Abnormal Molecular Masses and Van’t Hoff Factor: A solute (ionic or covalent), when dissolved in a polar or non-polar solvent, undergoes either association or dissociation. This affects the net molar mass of the solute in the solution. For example, if we dissolve one mole of \(\mathrm{KCl}(74.5 \mathrm{~g})\) in \(1 \mathrm{~kg}\) of water, there will be \(1\) mole of \(\text {K}^{+}\) ions and \(1\) mole of \(\text {Cl}^{-}\) ions in the resulting solution. This means a total of \(2\) moles of ions are present in the resulting solution. The experimentally determined molar mass for such solutes is always lower than the true value.

Similarly, few substances tend to associate in an aqueous state. Calculating the molar mass using the colligative properties for such solutes, it is found that the experimentally determined molar mass is always more than the true value. Let’s explore more about this concept and the correction factor used to fix this abnormality.

Abnormal Molar Mass

When ionic compounds dissolve in water, it dissociates into their corresponding cations and anions. Similarly, few substances tend to associate in an aqueous state. Hence, the number of ions/molecules present in the solution will be more than the actual number of molecules in case of dissociation and will be less in case of association.
This discrepancy in molar mass due to association or dissociation is known as abnormal molar mass.

The accurate values of molar masses can be obtained only under two conditions-

(i) The solutions should be diluted

In the concentrated solutions, the particles begin to interact with each other as well as with the solvent. As a result, the vapour pressure and, therefore, other colligative properties depend upon the nature of the solute and not just on the number of solute particles.

(ii) The non-volatile solute must not dissociate or associate in solution

Due to the association or the dissociation of the solute molecules in the solution, the number of molecules changes. The discrepancy in molar mass is called abnormal molar mass. Abnormal molar mass is the solution’s molar mass, which is either lower or higher than the expected or normal value.

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Van’t Hoff Factor

In \(1880\), Van’t Hoff introduced a factor (i) to account for the extent of dissociation or association of solute particles in a solution. The factor (i) is known as the Van’t Hoff factor, which is defined as:

\(\text {i}=\frac{\text { Normal Molar Mass }}{\text { Abnormal Molar Mass }}\ldots \ldots\text {Eqn}(1)\)

\(\text {i}=\frac{\text { Observed Colligative Property }}{\text { Calculated Colligative Property }}………..\text {Eqn}(2)\)

\(\text {i}=\frac{\text { Total number of moles of particles after association/dissociation }}{\text { Number of moles of particles before association/dissociation }} \ldots \ldots \ldots\text {Eqn}(3)\)

A solute or electrolyte dissociates or associates only up to a fraction. This is known as the degree of dissociation, which is expressed by the symbol \({\rm{\alpha }}\).

Van’t Hoff Factor and Dissociation of Solute Molecules

Van’t Hoff factor (i) can also be used to calculate the extent of dissociation of an ionic solute. This is done in terms of the degree of dissociation \(({\rm{\alpha }})\).

The degree of dissociation or is defined as the fraction of the ionic solute dissolved in a solvent undergoing dissociation into cations and anions.

\(\alpha=\frac{\text { Number of moles of the solute dissociated }}{\text { Total number of moles of the solute present in the solution }}\)

Suppose one mole of an ionic solute dissociates to give n number of ions, with a degree of dissociation \(({\rm{\alpha }})\).

At equilibrium:

Number of moles of undissociated solute left (after dissociation) \({\rm{ = 1 – \alpha }}\)

Number of moles of ions formed (after dissociation) \({\rm{ = n\alpha }}\)

Total number of moles of particles (after dissociation) \({\rm{ = 1 – \alpha + n\alpha }}\)

From Van’t Hoff factor definition and using Eqn (3), we have:

\(\text {i}=\frac{\text { Total number of moles of particles after dissociation }}{\text { Number of moles of particles before dissociation }}=\frac{1-\alpha+\text {n} \alpha}{1}\)

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

\(\Rightarrow \alpha=\frac{\text {i}-1}{\text {n}-1}\)

The value of \({\rm{\alpha }}\) can be calculated by knowing the value of (i) from observed molar mass and normal molar mass.

