## Colligative Properties

The colligative properties can be defined as the properties of solutions which is wholly determined by the ratio of the number of solute particles and the number of solvent molecules in a particular solution, and are completely independent of the nature of the chemical species present. The number ratio can be calculated by using various units that determine the concentration of solutions. The general assumption is that in the case of an ideal solution the properties are independent of the nature of solute particles present in it and are somewhat approximate for dilute real solutions. In other words, colligative properties can be considered as a set of properties of the solution that can be reasonably approached if we follow the assumption that the solution is ideal.

Here we have considered only those properties which are formed from the dissolution of a non-volatile solute in a volatile liquid solvent. Essentially the solvent properties are changed due to the presence of the solute. The solute particles actually displace some solvent molecules in the liquid phase and thus result in the reduction of the concentration of solvent. Thus we may conclude that the colligative properties are independent of the nature of the solute. The word colligative has been derived from the Latin colligates which means bound together.

### Colligative Properties of a Solution Include:

Relative lowering of vapor pressure

Elevation of boiling point

Depression of freezing point

Osmotic pressure

For a particular mass ratio of solute and solvent, all colligative properties should be inversely proportional to solute molar mass.

Relative molar masses can be determined by measuring colligative properties for a dilute solution of a non-ionized solute. Such solutions include urea or glucose in water or another solvent. This is applicable for both small molecules and for polymers. In an alternate manner, the percentage of dissociation taking place can be estimated by measuring ionized solutes.

Colligative properties are mostly applicable for dilute solutions as their behavior may often be approximated. This is because they are ideal solutions.

This unit will focus mainly on the relative lowering of vapor pressure.

### Relative Lowering of Vapour Pressure

On dissolving the non-volatile solute in a pure solvent, the vapour pressure of a pure solvent gets gradually decreased.

Let us assume that p is the vapour pressure of the solvent and ps is the vapour pressure of the solution, then the lowering of vapour pressure can be written as (p – ps). This lowering of vapour pressure relative to the vapour pressure of the pure solvent is called the Relative lowering of Vapour pressure. Thus,

Relative Lowering Vapour Pressure = \[\frac{p - p_s}{p}\]

After extensive experimentation, Raoult (1886) gave an empirical relation to establishing the connection between the relative lowering of vapor pressure and the concentration of the solute in a solution. This is now referred to as Raoult's Law. As per this law, the relative lowering of the vapor pressure of a dilute solution is equal to the mole fraction of the solute present in dilute solution.

### Mathematically Raoult’s Law can be Expressed in the Form:

where n = number of moles or molecules of solute

\[\frac{p - p_s}{p} = \frac{n}{n + N} \]

### Derivation of Raoult's Law

The vapor pressure of the pure solvent is the result caused by the number of molecules evaporating from its surface. When a non-volatile solute is dissolved in solution, due to the presence of solute molecules in the surface a fraction of the surface gets blocked and here no evaporation can take place.

By Lowering of vapor pressure by a non-volatile solute the particles of the solute prohibit the escape of solvent molecules from the surface of the solution. This finally results in the lowering of the vapor pressure. The vapor pressure of the solution is, therefore, dependent on the number of molecules of the solvent found at any time in the surface which is again proportional to the mole fraction. That is,

\[p_s \alpha \frac{n}{n + N}\]

where N = moles of solvent and n = moles of solute.

Or we can write it in the form

\[p_s = K \frac{N}{n + N}\]

k being a proportionality factor.

In case of pure solvent n = 0 and hence Mole fraction of solvent

\[p_s = \frac{N}{n + N} = \frac{N}{0 + N} = 1\]

Now from equation (1), the vapor pressure p = k Therefore the equation (1) assumes the form:

\[p_s = p \frac{N}{n + N}\]

\[\frac{p_s}{p} = \frac{N}{n + N}\]

\[1 - \frac{p_s}{p} = 1- \frac{N}{n + N}\]

\[\frac{p - p_s}{p} = \frac{n}{n + N}\]

This is Raoult’s law.

