## Unit 2 Solutions

In normal life we rarely come across pure substances. Most of these are mixtures containing two or more pure substances. Their utility or importance in life depends on their composition. For example, the properties of brass (mixture of copper and zinc) are quite different from those of German silver (mixture of copper, zinc and nickel) or bronze (mixture of copper and tin); 1 part per million $(\mathrm{ppm})$ of fluoride ions in water prevents tooth decay, while $1.5 \mathrm{ppm}$ causes the tooth to become mottled and high concentrations of fluoride ions can be poisonous (for example, sodium fluoride is used in rat poison); intravenous injections are always dissolved in water containing salts at particular ionic concentrations that match with blood plasma concentrations and so on.

In this Unit, we will consider mostly liquid solutions and their formation. This will be followed by studying the properties of the solutions, like vapour pressure and colligative properties. We will begin with types of solutions and then various alternatives in which concentrations of a solute can be expressed in liquid solution.

### 2.1 Types of Solutions

Solutions are homogeneous mixtures of two or more than two components. By homogenous mixture we mean that its composition and properties are uniform throughout the mixture. Generally, the component that is present in the largest quantity is known as solvent. Solvent determines the physical state in which solution exists. One or more components present in the solution other than solvent are called solutes. In this Unit we shall consider only binary solutions (i.e., consisting of two components). Here each component may be solid, liquid or in gaseous state and are summarised in Table 2.1.

Type of Solution | Solute | Solvent | Common Examples |
---|---|---|---|

Gaseous Solutions | Gas Liquid Solid |
Gas Gas Gas |
Mixture of oxygen and nitrogen gases Chloroform mixed with nitrogen gas Camphor in nitrogen gas |

Liquid Solutions | Gas Liquid Solid |
Liquid Liquid Liquid |
Oxygen dissolved in water Ethanol dissolved in water Glucose dissolved in water |

Solid Solutions | Gas Liquid Solid |
Solid Solid Solid |
Solution of hydrogen in palladium Amalgam of mercury with sodium Copper dissolved in gold |

### 2.2 Expressing Concentration of Solutions

Composition of a solution can be described by expressing its concentration. The latter can be expressed either qualitatively or quantitatively. For example, qualitatively we can say that the solution is dilute (i.e., relatively very small quantity of solute) or it is concentrated (i.e., relatively very large quantity of solute). But in real life these kinds of description can add to lot of confusion and thus the need for a quantitative description of the solution.

There are several ways by which we can describe the concentration of the solution quantitatively.

(i) Mass percentage $(\mathrm{w} / \mathrm{w})$ : The mass percentage of a component of a solution is defined as:

Mass \% of a component

$$ \begin{equation*} =\frac{\text { Mass of the component in the solution }}{\text { Total mass of the solution }} \times 100 \tag{2.1} \end{equation*} $$

For example, if a solution is described by $10 \%$ glucose in water by mass, it means that $10 \mathrm{~g}$ of glucose is dissolved in $90 \mathrm{~g}$ of water resulting in a $100 \mathrm{~g}$ solution. Concentration described by mass percentage is commonly used in industrial chemical applications. For example, commercial bleaching solution contains 3.62 mass percentage of sodium hypochlorite in water.

(ii) Volume percentage ( $\mathrm{V} / \mathrm{V}$) : The volume percentage is defined as:

Volume $\%$ of a component $=\frac{\text { Volume of the component }}{\text { Total volume of solution }} \times 100$

For example, $10 \%$ ethanol solution in water means that $10 \mathrm{~mL}$ of ethanol is dissolved in water such that the total volume of the solution is $100 \mathrm{~mL}$. Solutions containing liquids are commonly expressed in this unit. For example, a $35 \%(v / v)$ solution of ethylene glycol, an antifreeze, is used in cars for cooling the engine. At this concentration the antifreeze lowers the freezing point of water to $255.4 \mathrm{~K}\left(-17.6^{\circ} \mathrm{C}\right)$.

(iii) Mass by volume percentage (w/V): Another unit which is commonly used in medicine and pharmacy is mass by volume percentage. It is the mass of solute dissolved in $100 \mathrm{~mL}$ of the solution.

(iv) Parts per million: When a solute is present in trace quantities, it is convenient to express concentration in parts per million (ppm) and is defined as:

Parts per million $=$

$$ \begin{equation*} \frac{\text { Number of parts of the component }}{\text { Total number of parts of all components of the solution }} \times 10^{6} \tag{2.3} \end{equation*} $$

As in the case of percentage, concentration in parts per million can also be expressed as mass to mass, volume to volume and mass to volume. A litre of sea water (which weighs $1030 \mathrm{~g}$ ) contains about $6 \times 10^{-3} \mathrm{~g}$ of dissolved oxygen $\left(\mathrm{O_2}\right)$. Such a small concentration is also expressed as $5.8 \mathrm{~g}$ per $10^{6} \mathrm{~g}(5.8 \mathrm{ppm})$ of sea water. The concentration of pollutants in water or atmosphere is often expressed in terms of $\mu \mathrm{g} \mathrm{mL}^{-1}$ or ppm.

(v) Mole fraction: Commonly used symbol for mole fraction is $x$ and subscript used on the right hand side of $x$ denotes the component. It is defined as:

Mole fraction of a component $=$ Number of moles of the component/ Total number of moles of all the components (2.4)

For example, in a binary mixture, if the number of moles of A and B are $n_{\mathrm{A}}$ and $n_{\mathrm{B}}$ respectively, the mole fraction of $\mathrm{A}$ will be

$$ \begin{equation*} x_{\mathrm{A}}=\frac{n_{\mathrm{A}}}{n_{\mathrm{A}}+n_{\mathrm{B}}} \tag{2.5} \end{equation*} $$

For a solution containing i number of components, we have:

$$ \begin{equation*} x_{\mathrm{i}}=\frac{n_{\mathrm{i}}}{n_{1}+n_{2}+\ldots \ldots+n_{\mathrm{i}}}=\frac{n_{\mathrm{i}}}{\sum n_{\mathrm{i}}} \tag{2.6} \end{equation*} $$

It can be shown that in a given solution sum of all the mole fractions is unity, i.e.

$$ \begin{equation*} x_{1}+x_{2}+\ldots \ldots \ldots \ldots \ldots \ldots .+x_{i}=1 \tag{2.7} \end{equation*} $$

Mole fraction unit is very useful in relating some physical properties of solutions, say vapour pressure with the concentration of the solution and quite useful in describing the calculations involving gas mixtures.

