# The van’t Hoff Factor – Definition and How to Calculate It The van’t Hoff factor is a measure of the number of particles a solute forms in solution. (Anne Helmenstine)

The van’t Hoff factor (i) is the number of moles of particles formed in solution per mole of solute. It is a property of the solute and does not depend on concentration for an ideal solution. However, the van’t Hoff factor of a real solution may be lower than the calculated value for a real solution at high concentration values or when the solute ions associate with one another. The van’t Hoff factor is a positive number, but it isn’t always an integer value. It is equal to 1 for a solute that does not dissociate into ions, greater than 1 for most salts and acids, and less than 1 for solutes that form associations when dissolved.

The van’t Hoff factor applies to colligative properties and appears in the formulas for osmotic pressure, vapor pressure, freezing point depression, and boiling point elevation. The factor is named for Dutch chemist Jacobus Henricus van’t Hoff, a founder of the field of physical chemistry and the first winner of the Nobel Prize in Chemistry.

### van’t Hoff Factor Formula

There are a few different ways to write the formula to calculate the van’t Hoff factor. The most common equation is:

i = moles of particles in solution / moles dissolved solute

Because solutes don’t always fully dissociate in solution, there is another relation that is often used:

i = 1 + α(n – 1)

Here, α is the fraction of solute particles that dissociate in n number of ions.

### How to Find the van’t Hoff Factor

You can follow general rules to predict the ideal van’t Hoff factor:

#### Nonelectrolytes

For nonelectrolytes, the van’t Hoff factor is 1. Examples of nonelectrolytes include sucrose, glucose, sugars, and fats. Nonelectrolytes dissolve in water, but do not dissociate. For example:

sucrose(s) → sucrose (aq); i = 1 (one sucrose molecule)

#### Strong Electrolytes

For strong electrolytes, the ideal van’t Hoff factor is greater than 1 and equal to the number of ions formed in aqueous solution. Strong acids, strong bases, and salts are strong electrolytes. For example:

NaCl(s) → Na+(aq) + Cl(aq); i=2 (one Na+ plus one Cl)
CaCl2(s) → Ca2+(aq) + 2Cl(aq); i=3 (one Ca2+ plus two Cl)
Fe2(SO4)3(s) → 2Fe3+(aq) + 3SO42-(aq); i=5

Take care, however, because solubility affects measured van’t Hoff factor values. For example strontium hydroxide [Sr(OH)2] is a strong base that fully dissociates into its ions, but is has a low solubility in water. You might predict the van’t Hoff factor to be 3 (Sr2+, OH, OH), but the experimental value will be lower. Also, the van’t Hoff factor for concentrated solutions in always slightly lower than the value for an ideal solution.

#### Weak Electrolytes

Weak electrolytes do not fully dissociate in water, so the van’t Hoff factor won’t be the same as the number of ions formed. You’ll need to set up an ICE table (Initial, Change, Equilibrium) to determine the concentration of reactants and products and use the formula to calculate the van’t Hoff factor. Another way to find the van’t Hoff factor is to measure osmotic pressure, plug it into the van’t Hoff formula, and solve for i.

#### Solutes With Low Solubility

For any solute with low solubility, you can often use i=1 as a close approximation to the true value.

### Table of van’t Hoff Factor Values

For solutes that dissolve in water, the van’t Hoff factor is 1. For strong acids and soluble salts, the ideal value is a close approximation to the measured value in dilute solutions. But, ion pairing occurs to some extend in all electrolyte solutions, making the measured value slightly lower than the idea value. The deviation is greatest for solutes with multiple charges. Ideally, the van’t Hoff factor is a property of the solute, but the measured value may depend on the solvent. For example, carboxylic acids (e.g., benzoic acid and acetic acid) form dimers in benzene, resulting in van’t Hoff factor values less than 1.

### References

• Atkins, Peter W.; de Paula, Julio (2010). Physical Chemistry (9th ed.). Oxford University Press. ISBN 978-0-19-954337-3.
• Chisholm, Hugh, ed. (1911). “van’t Hoff, Jacobus Hendricus” . Encyclopædia Britannica (11th ed.). Cambridge University Press.
• Lewis, Gilbert Newton (1908). “The Osmotic Pressure of Concentrated Solutions and the Laws of the Perfect Solution”. Journal of the American Chemical Society. 30 (5): 668–683. doi:10.1021/ja01947a002
• McQuarrie, Donald, et al. (2011). “Colligative properties of Solutions”. General Chemistry. Mill Valley: Library of Congress. ISBN 978-1-89138-960-3.
• Voet, Donald; Judith Aadil; Charlotte W. Pratt (2001). Fundamentals of Biochemistry. New York: Wiley. ISBN 978-0-471-41759-0.

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