### Ideal Solutions and Deviations From Raoult’s Law

A solution which strictly follows Raoult's law strictly is called an ideal solution. A solution which shows even slight deviations from Raoult’s law is called a non-ideal or Real solution. Let us consider that the molecules of the solvent and solute are represented by A and B respectively. Now let γAB be the representation of the attractive force acting between A and B, and γAA between A and A. If

\[\gamma_{AB} = \gamma_{AA}\]

The solution will have the same vapor pressure as predicted by Raoult's law and it is an ideal solution. However, if

\[\gamma_{AB} > \gamma_{AA}\]

molecule A will escape comparatively less readily and the vapor pressure will then be less than that of the predicted one, which is calculated by obeying Raoult’s law (Such deviation is called Negative deviation). On the other hand, if

\[\gamma_{AB} < \gamma_{AA}\]

A molecule will escape from the solution surface more rapidly and then the vapor pressure of the solution will become higher than predicted by Raoult’s law (Such deviation is then called the Positive deviation). If we consider very dilute solutions of nonelectrolytes, the solvent and solute molecules are very much similar be it in terms of molecular size or be its molecular attractions. Thus, under such situations, the given solutions have the tendency to approach the ideal behavior and obey Raoult's law more or less accurately.

### Determination of Molecular Mass From Vapour Pressure Lowering

The molecular mass of a non-volatile solute can be calculated from the measurement of the lowering of vapor pressure (p – ps ) produced by dissolving a known weight of it in a known weight of the solvent. It is considered that w grams of solute is dissolved in W grams of the solvent, and let m and M are molecular masses of the solute and solvent respectively, we have :

No of Moles of solute (n) = \[\frac{w}{m}\]

No of Moles of solvent (N) = \[\frac{W}{M}\]

Substituting these values in Raoult’s law Equation

\[\frac{p - p_s}{p} = \frac{n}{n + N}\]

\[\frac{p - p_s}{p} = \frac{\frac{w}{m}}{\frac{w}{m} + \frac{W}{M} }\]

Considering an extremely diluted solution, the number of moles (molecules) of solute (w/m), is very very small, it can be then neglected in the denominator. The equation (1) can then be written as

\[\frac{p - p_s}{p} = \frac{wM}{mW} \]

With the known experimental value of \[\frac{p - p_s}{p} \], and the molecular mass of the solvent (M), the molecular weight of solute (m) can be calculated from the above equations.

### Measurement of Lowering of Vapour Pressure

Barometric Method:

The individual vapor pressure of a liquid was calculated by Raoult and then the same process was followed to calculate the vapor pressure of the solution as well. He poured the liquid or the solution into the Torricellian vacuum of a barometer tube and calculated the depression of the mercury level. This method was later found to be neither practicable nor accurate as the lowering of vapor pressure is almost negligible.

Manometric Method:

The vapor pressure of a liquid or solution can be fairly measured with the help of a manometer. Let us assume a bulb is charged with the liquid or solution. The air in the connecting tube of the instrument is then removed with a vacuum pump. With the stopcock being closed, the pressure inside is only due to the vapor evaporating from the solution or liquid. This method can be applied to aqueous solutions. The manometric liquid used can be either mercury or n-butyl phthalate which has low density and low volatility.

### Ostwald and Walker’s Dynamic Method (Gas Saturation Method):

In this method, the relative lowering of vapor pressure can be calculated in an easy simple Procedure. The apparatus used by Ostwald and Walker consist of two sets of bulbs : (a) Set A contains the solution (b) Set B contains the solvent. The weight of each set is calculated separately. A slow stream of dry air is then removed by a suction pump through the two sets of bulbs. At the end of the operation, the weight of these sets is again measured. From the weight loss in each of the two sets, the lowering of vapor pressure is measured. But here the temperature of the air, the solution, and the solvent must be kept constant all throughout. As the air bubbles through set A reaches saturation up to the vapor pressure ps of the solution and then up to vapor pressure p of solvent in set B, the amount of solvent taken up in set A becomes proportional to ps and the amount taken up in set B becomes proportional to (p – ps ).