Van’t Hoff Factor and Association of Solute Molecules

Acetic acid (ethanoic acid) self-associates with benzene to form dimers. This is an example of the association of molecules to form bigger molecules.

Van’t Hoff factor (i) can also be used to calculate the extent of association of a non-polar solute. This is done in terms of the degree of association \(({\rm{\alpha }})\).

The degree of association or \({\rm{\alpha }}\) is defined as the fraction of the total non-polar solute present in the solution that undergoes association.

\(\alpha=\frac{\text { Number of moles of the solute associated }}{\text { Total number of moles of the solute present in the solution }}\)

Suppose, \(\text {n}\) number of molecules of the solute associates to one mole of the associated solute, with a degree of association \({\rm{\alpha }}\).

At equilibrium:

Number of moles of unassociated solute left (after association) \({\rm{ = 1 – \alpha }}\)

Number of moles of associated molecules formed (after association) \({\rm{ = \alpha /n}}\)

Total number of moles of particles (after association) \({\rm{ = 1 – \alpha + \alpha /n}}\)

Using the definition of Van’t Hoff factor and using Eqn(3), we have:

\({\rm{i}} = \frac{{{\rm{ Total number of moles of particles after association }}}}{{{\rm{ Number of moles of particles before the association }}}} = \frac{{1 – {\rm{\alpha + \alpha /n}}}}{1}\)

\( \Rightarrow {\rm{i = 1 – \alpha + \alpha /n}}\)

\( \Rightarrow {\rm{\alpha }}(1 – {\rm{n}}) = {\rm{ni}} – {\rm{n}} = {\rm{n}}({\rm{i}} – 1)\)

\( \Rightarrow {\rm{\alpha }} = \frac{{{\rm{n}}({\rm{i}} – 1)}}{{1 – {\rm{n}}}}\)

The value of \({\rm{\alpha }}\) can be calculated by knowing the value of observed molar mass, normal molar mass and the number \((\text {n})\) of simple molecules that undergo association.

Value of Van’t Hoff Factor

Dissociation	\(\text {i} >1\)
Association	\(\text {i} < 1\)
No association or dissociation	\(\text {i} = 1\)

The equations for colligative properties with Van’t Hoff factor can be written as follows:

\(\text {Relative lowering of the vapour pressure of solvent}\, =\frac{\text {P}_{\text {A}}^{0}-\text {P}_{\text {A}}}{\text {P}^{\text {o}}{ }{\text {A}}}=\text {i} \frac{\text {n}_{\text {B}}}{\text {n}_{\text {A}}}\)

Elevation of boiling point, \(\Delta {{\rm{T}}_{\rm{b}}} = {\rm{i}}{{\rm{K}}_{\rm{b}}}\;{\rm{m}}\)

Depression ofzing point, \(\Delta {{\rm{T}}_{\rm{f}}} = {\rm{i}}{{\rm{K}}_{\rm{f}}}\;{\rm{m}}\)

The osmotic pressure of the solution, \(\Pi  = \frac{{{\rm{i}}{{\rm{n}}_{\rm{B}}}{\rm{RT}}}}{{\rm{V}}}\)

Where, \(\text {n}_{\text {A}}\) represents the number of moles of the solvent

\(\text {n}_{\text {B}}\) represents the number of moles of the solute, and

\(\text {P}_{\text {A}}^{\text {o}}\) represents the vapour pressure of the solvent in its pure form

The Van’t Hoff factor is, therefore, a measure of a deviation from ideal behaviour. The lower the van ‘t Hoff factor, the higher the deviation. The Van’t Hoff factor decreases with the increase in the concentration of the solute. This is because ionic compounds do not undergo dissociation in an aqueous solution completely.

