Van’t Hoff Factor Notes: Colligative properties such as the relative lowering of vapour pressure, depression in freezing point, elevation in boiling point, and osmotic pressure of a solution depend on the number of solute particles dissolved in the solvent. These properties are independent of the type of solute. When a solute (ionic or covalent) is dissolved in a solvent (polar or non-polar), it undergoes dissociation or association, which affects the molar mass of the solute in the solution. Let’s learn how the number of the solute particles affects the molar mass of the solute in the solution.

In this article, we have provided detailed information on Van’t Hoff factor, Van’t Hoff factor formula etc. Students can learn the concept easily through the Van’t Hoff factor class 12 notes provided by Embibe. Continue reading this article to learn more.

Abnormal Molar Mass

We know that ionic or polar compounds, on dissolution, dissociate into their corresponding cations and anions. For example, there will be \(1\) mole of \(\mathrm{Cl}^{-}\) ions and \(1\) mole of \({{\rm{K}}^{\rm{ + }}}\) ions in the resulting solution (a total of \(2\) moles of ions in the solution) if we dissolve one mole of \(\mathrm{KCl}(74.5 \mathrm{~g})\) in \(1 \mathrm{~kg}\) of water. Calculating the molar mass using the colligative properties for such solutes, it is found that the experimentally determined molar mass is always lower than its original molar mass.

On the other hand, few substances tend to associate in an aqueous state. The actual number of molecules for such solutes is always more than the number of ions/molecules present in the solution. 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.

The equations derived for measuring the colligative property are for non-electrolytes that do not undergo association or dissociation in any solution. The discrepancy in molar mass, also known as Abnormal Molar mass, arises either due to association or dissociation of solute particles. 

Abnormal molar mass is the molar mass of the solution which is either lower or higher than the expected or normal value.

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

\({\rm{i = }}\frac{{{\rm{ Normal\, Molar\, Mass }}}}{{{\rm{ Abnormal\, Molar\, Mass }}}}\quad {\rm{ \ldots  \ldots  \ldots  \ldots Eqn(1)}}\)

\({\rm{i = }}\frac{{{\rm{ Observed\, Colligative\, Property }}}}{{{\rm{ Calculated\, Colligative\, Property }}}}{\rm{, \ldots  \ldots  \ldots  \ldots Eqn(2)}}\)

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

The abnormal molar mass in Equation \((1)\) is the experimentally determined molar mass. The calculated colligative properties in Equation \((2)\) are obtained by assuming that the non-volatile solute is neither associated nor dissociated.

A solute or electrolyte never dissociates or associates completely when dissolved in a solvent, but only up to a fraction known as the degree of dissociation which is expressed by the symbol \(\alpha .\)

Van’t Hoff Factor and Dissociation of Solute Molecules

Van’t off factor is used to calculate the extent of dissociation of an ionic solute in terms of the degree of dissociation \(\alpha .\)

The degree of dissociation or \(\alpha .\) is defined as the fraction of the total ionic solute present in the solution that undergoes dissociation into cations and anions.

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

Suppose \(1\) mole of an electrolyte dissociates to give n number of ions, with a degree of dissociation \(\alpha\)

At equilibrium:

Number of moles of undisscoiated solute left (after dissociation) \(=1-\alpha\)

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

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

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

\({\rm{i = }}\frac{{{\rm{Total}}{\mkern 1mu} {\rm{number}}{\mkern 1mu} {\rm{of}}{\mkern 1mu} {\rm{moles}}{\mkern 1mu} \,{\rm{of}}{\mkern 1mu} {\rm{particles}}{\mkern 1mu} \,{\rm{after}}{\mkern 1mu} {\rm{dissociation}}}}{{{\rm{Number}}{\mkern 1mu} {\rm{of}}{\mkern 1mu} {\rm{moles}}{\mkern 1mu} \,{\rm{of}}{\mkern 1mu} {\rm{particles}}{\mkern 1mu} \,{\rm{before}}\,{\mkern 1mu} {\rm{dissociation}}}}{\rm{ = }}\frac{{{\rm{1 – \alpha + n\alpha }}}}{{\rm{1}}}\)

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

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

The value of \(\alpha\) can be calculated by knowing the value of Van’t Hoff factor (i) from observed molar mass and normal molar mass.