### 2.3 Solubility

Solubility of a substance is its maximum amount that can be dissolved in a specified amount of solvent at a specified temperature. It depends upon the nature of solute and solvent as well as temperature and pressure. Let us consider the effect of these factors in solution of a solid or a gas in a liquid.

#### 2.3.1 Solubility of a Solid in a Liquid

Every solid does not dissolve in a given liquid. While sodium chloride and sugar dissolve readily in water, naphthalene and anthracene do not. On the other hand, naphthalene and anthracene dissolve readily in benzene but sodium chloride and sugar do not. It is observed that polar solutes dissolve in polar solvents and non polar solutes in nonpolar solvents. In general, a solute dissolves in a solvent if the intermolecular interactions are similar in the two or we may say like dissolves like.

When a solid solute is added to the solvent, some solute dissolves and its concentration increases in solution. This process is known as dissolution. Some solute particles in solution collide with the solid solute particles and get separated out of solution. This process is known as crystallisation. A stage is reached when the two processes occur at the same rate. Under such conditions, number of solute particles going into solution will be equal to the solute particles separating out and a state of dynamic equilibrium is reached.

$$ \begin{equation*} \text { Solute }+ \text { Solvent } \rightleftharpoons \text { Solution } \tag{2.10} \end{equation*} $$

At this stage the concentration of solute in solution will remain constant under the given conditions, i.e., temperature and pressure. Similar process is followed when gases are dissolved in liquid solvents. Such a solution in which no more solute can be dissolved at the same temperature and pressure is called a saturated solution. An unsaturated solution is one in which more solute can be dissolved at the same temperature. The solution which is in dynamic equilibrium with undissolved solute is the saturated solution and contains the maximum amount of solute dissolved in a given amount of solvent. Thus, the concentration of solute in such a solution is its solubility.

Earlier we have observed that solubility of one substance into another depends on the nature of the substances. In addition to these variables, two other parameters, i.e., temperature and pressure also control this phenomenon.

**Effect of temperature**

The solubility of a solid in a liquid is significantly affected by temperature changes. Consider the equilibrium represented by equation 1.10 . This, being dynamic equilibrium, must follow Le Chateliers Principle. In general, if in a nearly saturated solution, the dissolution process is endothermic $(\left.\Delta_{\text {sol }} \mathrm{H}>0\right)$, the solubility should increase with rise in temperature and if it is exothermic $\left(\Delta_{\text {sol }} \mathrm{H}<0\right)$ the solubility should decrease. These trends are also observed experimentally.

**Effect of pressure**

Pressure does not have any significant effect on solubility of solids in liquids. It is so because solids and liquids are highly incompressible and practically remain unaffected by changes in pressure.

#### 2.3.2 Solubility of a Gas in a Liquid

Many gases dissolve in water. Oxygen dissolves only to a small extent in water. It is this dissolved oxygen which sustains all aquatic life. On the other hand, hydrogen chloride gas (HCl) is highly soluble in water. Solubility of gases in liquids is greatly affected by pressure and temperature. The solubility of gases increase with increase of pressure. For solution of gases in a solvent, consider a system as shown in Fig. 2.1 (a). The lower part is solution and the upper part is gaseous system at pressure p and temperature T. Assume this system to be in a state of dynamic equilibrium, i.e., under these conditions rate of gaseous particles entering and leaving the solution phase is the same. Now increase the pressure over the solution phase by compressing the gas to a smaller volume [Fig. 2.1 (b)]. This will increase the number of gaseous particles per unit volume over the solution and also the rate at which the gaseous particles are striking the surface of solution to enter it. The solubility of the gas will increase until a new equilibrium is reached resulting in an increase in the pressure of a gas above the solution and thus its solubility increases.

Henry was the first to give a quantitative relation between pressure and solubility of a gas in a solvent which is known as Henry’s law. The law states that at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas present above the surface of liquid or solution. Dalton, a contemporary of Henry, also concluded independently that the solubility of a gas in a liquid solution is a function of partial pressure of the gas. If we use the mole fraction of a gas in the solution as a measure of its solubility, then it can be said that the mole fraction of gas in the solution is proportional to the partial pressure of the gas over the solution. The most commonly used form of Henry’s law states that “the partial pressure of the gas in vapour phase (p) is proportional to the mole fraction of the gas $(\boldsymbol{x})$ in the solution" and is expressed as:

$$ \begin{equation*} p=K_{\mathrm{H}} x \tag{2.11} \end{equation*} $$

Here $K_H$ is the Henry’s law constant. If we draw a graph between partial pressure of the gas versus mole fraction of the gas in solution, then we should get a plot of the type as shown in Fig. 2.2.

Different gases have different KH values at the same temperature (Table 2.2). This suggests that KH is a function of the nature of the gas.

It is obvious from equation (2.11) that higher the value of $K_{\mathrm{H}}$ at a given pressure, the lower is the solubility of the gas in the liquid. It can be seen from Table 1.2 that $\mathrm{K_\mathrm{H}}$ values for both $\mathrm{N_2}$ and $\mathrm{O_2}$ increase with increase of temperature indicating that the solubility of gases increases with decrease of temperature. It is due to this reason that aquatic species are more comfortable in cold waters rather than in warm waters.

increases with decrease of temperature. It is due to this reason that aquatic species are more comfortable in cold waters rather than in warm waters.

Henry’s law finds several applications in industry and explains some biological phenomena. Notable among these are:

• To increase the solubility of CO2 in soft drinks and soda water, the bottle is sealed under high pressure.

