Understanding Osmotic Pressure: Equation and Explanation

[john16O]

can anyone explain osmotic pressure along with its equation because I am having a hard time with this concept. Thank you!

 

[Andy Resnick]

Osmotic pressure is a way of thinking about the effect of solutes. It's not much different from hydrostatic pressure- given a semi-permeable membrane, one can interchange one for the other, and compensate for one gradient with another.

As it happens, I'm currently working through "Water Movement Through Lipid Bilayers, Pores, and Plasma Membranes" by Alan Finkelstein (1987).

What equation are you referring to?

 

[john16O]

(pie)=nRT --> van't Hoff equation

 

[Andy Resnick]

Oh, ok. Here goes... hoping my tex skills are up to the challenge:

First, let's start by writing down the chemical potential of water: this is the amount of energy required to add a molecule (or a mole) of water to a solution:

\mu_{W} =\mu^{(0)}_{W} + RT ln X_{W} + P\overline{V_{W}}

Where \mu^{(0)}_{W} is the chemical potential defined at STP, X_{W} the mole fraction of water, \overline{V_{W}} the partial molar volume, R the gas constant, P the hydrostatic pressure, T the temperature.

If we have two compartments separated by a water-permeable membrane, such that one has solute and the other does not, both compartments are at internal equilbrium, then \mu(1)_{W} = \mu(2)_{W}. Substituting that big expression about for \mu_{W}, with the knowledge that X_W(2) is 1 (pure water) and that a hydrostatic pressure difference must exist to oppose the flow of water across the membrane, we get

\Pi\equiv[P(1) - P(2)] = -\frac{RT}{\overline{V_{W}}} lnX_{W}(1)

Now, for a dilute solution ln(X) = X, and doing a few other manipulations of X into n you end up with the van't Hoff expression.

How's that?

Edit: oops, made an error in tex formatting.

 

[john16O]

wow!, thank you so much, you have defiantly cleared it up for me. thank you again!