Osmosis Equation

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The osmosis equation is:

π = iMRT

π is not equal to 3.14159 in this situation. π stands for the osmotic pressure and is usually expressed in the pressure unit of atmospheres.

The definition of osmotic pressure: the amount of pressure required to stop the process of osmosis in your experimental set-up.

The lowercase letter "i" is called the van 't Hoff factor and it will be dealt with in the problems below. It is named for Jacobus Henricus van 't Hoff (Henry to his friends), who applied PV = nRT to solutions and figured out why "i" was needed and what it represents. The image just to the right is a 23K GIF of him.

He was awarded the first Nobel prize in chemistry in 1901 and the ChemTeam thinks this is the official portrait selected from the many pictures taken at the photo session. Love that hair! From the late 1870's to the turn of the century, van 't Hoff was one of the premier chemists in the world.

M is molarity: good old moles per liter.

R is the gas constant and we will be using the same value as in the gas laws unit: 0.08206 L atm/mol K. Now, you may ask what a "gas" constant is doing in a discussion of solutions. Well, for one thing it's called the "gas" constant because it was discovered in the course of research on gases.

Also, van 't Hoff's insight was to see that PV = nRT applied to molecules of solute moving though the solvent. (There is an article called How the Theory of Solutions Arose, which is about his insight. It is in the Classic Papers section of the ChemTeam.) In essence, the molecules of solute are a "gas," dispersed through the "universe" of solvent molecules. If I were to move V to the right side, I would get:

P = (n / V)RT

(n / V) is moles divided by liters and that is molarity.

T is temperature, measured as usual in Kelvins.

What is the osmotic pressure of a 1.00 M solution of sucrose at 25°C?

When we insert into the equation, we have:

π = i (1.00 mol/L) (0.08206 L atm / mol K) (298 K)

However, there are two unknowns: π, the one we want and i. What is i?

Once again, i is called the van 't Hoff factor.

The van 't Hoff factor is a unitless, empirical constant related to the degree of dissociation of the solute.


OK, OK. The value is unitless. That means it is just a number like 1 or 2. Empirical means we must determine it by experiment. You can predict what a theoretical value for i might be, but the real value is only found in an experiment. The explanation follow shortly as to why.

The key is "degree of dissociation." This refers to the fact that some molecules ionize in solution (they split into their positive and negative ions) and other do not. This idea was put forth by Svante Arrhenius in 1884 in his Ph.D. dissertation and it was soundly rejected. (Arrhenius on electrolytic dissociation links to an excerpt from his article which announced this concept to the world.) In 1903, he was awarded the Nobel Prize in chemistry for it. Today, it's part of the common ordinary high school chemistry curriculum.

The van 't Hoff factor for sucrose is 1, since sucrose does not ionize in solution. It remains as whole molecules.

So the answer is 24.4 atm.

What is the osmotic pressure (at 25°C) of seawater? It contains approximately 35.0 grams of NaCl per liter. (Seawater contains other stuff, but we'll ignore it.)

Convert grams to moles:

35.0 g/L ÷ 58.443 g/mol = 0.599 mol/L

Now, plug into the equation:

π = (i) (0.599 mol/L) (0.08206 L atm / mol K) (298 K)

There's that pesky van 't Hoff factor. What is its value for NaCl?

When NaCl ionizes in solution it produces Na+ ions and Cl¯ ions. One mole of NaCl produces 1 mole of each type of ion. So the van 't Hoff factor is, theoretically, equal to 2. However, we will use 1.8 and I'll explain that in a moment.

So, plug again and then solve:

π = (1.8) (0.599 mol/L) (0.08206 L atm / mol K) (298 K)

π = 26.4 atm

Why did I use 1.8 for the van 't Hoff factor for NaCl rather than 2?

This has to do with a concept called ion pairing. In solution, a certain number of Na+ ions and Cl¯ ions will randomly come together and form NaCl ion pairs. This reduces the total number of particles in solution, hereby reducing the van 't Hoff factor.

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