• Scuba divers must cope with high concentrations of dissolved gases while breathing air at high pressure underwater. Increased pressure increases the solubility of atmospheric gases in blood. When the divers come towards surface, the pressure gradually decreases. This releases the dissolved gases and leads to the formation of bubbles of nitrogen in the blood. This blocks capillaries and creates a medical condition known as bends, which are painful and dangerous to life.

To avoid bends, as well as, the toxic effects of high concentrations of nitrogen in the blood, the tanks used by scuba divers are filled with air diluted with helium (11.7% helium, 56.2% nitrogen and 32.1% oxygen).

• At high altitudes the partial pressure of oxygen is less than that at the ground level. This leads to low concentrations of oxygen in the blood and tissues of people living at high altitudes or climbers. Low blood oxygen causes climbers to become weak and unable to think clearly, symptoms of a condition known as anoxia.

**Effect of Temperature**

Solubility of gases in liquids decreases with rise in temperature. When dissolved, the gas molecules are present in liquid phase and the process of dissolution can be considered similar to condensation and heat is evolved in this process. We have learnt in the last Section that dissolution process involves dynamic equilibrium and thus must follow Le Chatelier’s Principle. As dissolution is an exothermic process, the solubility should decrease with increase of temperature.

### 2.4 Vapour Pressure of Liquid Solutions

Liquid solutions are formed when solvent is a liquid. The solute can be a gas, a liquid or a solid. Solutions of gases in liquids have already been discussed in Section 2.3.2. In this Section, we shall discuss the solutions of liquids and solids in a liquid. Such solutions may contain one or more volatile components. Generally, the liquid solvent is volatile. The solute may or may not be volatile. We shall discuss the properties of only binary solutions, that is, the solutions containing two components, namely, the solutions of (i) liquids in liquids and (ii) solids in liquids.

#### 2.4.1 Vapour Pressure of Liquid- Liquid Solutions

Let us consider a binary solution of two volatile liquids and denote the two components as 1 and 2. When taken in a closed vessel, both the components would evaporate and eventually an equilibrium would be established between vapour phase and the liquid phase. Let the total vapour pressure at this stage be ptotal and p1 and p2 be the partial vapour pressures of the two components 1 and 2 respectively. These partial pressures are related to the mole fractions x1 and x2 of the two components 1 and 2 respectively.

The French chemist, Francois Marte Raoult (1886) gave the quantitative relationship between them. The relationship is known as the Raoult’s law which states that for a solution of volatile liquids, the partial vapour pressure of each component of the solution is directly proportional to its mole fraction present in solution.

Thus, for component 1

$$ \begin{aligned} & p_{1} \propto x_{1} \\ & \text { and } \quad p_{1}=p_{1}^{0} x_{1} \end{aligned} $$

where $p_{1}^{0}$ is the vapour pressure of pure component 1 at the same temperature.

Similarly, for component 2

$$ \begin{equation*} p_{2}=p_{2}{ }^{0} x_{2} \tag{2.13} \end{equation*} $$

where $p_{2}{ }^{0}$ represents the vapour pressure of the pure component 2 .

According to Dalton’s law of partial pressures, the total pressure

$(\left.p_{\text {total }}\right)$ over the solution phase in the container will be the sum of the partial pressures of the components of the solution and is given as:

$$ \begin{equation*} p_{\text {total }}=p_{1}+p_{2} \tag{2.14} \end{equation*} $$

Substituting the values of $p_{1}$ and $p_{2}$, we get

$$ \begin{align*} p_{\text {total }} & =x_{1} p_{1}{ }^{0}+x_{2} p_{2}{ }^{0} \\ & =\left(1-x_{2}\right) p_{1}{ }^{0}+x_{2} p_{2}{ }^{0} \tag{2.15}\\ & =p_{1}{ }^{0}+\left(p_{2}{ }^{0}-p_{1}{ }^{0}\right) x_{2} \tag{2.16} \end{align*} $$

Following conclusions can be drawn from equation (2.16). (i) Total vapour pressure over the solution can be related to the mole fraction of any one component. (ii) Total vapour pressure over the solution varies linearly with the mole fraction of component 2.

(iii) Depending on the vapour pressures of the pure components 1 and 2, total vapour pressure over the solution decreases or increases with the increase of the mole fraction of component 1.

A plot of $p_{1}$ or $p_{2}$ versus the mole fractions $x_{1}$ and $x_{2}$ for a solution gives a linear plot as shown in Fig. 2.3. These lines (I and II) pass through the points for which $x_{1}$ and $x_{2}$ are equal to unity. Similarly the plot (line III) of $p_{\text {total }}$ versus $x_{2}$ is also linear (Fig. 2.3). The minimum value of $p_{\text {total }}$ is $p_{1}{ }^{0}$ and the maximum value is $p_{2}{ }^{0}$, assuming that component 1 is less volatile than component 2 , i.e., $p_{1}{ }^{0}<p_{2}{ }^{0}$.

The composition of vapour phase in equilibrium with the solution is determined by the partial pressures of the components. If $y_{1}$ and $y_{2}$ are the mole fractions of the components 1 and 2 respectively in the vapour phase then, using Dalton’s law of partial pressures:

$$ \begin{align*} & p_{1}=y_{1} p_{\text {total }} \tag{2.17}\\ & p_{2}=y_{2} p_{\text {total }} \tag{2.18} \end{align*} $$

In general

$$ \begin{equation*} p_{\mathrm{i}}=y_{\mathrm{i}} \quad p_{\text {total }} \tag{2.19} \end{equation*} $$

Vapour pressure of chloroform $\left(\mathrm{CHCl_3}\right)$ and dichloromethane $\left(\mathrm{CH_2} \mathrm{Cl_2}\right)$ at $298 \mathrm{~K}$ are $200 \mathrm{~mm} \mathrm{Hg}$ and $415 \mathrm{~mm} \mathrm{Hg}$ respectively. (i) Calculate the vapour pressure of the solution prepared by mixing $25.5 \mathrm{~g}$ of $\mathrm{CHCl_3}$ and $40 \mathrm{~g}$ of $\mathrm{CH_2} \mathrm{Cl_2}$ at $298 \mathrm{~K}$ and, (ii) mole fractions of each component in vapour phase.

