligned, and energy must be added to overcome this low entropy and break the ridged crystal structure. Any solute in this frozen system actually increases the entropy because mixtures always have more entropy that pure substances, thus to stay frozen the system will have to be even colder to overcome the extra energy from the entropy. Now lets consider freezing a solution. As one slowly makes the solution colder and colder, there comes a point when the energy in the system is so low it can overcome the energy from entropy and thus allow it to freeze. But if the entropy is higher than normal because of lots of solute, we will have to lower the temperature even more to reach the phase change. This depression is the difference between the freezing point of the solvent in a solution and the freezing point of the pure solvent.

$\Delta\mbox{T}_{f} = -\mbox{K}_{f}\mbox{m}_{solute}i$

Where Kf is the cryoscopic constant, which is a constant unique to the solvent, m is molality, and i is the van 't Hoff factor, which accounts for the number of individual particles (typically ions) formed by a compound in solution (for instance, NaCl produces two molecules for everyone one NaCl molecule, thus its i factor is 2).

Like boiling point elevation, this formula gives the change from the normal temperature, and that the negative sign denotes that it is a lower temperature than the original.

+5.4C

+3.7C

-3.7C

-5.4C

→

Firstly, 90 grams of Glucose is equal to 0.5 moles (since 90grams / 180 g/mol = 0.5 moles). Secondly, 250 grams of water is equal to 0.250kg. Thirdly, glucose does not dissociate thus we can ignore the i term and assume it is equal to 1. Now we can use the formula above to solve for the freezing point depression, dT = -(1.86)(0.5/.25)(1). = -3.72 degrees. Since the freezing point of pure water is zero, our solution's freezing point will be -3.72C.

NaCl

MgCl₂

AlCl₃

KCl

→

AlCl₃ produces the most moles of dissociated ions per mole salt compared to the others, thus its i factor is 4, compared to the others (which are NaCl -> 2, MgCl2 -> 3, and KCl -> 2, respectively) thus it would reduce the freezing point the most if equal mole amounts of salt were used.

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[edit] Osmotic Pressure

Osmotic pressure is the effect of waters natural tendency to want to saturate molecules. This can be observed in cells where water generally tries to flow into them (due to the large amount of solute in them) and the cells must continuously pump water out or they will lyse. A more experimental observation of osmotic pressure however is generally carried out using a container with a semipermeable membrane which can selectively allow only certain molecules through. As an example, consider this apparatus filled with water on both sides. The membrane is designed to allow water to pass through but excludes glucose. What would happen if glucose where added to the left side only? The answer is simple, water will begin moving to the left side in an attempt to saturate (or dilute out) the glucose. There will become a point however where no more water can freely move to the left because of the increasing resistance. Often, the water will keep flowing to dilute the solution until the force of gravity pulling down on the column of this solution balances the osmotic pressure pushing the water through the membrane.

Interestingly, osmotic pressure ( $\pi$ ) follows a relationship analogous to the ideal gas equation,

$\pi\mbox{V} =n\mbox{R}\mbox{T}$

$\pi =\frac{n\mbox{R}\mbox{T}}{\mbox{V}}$

which has a term n/V, which is in essence the same as molality and thus why we consider osmotic pressure to be a colligative property (because this pressure depends on the ratio of the number of solute particles to the volume of the solution -- n/V -- not the identity of the solute particles).

1 mole of NaCl

→

NaCl will dissolve into Na⁺ and Cl^-, thus it has the effect of ~2 moles of molecules.

1 mole of H₃PO₄

→

H₃PO₄ is triprotic and those able to dissociate up to three times produce almost 3 moles of H⁺ and 1 mole of PO₄

1 mole of Glucose

→

Glucose, while it dissolves, does not dissociate into anything, thus it is equivalent to one mole of molecules.

1 mole MgCl₂

→

MgCl₂ dissolves into one mole of Mg²⁺ and 2 moles of Cl^-.

→

Overall then, H₃PO₄ produces the most moles of solute and thus will result in the strongest osmotic pressure.

left; the right column will contain more water.

left; the left column will contain more water.

right; the right column will contain more water.

right; the left column will contain more water.

→

Initially the left side has 1.5 moles of solute while the right side has 1 mole of solute, thus water will be driven to the side with a higher solute concentration, or the left side. After some time however, Sucrase, the enzyme that catalyses the break down of sucrose into glucose and fructose will have converted the 1 mole on the right into 2 moles (1 mole glucose, 1 mole fructose). Thus, the right side will then contain more solute and the water will push to fill this side instead.

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