Solved Examples

Q.1. The Van’t Hoff factor for \(\mathrm{BaCl}_{2}\) at \(0.001\, \mathrm{M}\) concentration is \(1.98\). What is the percentage dissociation of \(\mathrm{BaCl}_{2}\) ?
Ans: Given, \(\mathrm{i}=1.98, \mathrm{c}=0.001 \mathrm{M}\)
\(\mathrm{BaCl}_{2} \rightleftharpoons 1 \mathrm{Ba}^{2+}+2 \mathrm{Cl}^{-} \quad ; \text {n}=3\)
\(\alpha=\frac{\text {i}-1}{\text {n}-1}\)
\(\alpha=\frac{1.98-1}{3-1}=\frac{0.98}{2}=0.49\)
\(\%\) dissociation \(=\alpha \times 100=49 \%\)

Q.2. The (i) of a \(0.1 \mathrm{M} \,\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) solution is \(4.20\). Calculate the degree of dissociation of \(\text {Al}_{2}\left(\text {SO}_{4}\right)_{3}\)?
Ans: Given, \(\text {i} = 4.20, \text {c} = 0.1 \text {M}\)
Let be the degree of dissociation.

\(\begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{A}}{{\text{l}}_2}{\left( {{\text{S}}{{\text{O}}_4}} \right)_3} \rightleftharpoons 2{\text{A}}{{\text{l}}^{3 + }} + 3{\text{SO}}_4^{2 – } \hfill \\ {\text{Before}}\,{\text{dissociation: }}\,\,\,\,\,{\text{1}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ {\text{After}}\,{\text{dissociation:}}\,\,\,\,\,\,\,1 – \alpha \,\,\,\,\,\,\,\,\,\,\,2\alpha \,\,\,\,\,\,\,\,\,3\alpha \hfill \\ \end{gathered} \)

\(\mathrm{i}=\frac{1-\alpha+2 \alpha+3 \alpha}{1}\)
\(4.20=1+4 \alpha\)
\(\alpha=\frac{3.20}{4}=0.8\)

Summary 

The number of solute particles in a solution affects the colligative properties of a solution. The solute particles either undergo dissociation into their ions or associate to form dimers. The Van’t Hoff factor measures the extent to which association or dissociation of solute particles affects the colligative properties of a solution. Hence, it’s essential to take this factor into account while dealing with solutions. 

FAQs

Q.1. What is Van’t Hoff’s Factor?
Ans: The relationship between the actual number of moles of solute added to form a solution and the apparent number determined by colligative properties is called the Van’t Hoff factor.

Q.2. How does Van’t Hoff factor affect molar mass?
Ans: Van’t Hoff factor denoted by (i) is used to correct the abnormality observed in molar masses of solutes that undergo either association or dissociation when dissolved in a solvent.

Q.3. What is the cause of abnormal molecular mass?
Ans: Due to the association or the dissociation of the solute molecules in the solution, the number of molecules changes. The discrepancy in molar mass is called abnormal molar mass. The abnormal molar masses is due to the association or dissociation of solute particles.

Q.4. Why do electrolytes show abnormal molecular masses?
Ans: The electrolytes show abnormal molar masses because of dissociation. The electrolytes dissociate in solution.

Q.5. How do molar masses affect the colligative properties?
Ans: The molar mass of a solute is inversely proportional to the colligative properties. Thus a decrease in colligative properties will lead to an increase in the molecular masses.

Q.6. Why do we get abnormal molecular mass? Explain with example.
Ans: A solute (ionic or covalent), when dissolved in a polar or non-polar solvent, undergoes either association or dissociation. This affects the net molar mass of the solute in the solution. For example, if we dissolve one mole of \(\mathrm{KCl}(74.5 \mathrm{~g})\) in \(1 \mathrm{~kg}\) of water, there will be \(1\) mole of \(\text {K}^{+}\) ions and \(1\) mole of \(\mathrm{Cl}^{-}\) ions in the resulting solution. This means a total of \(2\) moles of ions are present in the resulting solution. The experimentally determined molar mass for such solutes is always lower than the true value.
Similarly, few substances tend to associate in an aqueous state-for example, acetic acid (ethanoic acid) self-associates in benzene to form dimers. The experimentally determined molar mass for such solutes is always more than the true value.

Q.7. How do you find the Van’t Hoff factor for dissociation?
Ans: The Van’t Hoff factor for dissociation can be calculated as-
\(\alpha=\frac{\text {i}-1}{\text {n}-1}\)

Learn About Van’t Hoff Factor Here

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