(i) Molar mass of $\mathrm{CH_2} \mathrm{Cl_2}=12 \times 1+1 \times 2+35.5 \times 2=85 \mathrm{~g} \mathrm{~mol}^{-1}$

Molar mass of $\mathrm{CHCl_3}=12 \times 1+1 \times 1+35.5 \times 3=119.5 \mathrm{~g} \mathrm{~mol}^{-1}$

Moles of $\mathrm{CH_2} \mathrm{Cl_2} \quad=\frac{40 \mathrm{~g}}{85 \mathrm{~g} \mathrm{~mol}^{-1}}=0.47 \mathrm{~mol}$

Moles of $\mathrm{CHCl_3} \quad=\frac{25.5 \mathrm{~g}}{119.5 \mathrm{~g} \mathrm{~mol}^{-1}}=0.213 \mathrm{~mol}$

Total number of moles $=0.47+0.213=0.683 \mathrm{~mol}$

$$ \begin{aligned} & x_{\mathrm{CH_2} \mathrm{Cl_2}}=\frac{0.47 \mathrm{~mol}}{0.683 \mathrm{~mol}}=0.688 \\ & x_{\mathrm{CHCl_3}}=1.00-0.688=0.312 \end{aligned} $$

Using equation (2.16),

$$ \begin{aligned} p_{\text {total }} & =p_{1}^{0}+\left(p_{2}{ }^{0}-p_{1}{ }^{0}\right) x_{2}=200+(415-200) \times 0.688 \\ & =200+147.9=347.9 \mathrm{~mm} \mathrm{Hg} \end{aligned} $$

(ii) Using the relation (2.19), $y_{\mathrm{i}}=p_{\mathrm{i}} / p_{\text {total }}$, we can calculate the mole fraction of the components in gas phase $\left(y_{\mathrm{i}}\right)$.

$$ \begin{aligned} & p_{\mathrm{CH_2} \mathrm{Cl_2}}=0.688 \times 415 \mathrm{~mm} \mathrm{Hg}=285.5 \mathrm{~mm} \mathrm{Hg} \\ & p_{\mathrm{CHCl_3}}=0.312 \times 200 \mathrm{~mm} \mathrm{Hg}=62.4 \mathrm{~mm} \mathrm{Hg} \\ & y_{\mathrm{CH_2} \mathrm{Cl_2}}=285.5 \mathrm{~mm} \mathrm{Hg} / 347.9 \mathrm{~mm} \mathrm{Hg}=0.82 \\ & y_{\mathrm{CHCl_3}}=62.4 \mathrm{~mm} \mathrm{Hg} / 347.9 \mathrm{~mm} \mathrm{Hg}=0.18 \end{aligned} $$

**Note**: Since, $\mathrm{CH_2} \mathrm{Cl_2}$ is a more volatile component than $\mathrm{CHCl_3}$, $\left[p_{\mathrm{CH_2} \mathrm{Cl_2}}^{0}=\right.$ $415 \mathrm{~mm} \mathrm{Hg}$ and $p_{\mathrm{CHCl_3}}^{0}=200 \mathrm{~mm} \mathrm{Hg}]$ and the vapour phase is also richer in $\mathrm{CH_2} \mathrm{Cl_2}\left[y_{\mathrm{CH_2} \mathrm{Cl_2}}=0.82\right.$ and $\left.y_{\mathrm{CHCl_3}}=0.18\right]$, it may thus be concluded that at equilibrium, vapour phase will be always rich in the component which is more volatile.

#### 2.4.2 Raoult’s Law as a special case of Henry’s Law

According to Raoult’s law, the vapour pressure of a volatile component in a given solution is given by $p_{\mathrm{i}}=x_{\mathrm{i}} p_{\mathrm{i}}{ }^{0}$. In the solution of a gas in a liquid, one of the components is so volatile that it exists as a gas and we have already seen that its solubility is given by Henry’s law which states that

$$ p=K_{\mathrm{H}} x $$

If we compare the equations for Raoult’s law and Henry’s law, it can be seen that the partial pressure of the volatile component or gas is directly proportional to its mole fraction in solution. Only the proportionality constant $K_{\mathrm{H}}$ differs from $p_{1}{ }^{0}$. Thus, Raoult’s law becomes a special case of Henry’s law in which $K_{\mathrm{H}}$ becomes equal to $p_{1}{ }^{0}$

#### 2.4.3 Vapour Pressure of Solutions of Solids in Liquids

Another important class of solutions consists of solids dissolved in liquid, for example, sodium chloride, glucose, urea and cane sugar in water and iodine and sulphur dissolved in carbon disulphide. Some physical properties of these solutions are quite different from those of pure solvents. For example, vapour pressure. We have learnt in Unit 5, Class XI, that liquids at a given temperature vapourise and under equilibrium conditions the pressure exerted by the vapours of the liquid over the liquid phase is called vapour pressure [Fig. 2.4 (a)]. In a pure liquid the entire surface is occupied by the molecules of the liquid. If a non-volatile solute is added to a solvent to give a solution [Fig. 2.4.(b)], the vapour pressure of the solution is solely from the solvent alone. This vapour pressure of the solution at a given temperature is found to be lower than the vapour pressure of the pure solvent at the same temperature. In the solution, the surface has both solute and solvent molecules; thereby the fraction of the surface covered by the solvent molecules gets reduced. Consequently, the number of solvent molecules escaping from the surface is correspondingly reduced, thus, the vapour pressure is also reduced.

The decrease in the vapour pressure of solvent depends on the quantity of non-volatile solute present in the solution, irrespective of its nature. For example, decrease in the vapour pressure of water by adding $1.0 \mathrm{~mol}$ of sucrose to one $\mathrm{kg}$ of water is nearly similar to that produced by adding $1.0 \mathrm{~mol}$ of urea to the same quantity of water at the same temperature.