Van’t Hoff Factor and Association of Solute Molecules

Generally, non-polar solutes undergo association to form bigger molecules. For example, ethanoic acid (acetic acid) self-associates in benzene and forms dimers. This is shown below:

Van’t off factor can be used to calculate the extent of association of a non-polar solute in terms of the degree of association \((\alpha)\)

The degree of association 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 we have \(1\) mole of the solute which undergoes association as shown:

\({\rm{nA = }}{{\rm{A}}_{\rm{n}}}\)

Let the degree of association be \(\alpha\)

At equilibrium:

Number of moles of unassociated solute left after association \(=1-\alpha\)

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

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

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

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

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

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

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

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

Value of Van’t Hoff Factor

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

The inclusion of the Van’t Hoff factor modifies the equations for colligative properties as follows: 

Relative lowering of the vapour pressure of solvent \({\rm{ = }}\frac{{{\rm{P}}_{\rm{A}}^{\rm{o}}{\rm{ – }}{{\rm{P}}_{\rm{A}}}}}{{{\rm{P}}_{\rm{A}}^{\rm{o}}}}{\rm{ = i}}\frac{{{{\rm{n}}_{\rm{B}}}}}{{{{\rm{n}}_{\rm{A}}}}}\)

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

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

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

Where,\({{\rm{n}}_{\rm{B}}}\) is the number of moles of the solute and \({{\rm{n}}_{\rm{A}}}\) is the number of moles of the solvent and where \({\rm{P}}_{\rm{A}}^{\rm{^\circ }}\) is the vapour pressure of the solvent in pure form.

The Van’t Hoff factor is, therefore, a measure of a deviation from ideal behaviour. The greater the deviation from the ideal behaviour, the lower is the Van ‘t Hoff factor.  On increasing the concentration of the solute, the Van’t Hoff factor decreases. This is because ionic compounds generally do not dissociate 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, \(\mathrm{i}=1.98, \mathrm{c}=0.001 \mathrm{M}\)
\(\mathrm{BaCl}_{2} \rightleftharpoons 1 \mathrm{Ba}^{2+}+2 \mathrm{Cl}^{-}\); \({\rm{n = 3}}\)
\({\rm{\alpha = }}\frac{{{\rm{i – 1}}}}{{{\rm{n – 1}}}}\)
\(\alpha=\frac{1.98-1}{3-1}=\frac{0.98}{2}=0.49\)
% dissociation \(=\alpha \times 100=49 \%\)

Q.2. The Van’t Hoff factor of a \(0.1 \mathrm{M}\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) the solution is \(4.20\). Calculate the degree of dissociation of \(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} .\)
Ans: Given, \(\mathrm{i}=4.20, \mathrm{c}=0.1 \mathrm{M}\)
Let \(\alpha\) be the degree of dissociation.
\(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \rightleftharpoons 2 \mathrm{Al}^{3+}+3 \mathrm{SO}_{4}^{2-}\)
Before dissociation: \(1 0 0\)
After dissociation: \({\rm{1 – \alpha ,2\alpha ,3\alpha }}\)
\(\mathrm{i}=\frac{1-\alpha+2 \alpha+3 \alpha}{1}\)
\(4.20=1+4 \alpha\)
\(\alpha=\frac{3.20}{4}=0.8\)

Summary 

The presence of solute particles in a solution tends to affect the colligative properties of a solution. The solute particles either undergo self linkage or dissociate into their respective ions. The Van’t Hoff factor measures the extent of association or dissociation of solute particles in a solution. Hence, it’s important that we learn about it while dealing with solutions. 

Frequently Asked Questions

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

Q.2. What is the value of the Van’t Hoff factor?
Ans: The value of Van’t Hoff factor:

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

Q.3. How do you find the Van’t Hoff factor?
Ans: The Van’t Hoff factor’ i’ is calculated by using the following formula:
\({\rm{i = }}\frac{{{\rm{ Normal\, Molar\, Mass }}}}{{{\rm{ Abnormal\, Molar\, Mass }}}}\quad {\rm{ \ldots  \ldots  \ldots  \ldots Eqn(1)}}\)
\({\rm{i = }}\frac{{{\rm{ Observed\, Colligative\, Property }}}}{{{\rm{ Calculated\, Colligative\, Property }}}}{\rm{, \ldots  \ldots  \ldots  \ldots Eqn(2)}}\)
\({\rm{i = }}\frac{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{moles}}\,{\rm{of}}\,{\rm{particles}}\,{\rm{after}}\,{\rm{association/dissociation}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{moles}}\,{\rm{of}}\,{\rm{particles}}\,{\rm{before}}\,{\rm{association/dissociation}}}} \ldots \ldots {\rm{Eqn}}({\rm{3}})\)

Q.4. Why is the Van’t Hoff factor used?
Ans: The association or dissociation of solute particles in a solution results in a decrease or increase in the molar mass of the solute particles. This results in abnormal molar mass. Van’t Hoff’s “i” factor is used to express this extent of association or dissociation of solutes in solution.

Q.5. How do you calculate the molality of a solution using the Van’t Hoff factor?
Ans: The molality (m)of a solution and Van’t Hoff factor (i) can be calculated by using the following formula-
Elevation of Boiling point, \({\rm{\Delta }}{{\rm{T}}_{\rm{b}}}{\rm{ = i}}{{\rm{K}}_{\rm{b}}}{\rm{m}}\)
Depression of Freezing point, \({\rm{\Delta }}{{\rm{T}}_{\rm{f}}}{\rm{ = i}}{{\rm{K}}_{\rm{f}}}{\rm{m}}\)