Ostwald-Walker method of measuring the relative lowering of vapor pressure

If w1 and w2 ne the loss of weight in set A and B respectively,

\[w_1\] \[\alpha\] \[p_s\]

\[w_2\] \[\alpha\] \[p - p_s\]

Adding (1) and (2), we have

\[w_1 + w_2\] \[\alpha\] \[p_s + p - p_s\]

\[\alpha\] \[p_s + p - p_s\]

Dividing (2) by (3), we can write

\[\frac{p - p_2}{p} = \frac{w_2}{w_1 + w_2}\]

Knowing the loss of mass in set B (w2) and the net loss of mass in the two sets (w1 + w2), we can find the relative lowering of vapor pressure. If we use water as the solvent, a set of calcium chloride tubes (or a set of bulbs containing conc. H2SO4) is linked to the end of the apparatus to capture the escaping water vapor. Therefore, the gain in mass of the CaCl2-tubes will be equal to (w1 + w2), the total loss of mass in sets A and B.

A colligative property is a property of an answer that is reliant upon the proportion between the absolute number of solute particles (in the answer for) the complete number of dissolvable particles. Colligative properties are not reliant upon the compound idea of the arrangement's parts. Subsequently, colligative properties can be connected to a few amounts that express the convergence of an answer, like molarity, ordinariness, and molality. The four colligative properties that can be displayed by an answer are:

Limit rise

Edge of freezing over gloom

Relative bringing down of fume pressure Osmotic strain

"Colligative" has been adjusted or taken from the Latin word "colligatus" which means "bound together". With regards to characterizing an answer, colligative properties assist us with seeing how the properties of the arrangement are connected to the grouping of solute in the arrangement.

### What are Colligative Properties?

Weaken arrangements containing non-unpredictable solutes display a few properties which rely just upon the quantity of solute particles present and not on the kind of solute present. These properties are called colligative properties. These properties are for the most part seen in weakened arrangements.

We can additionally consider colligative properties as those properties that are acquired by the disintegration of a non-unpredictable solute in an unstable dissolvable. By and large, the dissolvable properties are changed by the solute where its particles eliminate a portion of the dissolvable atoms in the fluid stage. This likewise brings about the decrease of the convergence of the dissolvable.

In the meantime, when we talk about the given solute-dissolvable mass proportion, colligative properties are supposed to be conversely corresponding to the solute molar mass.

### Colligative Properties Examples

We can notice the colligative properties of arrangements by going through the accompanying models. Assuming we add a touch of salt to a glass brimming with water its frigid temperature is brought down extensively than the typical temperature. On the other hand, its bubbling temperature is additionally expanded and the arrangement will have a lower fume pressure. There are changes in its osmotic tension too.

Likewise, assuming we add liquor to water, the arrangement's edge of freezing over goes down underneath the ordinary temperature that is noticed for either unadulterated water or liquor.

### Various Types of Colligative Properties of Solution

There are various kinds of colligative properties of an answer. These incorporate, fume pressure bringing down, edge of boiling over height, edge of freezing over gloom and osmotic strain.

In an unadulterated dissolvable, the whole surface is involved by the atoms of the dissolvable. Assuming a non-unstable solute is added to the dissolvable, the surface presently has both solute and dissolvable particles; along these lines part of the surface covered by dissolvable atoms gets diminished. Since the fume strain of the arrangement is exclusively because of dissolvable alone, at a similar temperature the fume tension of the arrangement is viewed as lower than that of the unadulterated dissolvable.

### 1. Bringing Down of Vapor Pressure

On the off chance that P0 is the fume strain of unadulterated dissolvable and Ps is the fume tension of the arrangement. The distinction Po – Ps is named as bringing down in fume pressure. The proportion (Po – Ps)/Po is known as the overall bringing down of fume pressure.

Raoult, in 1886, set up a connection between relatives bringing down in fume tension and mole division. The relationship is known as Raoult's law. It expresses that the overall bringing down in fume strain of a weaken arrangement is equivalent to the mole part of the solute present in the arrangement.

On the off chance that n moles of solute is broken down in N moles of the dissolvable, then, at that point, as per Raoult's law

\[ \frac{P_o – P_s}{Po} = \frac{n}{n + N}\]

### 2. Height in Boiling Point

The limit of a fluid is the temperature at which the fume pressure is equivalent to climatic tension. We realize that on the expansion of a non-unpredictable fluid to an unadulterated dissolvable, the fume tension of an answer declines. Consequently to make fume pressure equivalent to barometrical tension we need to expand the temperature of the arrangement. The distinction in the limit of the arrangement and the edge of boiling over of the unadulterated dissolvable is named as height in edge of boiling over.