Raoult’s law in its general form can be stated as, for any solution the partial vapour pressure of each volatile component in the solution is directly proportional to its mole fraction.

In a binary solution, let us denote the solvent by 1 and solute by 2. When the solute is non-volatile, only the solvent molecules are present in vapour phase and contribute to vapour pressure. Let p1 be the vapour pressure of the solvent, $x_{1}$ be its mole fraction, $p_{i}^{0}$ be its vapour pressure in the pure state. Then according to Raoult’s law

$$ \begin{align*} & p_{1} \propto x_{1} \ & \text { and } \quad p_{1}=x_{1} p_{1}^{0} \tag{2.20} \end{align*} $$

The proportionality constant is equal to the vapour pressure of pure solvent, $p_{1}^{0}$. A plot between the vapour pressure and the mole fraction of the solvent is linear (Fig. 2.5).

### 2.5 Ideal and Nonideal Solutions

Liquid-liquid solutions can be classified into ideal and non-ideal solutions on the basis of Raoult’s law.

#### 2.5.1 Ideal Solutions

The solutions which obey Raoult’s law over the entire range of concentration are known as ideal solutions. The ideal solutions have two other important properties. The enthalpy of mixing of the pure components to form the solution is zero and the volume of mixing is also zero, i.e., $$ \begin{equation*} \Delta_{\text {mix }} H=0, \quad \Delta_{\text {mix }} V=0 \tag{2.21} \end{equation*} $$

It means that no heat is absorbed or evolved when the components are mixed. Also, the volume of solution would be equal to the sum of volumes of the two components. At molecular level, ideal behaviour of the solutions can be explained by considering two components A and B. In pure components, the intermolecular attractive interactions will be of types A-A and B-B, whereas in the binary solutions in addition to these two interactions, A-B type of interactions will also be present. If the intermolecular attractive forces between the A-A and B-B are nearly equal to those between A-B, this leads to the formation of ideal solution. A perfectly ideal solution is rare but some solutions are nearly ideal in behaviour. Solution of n-hexane and n-heptane, bromoethane and chloroethane, benzene and toluene, etc. fall into this category.

#### 2.5.2 Non-ideal Solutions

When a solution does not obey Raoult’s law over the entire range of concentration, then it is called non-ideal solution. The vapour pressure of such a solution is either higher or lower than that predicted by Raoult’s law (equation 2.16). If it is higher, the solution exhibits positive deviation and if it is lower, it exhibits negative deviation from Raoult’s law. The plots of vapour pressure as a function of mole fractions for such solutions are shown in Fig. 2.6.

The cause for these deviations lie in the nature of interactions at the molecular level. In case of positive deviation from Raoult’s law, A-B interactions are weaker than those between A-A or B-B, i.e., in this case the intermolecular attractive forces between the solute-solvent molecules are weaker than those between the solute-solute and solvent-solvent molecules. This means that in such solutions, molecules of A (or B) will find it easier to escape than in pure state. This will increase the vapour pressure and result in positive deviation. Mixtures of ethanol and acetone behave in this manner. In pure ethanol, molecules are hydrogen bonded. On adding acetone, its molecules get in between the host molecules and break some of the hydrogen bonds between them. Due to weakening of interactions, the solution shows positive deviation from Raoult’s law [Fig. 2.6 (a)]. In a solution formed by adding carbon disulphide to acetone, the dipolar interactions between solute-solvent molecules are weaker than the respective interactions among the solute-solute and solvent-solvent molecules. This solution also shows positive deviation.

In case of negative deviations from Raoult’s law, the intermolecular attractive forces between A-A and B-B are weaker than those between A-B and leads to decrease in vapour pressure resulting in negative deviations. An example of this type is a mixture of phenol and aniline. In this case the intermolecular hydrogen bonding between phenolic proton and lone pair on nitrogen atom of aniline is stronger than the respective intermolecular hydrogen bonding between similar molecules. Similarly, a mixture of chloroform and acetone forms a solution with negative deviation from Raoult’s law. This is because chloroform molecule is able to form hydrogen bond with acetone molecule as shown.

This decreases the escaping tendency of molecules for each component and consequently the vapour pressure decreases resulting in negative deviation from Raoult’s law [Fig. 2.6. (b)].

Some liquids on mixing, form azeotropes which are binary mixtures having the same composition in liquid and vapour phase and boil at a constant temperature. In such cases, it is not possible to separate the components by fractional distillation. There are two types of azeotropes called minimum boiling azeotrope and maximum boiling azeotrope. The solutions which show a large positive deviation from Raoult’s law form minimum boiling azeotrope at a specific composition.

For example, ethanol-water mixture (obtained by fermentation of sugars) on fractional distillation gives a solution containing approximately 95% by volume of ethanol. Once this composition, known as azeotrope composition, has been achieved, the liquid and vapour have the same composition, and no further separation occurs.

The solutions that show large negative deviation from Raoult’s law form maximum boiling azeotrope at a specific composition. Nitric acid and water is an example of this class of azeotrope. This azeotrope has the approximate composition, 68% nitric acid and 32% water by mass, with a boiling point of 393.5 K.

### 2.6 Colligative Properties and Determination of Molar Mass

We have learnt in Section 2.4.3 that the vapour pressure of solution decreases when a non-volatile solute is added to a volatile solvent. There are many properties of solutions which are connected with this decrease of vapour pressure. These are: (1) relative lowering of vapour pressure of the solvent (2) depression of freezing point of the solvent (3) elevation of boiling point of the solvent and (4) osmotic pressure of the solution. All these properties depend on the number of solute particles irrespective of their nature relative to the total number of particles present in the solution. Such properties are called colligative properties (colligative: from Latin: co means together, ligare means to bind). In the following Sections we will discuss these properties one by one.

