### Acid base calculations using osmotic pressure data

Problem #1: What is the Ka of a monoprotic acid in which an 0.0100 M solution exerts an osmotic pressure of 200.0 torr at 25.0 °C

Solution:

1) Determine the van 't Hoff factor:

(200.0 / 760.0) = i (0.0100) (0.08206) (298.0)

i = 1.07614 (I'll carry a few guard digits.)

2) Determine the concentrations of all three species in solution:

a) We can assign concentrations using one unknown:
[H+] = [A¯] = x
[HA] = 0.0100 - x
b) The sum of all three concentrations is 0.0107614 M (reasoning from the van 't Hoff factor). Therefore:
x + x + (0.0100 - x) = 0.0107614 M
x = 0.0007614 M
3) Calculate the Ka:
Ka = ([H+] [A¯]) / [HA]

Ka = [(0.0007614) (0.0007614)] / (0.01 - 0.0007614)

Ka = 6.27 x 10¯5

A personal note about this problem: I presented the solution to a problem very much like this to a study group of about 25 high school chemistry teachers. (There were about 50 in the month-long workshop, so two study groups.) One teacher complained about my solution, pointing to a "problem" in the van't Hoff equation. He pointed out that the liters in the molarity comes from a solution-based theory and that the liters in the gas constant came from a gas-based theory. His claim was that those two units could not cancel.

Quite frankly, I didn't know what to do. This teacher did not know that one of van 't Hoff's contributions was to apply the ideal gas equation to solutions, showing that the same principles govern both gases and solutions. My personal struggle was to not hold him up to ridicule in front of his peers. I finally said we could discuss this later and proceeded with the solution.

Here's the funny part: the problem that I was solving was from Steven Zumdahl's general chemistry textbook (the second edition, it was about 1992, if I remember correctly) and Steve was leading the study group. When I came to the board, he moved to the back of the room, and wound up laughing quietly at my discomfort. He later came over to me and "apologized" for laughing at me, but I had not taken offense.

That other teacher and I talked after the workshop, but I was unable to convince him of his error. I later saw him talking to Steve, but I don't remember if it was about my presentation.

Problem #2: A 0.0350 M aqueous nitrous acid (HNO2) solution has an osmotic pressure of 0.930 atm at 22.0 °C. Calculate the percent ionization of the acid.

Solution:

1) Determine the van 't Hoff factor:

0.930 = i (0.0350) (0.08206) (295.0)

i = 1.09764367 (I'll carry a few guard digits.)

2) Determine the concentrations of all three species in solution:

a) We can assign concentrations using one unknown:
[H+] = [A¯] = x
[HA] = 0.0100 - x
b) The sum of all three concentrations is 0.03841753 M (reasoning from the concentration times the van 't Hoff factor). Therefore:
x + x + (0.0350 - x) = 0.03841753 M
x = 0.003417528 M

3) Calculate the percent ionization:

(0.003417528 M / 0.035 M) x 100 = 9.76%

Note that the percent ionization can be found in the van 't Hoff factor (1.09764367). In fact, to the limit of my calculator display, the van 't Hoff factor contains the percent ionization.

Problem #3: An aqueous solution that is 0.015 M in acetic acid is 3.5% ionized at 25.0 °C. Calculate the osmotic pressure of this solution.

Solution:

1) Based on the 3.5%, we have the following:
[H+] = [Ac-] = (0.015) (0.035) = 0.000525 M

2) The actual concentration of all species in solution (H+, Ac- and HAc) is this:

0.000525 + 0.000525 + (0.015 - 0.000525) = 0.015525 M

3) Which leads to this van 't Hoff factor:

0.015525 / 0.015 = 1.035

4) Now, for the osmotic pressure:

π = (1.035) (0.015 mol/L) (0.08206 L atm / mol K) (298 K)

π = 0.38 atm (to two sig figs)

Did you notice the percent ionization value reflected in the van 't Hoff factor?