Osmometry

The first characterization methods we will address are those that can help us figure out the polymer number average molar mass. $M_n$ was one of the first characteristics of polymers we learned about in this class. Recall this fundamental relationship:

$${\overline{M}}_n={\overline{M}}_0{\overline{x}}_n$$

The number average molar mass is just a function of the molar mass of the repeat unit and how many repeat units there are (i.e., the degree of polymerization). We, of course, know the molar mass of the repeat unit, that just is a function of what polymer we are trying to make. So, how do we figure out $M_n$? Well, if we could just figure out $x_n$ that would be easy….

Now, recall this expression we learned very early on as well, that related the total number of molecules at the start and end of the reaction to degree of polymerization:

$${\overline{x}}_n=\frac{N_0}{N}$$

We can easily find $N_0$ because this is just the total number of monomers we started with. If we could just figure out $N$, the number of molecules at the end of the reaction, then this would allow us to solve for degree of polymerization. And then we could use degree of polymerization to solve for $M_n$!

It turns out, there are a number of measurable physical properties of solutions that depend primarily on the number of solute molecules per unit volume; these are called colligative properties. Examples include boiling point elevation, freezing point depression, osmotic pressure, and vapor pressure. If we can use these techniques to quantify the number of polymer molecules in solution, then we can use these to determine $x_n$ and $M_n$. We will focus on osmotic pressure, because this is the only colligative property that is most accurately measured for polymers of higher molar mass. But there is still a limit – if the polymer is too big and solvent too dilute, it’s hard to measure osmotic pressure, and if the polymer is too small, then the membrane may have a hard time keeping out the polymer solute. But those concerns aside, we will be considering osmometry in depth first!

As a quick review, recall that osmosis is the movement of molecules through a semi-permeable membrane from a region of low solute concentration to a region of high solute concentration. Osmotic pressure is then the minimum pressure that is required to prevent the flow of solvent across the membrane, and in effect is a measure of the tendency of the solution to take up solvent. Review the Wikipedia: Osmotic Pressure page for more information.

Figure 11.02: A simplified osmotic pressure experiment

Source: Figure 11.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

The difference in height between the solutions in the two sides of the chamber, $h$ is related to the osmotic pressure ($\pi$), where $g$ is acceleration due to gravity and ${\rho }_0$ is the solvent density:

$$\pi =h{\rho }_{0}g$$

And we can relate the osmotic pressure to the Flory-Huggins parameter ($\chi$) where $c$ is the polymer concentration (mass/volume) in the mixture, $V_1$ is molar volume of solvent, and ${\rho }_2$ is polymer density:

$$\frac{\pi }{c}=\frac{RT}{{\overline{M}}_n}+\left( \frac{RT}{V_1{\rho }_2^2}\right)\left( \frac{1}{2}-\chi \right)c$$

The quantity $\frac{\pi }{c}$ is called the reduced osmotic pressure.

PROBLEM

Consider the experimental data plotted below and the relationship between reduced osmotic pressure and concentration of polymer. What does the Y intercept of the plot represent?

Source: Figure 11.2 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.
(Data taken from Kamide, K. et al., Br. Polym. J. 15,91, 1983.)

1. The density of polymer
2. χ
3. $\frac{RT}{{\overline{M}}_n}$
4. The molar volume of polymer

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ANSWER

C. $\frac{RT}{{\overline{M}}_n}$

The plot has reduced osmotic pressure, $\frac{\pi }{c}$, on the Y axis and $c$ is on the X axis. Looking at the equation:

$$\frac{\pi }{c}=\frac{RT}{{\overline{M}}_n}+\left( \frac{RT}{V_1{\rho }_2^2}\right)\left( \frac{1}{2}-\chi \right)c$$

We see that the quantity $\frac{RT}{{\overline{M}}_n}$ thus corresponds to the Y intercept and $\left( \frac{RT}{V_1{\rho }_2^2}\right)\left( \frac{1}{2}-\chi \right)$ corresponds to the slope. We can use the slope to therefore measure the Flory Huggins parameter and the Y intercept to measure the number average molar mass.

PROBLEM 2

Consider the experimental data plotted below for two different sizes of polystyrene in toluene, and the relationship between osmotic pressure and concentration of polymer. Which sample has higher $M_n$?

Source: Figure 11.2 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.
(Data taken from Kamide, K. et al., Br. Polym. J. 15,91, 1983.)

1. Sample (a)
2. Sample (b)

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ANSWER 2

A. Sample (b)

We know that $\frac{RT}{{\overline{M}}_n}$ is the Y intercept. Thus, as $M_n$ increases, the intercept decreases. Sample (b) has the lower intercept, and higher $M_n$ Thus, the intercept helps us measure number average molar mass.