#### 2.6.1 Relative Lowering of Vapour Pressure

We have learnt in Section 2.4.3 that the vapour pressure of a solvent in solution is less than that of the pure solvent. Raoult established that the lowering of vapour pressure depends only on the concentration of the solute particles and it is independent of their identity. The equation (2.20) given in Section 2.4.3 establishes a relation between vapour pressure of the solution, mole fraction and vapour pressure of the solvent, i.e.,

$$ \begin{equation*} p_{1}=x_{1} p_{1}^{0} \tag{2.22} \end{equation*} $$

The reduction in the vapour pressure of solvent $\left(\Delta p_{1}\right)$ is given as:

$$ \begin{align*} \Delta p_{1} & =p_{1}^{0}-p_{1}=p_{1}^{0}-p_{1}^{0} x_{1} \\ & =p_{1}^{0}\left(1-x_{1}\right) \tag{2.23} \end{align*} $$

Knowing that $x_{2}=1-x_{1}$, equation (1.23) reduces to

$$ \begin{equation*} \Delta p_{1}=x_{2} p_{1}{ }^{0} \tag{2.24} \end{equation*} $$

In a solution containing several non-volatile solutes, the lowering of the vapour pressure depends on the sum of the mole fraction of different solutes. Equation (2.24)

$$ \begin{equation*} \frac{\Delta p_{1}}{p_{1}^{0}}=\frac{p_{1}^{0}-p_{1}}{p_{1}^{0}}=x_{2} \tag{2.25} \end{equation*} $$

The expression on the left hand side of the equation as mentioned earlier is called relative lowering of vapour pressure and is equal to the mole fraction of the solute. The above equation can be written as:

$$ \begin{equation*} \frac{p_{1}^{0}-p_{1}}{p_{1}^{0}}=\frac{n_{2}}{n_{1}+n_{2}}\left(\text { since } x_{2}=\frac{n_{2}}{n_{1}+n_{2}}\right) \tag{2.26} \end{equation*} $$

Here $n_{1}$ and $n_{2}$ are the number of moles of solvent and solute respectively present in the solution. For dilute solutions $n_{2}<n_{1}$, hence neglecting $n_{2}$ in the denominator we have

$$ \begin{align*} \frac{p_{1}^{0}-p_{1}}{p_{1}^{0}} & =\frac{n_{2}}{n_{1}} \tag{2.27}\\ \text { or } \frac{p_{1}^{0}-p_{1}}{p_{1}^{0}} & =\frac{\mathrm{w_2} \times M_{1}}{M_{2} \times \mathrm{w_1}} \tag{2.28} \end{align*} $$

Here $w_{1}$ and $w_{2}$ are the masses and $M_{1}$ and $M_{2}$ are the molar masses of the solvent and solute respectively.

From this equation (2.28), knowing all other quantities, the molar mass of solute (M2) can be calculated.

#### 2.6.2 Elevation of Boiling Point

The vapour pressure of a liquid increases with increase of temperature. It boils at the temperature at which its vapour pressure is equal to the atmospheric pressure. For example, water boils at $373.15 \mathrm{~K}\left(100^{\circ} \mathrm{C}\right)$ because at this temperature the vapour pressure of water is 1.013 bar ( 1 atmosphere). We have also learnt in the last section that vapour pressure of the solvent decreases in the presence of non-volatile solute. Fig. 2.7 depicts the variation of vapour pressure of the pure solvent and solution as a function of temperature. For example, the vapour pressure of an aqueous solution of sucrose is less than 1.013 bar at $373.15 \mathrm{~K}$. In order to make this solution boil, its vapour pressure must be increased to 1.013 bar by raising the temperature above the boiling temperature of the pure solvent (water). Thus, the boiling point of a solution is prepared as shown in Fig. 2.7: The vapour pressure curve for solution lies below the curve for pure water. The diagram shows that $\Delta T_{\mathrm{b}}$ denotes the elevation of boiling point of a solvent in solution. always higher than that of the boiling point of the pure solvent in which the solution is prepared as shown in Fig. 2.7. Similar to lowering of vapour pressure, the elevation of boiling point also depends on the number of solute molecules rather than their nature. A solution of $1 \mathrm{~mol}$ of sucrose in $1000 \mathrm{~g}$ of water boils at $373.52 \mathrm{~K}$ at one atmospheric pressure.

Let $T_{\mathrm{b}}^{0}$ be the boiling point of pure solvent and $T_{\mathrm{b}}$ be the boiling point of solution. The increase in the boiling point $\Delta T_{\mathrm{b}}=T_{\mathrm{b}}-T_{\mathrm{b}}^{0}$ is known as elevation of boiling point.

Experiments have shown that for dilute solutions the elevation of boiling point $\left(\Delta T_{\mathrm{b}}\right)$ is directly proportional to the molal concentration of the solute in a solution. Thus

$$ \begin{align*} & \Delta T_{\mathrm{b}} \propto \mathrm{m} \tag{2.29}\\ \text { or } \quad \Delta T_{\mathrm{b}} & =K_{\mathrm{b}} \mathrm{m} \tag{2.30} \end{align*} $$

Here $m$ (molality) is the number of moles of solute dissolved in $1 \mathrm{~kg}$ of solvent and the constant of proportionality, $K_{\mathrm{b}}$ is called Boiling Point Elevation Constant or Molal Elevation Constant (Ebullioscopic Constant). The unit of $K_{\mathrm{b}}$ is $\mathrm{K} \mathrm{kg} \mathrm{mol}^{-1}$. Values of $K_{\mathrm{b}}$ for some common solvents are given in Table 1.3. If $\mathrm{w_2}$ gram of solute of molar mass $M_{2}$ is dissolved in $\mathrm{w_1}$ gram of solvent, then molality, $\mathrm{m}$ of the solution is given by the expression:

$$ \begin{equation*} \mathrm{m}=\frac{w_{2} / \mathrm{M_2}}{w_{1} / 1000}=\frac{1000 \times w_{2}}{M_{2} \times w_{1}} \tag{2.31} \end{equation*} $$

Substituting the value of molality in equation (1.30) we get

$$ \begin{align*} \Delta T_{\mathrm{b}} & =\frac{K_{\mathrm{b}} \times 1000 \times w_{2}}{M_{2} \times w_{1}} \tag{2.32}\\ M_{2} & =\frac{1000 \times w_{2} \times K_{\mathrm{b}}}{\Delta T_{\mathrm{b}} \times w_{1}} \tag{2.33} \end{align*} $$

Thus, in order to determine $M_{2}$, molar mass of the solute, known mass of solute in a known mass of the solvent is taken and $\Delta T_{\mathrm{b}}$ is determined experimentally for a known solvent whose $K_{\mathrm{b}}$ value is known.