In the event that T0b is the limit of the unadulterated dissolvable and Tb is the edge of boiling over of the arrangement then rise in edge of boiling over is given as

∆Tb = T0b - Tb

Trial results show that there is a connection between rise in edge of boiling over and molality 'm' of the solute present in arrangement

∆Tb ∝ m

∆Tb = kb m

Where,

kb = molal rise steady

Subbing the worth of 'm' in the above connection we get

∆Tb = \[\frac{1000 \times kb \times m_2}{M_2 \times m_1}\]

Where,

m2 = mass of dissolvable in g

M1 = mass of dissolvable in kg

M2 = molar mass of solute

### 3. Sadness in Freezing Point

The edge of freezing over a substance is characterized as the temperature at which the fume tension of its fluid is equivalent to the fume of the comparing strength. As indicated by Raoult's law when a non-unpredictable strong is added to the dissolvable its fume pressure diminishes and presently it would become equivalent to that of strong dissolvable at a lower temperature. The contrast between the edge of freezing over of the unadulterated dissolvable and its answer is called wretchedness in edge of freezing over.

In the event that T0f is the limit of the unadulterated dissolvable and Tf is the edge of boiling over of the arrangement then wretchedness in edge of freezing over is given as

∆Tf = T0f - Tf

Very much like height at the edge of boiling over, sadness at the edge of freezing over is likewise straightforwardly connected with the morality 'm'.

∆Tf = \[\frac{1000 \times kf \times m_2}{M_2 \times m_1}\]

Where,

k f = molal gloom consistent

m2 = mass of dissolvable in g

M1 = mass of dissolvable in kg

M2 = molar mass of solute

### 4. Osmotic Pressure

At the point when a semipermeable layer is put between an answer and dissolvable, it is seen that dissolvable particles enter the arrangement through the semipermeable film and the volume of the arrangement increases. The semi-porous film permits just dissolvable atoms to go through it however forestalls the section of greater particles like solute. This peculiarity of the unconstrained progression of dissolvable atoms through a semipermeable film from an unadulterated dissolvable to an answer or from a weakened to a concentrated arrangement is called assimilation.

The progression of dissolvable particles through the semipermeable layer can be halted on the off chance that some additional strain is applied from the arrangement side. This strain that simply stops the progression of dissolvable is called osmotic tension of the arrangement.

Osmotic strain is a colligative property as it relies upon the quantity of solute present and not on the idea of the solute. Tentatively it was demonstrated that osmotic tension (⫪) is straightforwardly relative to molarity(C) and temperature(T).

Numerically, ℼ = CRT where R is the gas consistent.

\[\Rightarrow \pi = (\frac{n_2 RT}{V})\]

Here, V is the volume of arrangement in liters and n2 is moles of solute

Assuming m2 is the heaviness of solute and M2 molar mass of solute, then, at that point, \[n_2 = \frac{m_2}{M_2}\]

\[\pi = \frac{W_2 RT}{M_2V} \]

Accordingly by knowing the upsides of ℼ,w2, T and V we can work out the molar mass of the solute.

### Various Solutions

Isotonic arrangement: Two arrangements having a similar osmotic tension at a given temperature are known as an isotonic arrangement. At the point when such arrangements are isolated by a semi-porous film then there is no assimilation.

Hypotonic arrangement: A hypotonic arrangement has a lower osmotic strain than that of the encompassing i.e, the convergence of solute particles is not exactly that of the encompassing. On the off chance that the hypotonic arrangement is isolated by a semipermeable layer then, at that point, water moves out of the hypotonic arrangement.

Hypertonic arrangement: A hypertonic arrangement has a higher osmotic strain than that of the encompassing i.e, the centralization of solute particles is more than that of the encompassing. Assuming the hypertonic arrangement is isolated by semipermeable film then, at that point, water moves inside the hypertonic arrangement.