#### 2.6.3 Depression of Freezing Point

The lowering of vapour pressure of a solution causes a lowering of the freezing point compared to that of the pure solvent (Fig. 2 8). We know that at the freezing point of a substance, the solid phase is in dynamic equilibrium with the liquid phase. Thus, the freezing point of a substance may be defined as the temperature at which the vapour pressure of the substance in its liquid phase is equal to its vapour pressure in the solid phase. A solution will freeze when its vapour pressure equals the vapour pressure of the pure solid solvent as is clear from Fig. 2.8. According to Raoult’s law, when a non-volatile solid is added to the solvent its vapour pressure decreases and now it would become equal to that of solid solvent at lower temperature. Thus, the freezing point of the solvent decreases.

Let $T_{\mathrm{f}}^{0}$ be the freezing point of pure solvent and $T_{\mathrm{f}}$ be its freezing point when non-volatile solute is dissolved in it. The decrease in freezing point.

$$ \Delta T_{\mathrm{f}}=T_{\mathrm{f}}^{0}-T_{\mathrm{f}} \text { is known as depression in } $$

freezing point.

Similar to elevation of boiling point, depression of freezing point $\left(\Delta T_{\mathrm{f}}\right)$ for dilute solution (ideal solution) is directly proportional to molality, $\mathrm{m}$ of the solution. Thus,

$$ \Delta T_{\mathrm{f}} \propto \mathrm{m} $$

$$ \begin{equation*} \text { or } \quad \Delta T_{\mathrm{f}}=K_{\mathrm{f}} \mathrm{m} \tag{2.34} \end{equation*} $$

The proportionality constant, $\mathrm{K_\mathrm{f}}$, which depends on the nature of the solvent is known as Freezing Point Depression Constant or Molal

Depression Constant or Cryoscopic Constant. The unit of $\mathrm{K_\mathrm{f}}$ is $\mathrm{K} \mathrm{kg}$ $\mathrm{mol}^{-1}$. Values of $K_{\mathrm{f}}$ for some common solvents are listed in Table 2.3.

If $\mathrm{w_2}$ gram of the solute having molar mass as $M_{2}$, present in $\mathrm{w_1}$ gram of solvent, produces the depression in freezing point $\Delta T_{\mathrm{f}}$ of the solvent then molality of the solute is given by the equation (2.31).

$$ \begin{equation*} \mathrm{m}=\frac{w_{2} / M_{2}}{w_{1} / 1000} \tag{2.31} \end{equation*} $$

Substituting this value of molality in equation (2.34) we get:

$$ \begin{align*} \Delta T_{\mathrm{f}} & =\frac{K_{\mathrm{f}} \times w_{2} / M_{2}}{w_{1} / 1000} \\ \Delta T_{\mathrm{f}} & =\frac{K_{\mathrm{f}} \times w_{2} \times 1000}{M_{2} \times w_{1}} \tag{2.35}\\ M_{2} & =\frac{K_{\mathrm{f}} \times w_{2} \times 1000}{\Delta T_{\mathrm{f}} \times w_{1}} \tag{2.36} \end{align*} $$

Thus for determining the molar mass of the solute we should know the quantities $\mathrm{w_1}, \mathrm{w_2}, \Delta T_{\mathrm{f}}$, along with the molal freezing point depression constant. The values of $K_{f}$ and $K_{b}$, which depend upon the nature of the solvent, can be ascertained from the following relations.

$$ \begin{align*} K_{\mathrm{f}} & =\frac{R \times M_{1} \times T_{\mathrm{f}}^{2}}{1000 \times \Delta_{\text {fus }} H} \tag{2.37}\\ K_{\mathrm{b}} & =\frac{R \times M_{1} \times T_{\mathrm{b}}^{2}}{1000 \times \Delta_{\text {vap }} H} \tag{2.38} \end{align*} $$

Here the symbols $R$ and $M_{1}$ stand for the gas constant and molar mass of the solvent, respectively and $T_{\mathrm{f}}$ and $T_{\mathrm{b}}$ denote the freezing point and the boiling point of the pure solvent respectively in kelvin. Further, $\Delta_{\text {fus }} H$ and $\Delta_{\text {vap }} H$ represent the enthalpies for the fusion and vapourisation of the solvent, respectively.

#### 2.6.4 Osmosis and Osmotic Pressure

There are many phenomena which we observe in nature or at home. For example, raw mangoes shrivel when pickled in brine (salt water); wilted flowers revive when placed in fresh water, blood cells collapse when suspended in saline water, etc. If we look into these processes we find one thing common in all, that is, all these substances are bound by membranes. These membranes can be of animal or vegetable origin and these occur naturally such as pig’s bladder or parchment or can be synthetic such as cellophane. These membranes appear to be continuous sheets or films, yet they contain a network of submicroscopic holes or pores. Small solvent molecules, like water, can pass through these holes but the passage of bigger molecules like solute is hindered. Membranes having this kind of properties are known as semipermeable membranes (SPM).

Assume that only solvent molecules can pass through these semipermeable membranes. If this membrane is placed between the solvent and solution as shown in Fig. 2.9, the solvent molecules will flow through the membrane from pure solvent to the solution. This process of flow of the solvent is called osmosis.

The flow will continue till the equilibrium is attained. The flow of the solvent from its side to solution side across a semipermeable membrane can be stopped if some extra pressure is applied on the solution. This pressure that just stops the flow of solvent is called osmotic pressure of the solution. The flow of solvent from dilute solution to the concentrated solution across a semipermeable membrane is due to osmosis. The important point to be kept in mind is that solvent molecules always flow from lower concentration to higher concentration of solution. The osmotic pressure has been found to depend on the concentration of the solution.

The osmotic pressure of a solution is the excess pressure that must be applied to a solution to prevent osmosis, i.e., to stop the passage of solvent molecules through a semipermeable membrane into the solution. This is illustrated in Fig. 2.10. Osmotic pressure is a colligative property as it depends on the number of solute molecules and not on their identity. For dilute solutions, it has been found experimentally that osmotic pressure is proportional to the molarity, C of the solution at a given temperature T. Thus:

$$ \begin{equation*} \Pi=C R T \tag{2.39} \end{equation*} $$

Here $\Pi$ is the osmotic pressure and $\mathrm{R}$ is the gas constant.

$$ \begin{equation*} \Pi=\left(n_{2} / V\right) R T \tag{2.40} \end{equation*} $$

Here $V$ is volume of a solution in litres containing $\mathrm{n_2}$ moles of solute. If $\mathrm{w_2}$ grams of solute, of molar mass, $M_{2}$ is present in the solution, then $n_{2}=\mathrm{w_2} / M_{2}$ and we can write,

$$ \begin{array}{r} \Pi V=\frac{\mathbf{w_2} R T}{M_{2}} \\ \text { or } M_{2}=\frac{\mathbf{w_2} R T}{\Pi V} \tag{2.42} \end{array} $$

Thus, knowing the quantities $\mathrm{w_2}, T, \Pi$ and $V$ we can calculate the molar mass of the solute.

Measurement of osmotic pressure provides another method of determining molar masses of solutes. This method is widely used to determine molar masses of proteins, polymers and other macromolecules. The osmotic pressure method has the advantage over other methods as pressure measurement is around the room temperature and the molarity of the solution is used instead of molality. As compared to other colligative properties, its magnitude is large even for very dilute solutions. The technique of osmotic pressure for determination of molar mass of solutes is particularly useful for biomolecules as they are generally not stable at higher temperatures and polymers have poor solubility.

Two solutions having same osmotic pressure at a given temperature are called isotonic solutions. When such solutions are separated by semipermeable membrane no osmosis occurs between them. For example, the osmotic pressure associated with the fluid inside the blood cell is equivalent to that of 0.9% (mass/ volume) sodium chloride solution, called normal saline solution and it is safe to inject intravenously. On the other hand, if we place the cells in a solution containing more than 0.9% (mass/volume) sodium chloride, water will flow out of the cells and they would shrink. Such a solution is called hypertonic. If the salt concentration is less than 0.9% (mass/volume), the solution is said to be hypotonic. In this case, water will flow into the cells if placed in this solution and they would swell.

The phenomena mentioned in the beginning of this section can be explained on the basis of osmosis. A raw mango placed in concentrated salt solution loses water via osmosis and shrivel into pickle. Wilted flowers revive when placed in fresh water. A carrot that has become limp because of water loss into the atmosphere can be placed into the water making it firm once again. Water will move into its cells through osmosis. When placed in water containing less than 0.9% (mass/ volume) salt, blood cells swell due to flow of water in them by osmosis. People taking a lot of salt or salty food experience water retention in tissue cells and intercellular spaces because of osmosis. The resulting puffiness or swelling is called edema. Water movement from soil into plant roots and subsequently into upper portion of the plant is partly due to osmosis. The preservation of meat by salting and of fruits by adding sugar protects against bacterial action. Through the process of osmosis, a bacterium on salted meat or candid fruit loses water, shrivels and dies.

#### 2.6.5 Reverse Osmosis and Water Purification

The direction of osmosis can be reversed if a pressure larger than the osmotic pressure is applied to the solution side. That is, now the pure solvent flows out of the solution through the semi permeable membrane. This phenomenon is called reverse osmosis and is of great practical utility. Reverse osmosis is used in desalination of sea water. A schematic set up for the process is shown in Fig. 2.11. When pressure more than osmotic pressure is applied, pure water is squeezed out of the sea water through the membrane. A variety of polymer membranes are available for this purpose.

The pressure required for the reverse osmosis is quite high. A workable porous membrane is a film of cellulose acetate placed over a suitable support. Cellulose acetate is permeable to water but impermeable to impurities and ions present in sea water. These days many countries use desalination plants to meet their potable water requirements.

### Summary

A solution is a homogeneous mixture of two or more substances. Solutions are classified as solid, liquid and gaseous solutions. The concentration of a solution is expressed in terms of mole fraction, molarity, molality and in percentages. The dissolution of a gas in a liquid is governed by Henry’s law, according to which, at a given temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas. The vapour pressure of the solvent is lowered by the presence of a non-volatile solute in the solution and this lowering of vapour pressure of the solvent is governed by Raoult’s law, according to which the relative lowering of vapour pressure of the solvent over a solution is equal to the mole fraction of a non-volatile solute present in the solution. However, in a binary liquid solution, if both the components of the solution are volatile then another form of Raoult’s law is used. Mathematically, this form of the Raoult’s law is stated as: $p_{\text {total }}=p_{1}^{0} x_{1}+p_{2}^{0} x_{2}$. Solutions which obey Raoult’s law over the entire range of concentration are called ideal solutions. Two types of deviations from Raoult’s law, called positive and negative deviations are observed. Azeotropes arise due to very large deviations from Raoult’s law.

The properties of solutions which depend on the number of solute particles and are independent of their chemical identity are called colligative properties. These are lowering of vapour pressure, elevation of boiling point, depression of freezing point and osmotic pressure. The process of osmosis can be reversed if a pressure higher than the osmotic pressure is applied to the solution. Colligative properties have been used to determine the molar mass of solutes. Solutes which dissociate in solution exhibit molar mass lower than the actual molar mass and those which associate show higher molar mass than their actual values.

Quantitatively, the extent to which a solute is dissociated or associated can be expressed by van’t Hoff factor i. This factor has been defined as ratio of normal molar mass to experimentally determined molar mass or as the ratio of observed colligative property to the calculated colligative property.