Thermodynamics and Mixing in Polymer Solutions

Thermodynamics and Mixing in Polymer Solutions jls164 Wed, 02/26/2014 - 12:24

Now that you have a sense of the various conformations that a polymer may assume in solution, let’s consider the thermodynamics of that mixing process. Will the polymer actually dissolve in the solvent? How do the solvent-polymer interactions affect the chain conformations? What about polymer-polymer interactions?

We begin to think about the thermodynamics of polymers in solution by starting with a far simpler solution, an ideal solution. This is a simplistic way of thinking of a solution, in which we mix a solute and solvent, but every molecule of solvent and solute is exactly the same in terms of size and intermolecular interactions. This is of course very idealistic and not representative of real solutions, but it’s a good place to start in terms of building a framework for thinking about enthalpy and entropy of mixing.

diagram of an ideal solution as described in the text

Figure 10.1: Ideal solutions

Source: Lauren Zarzar

Brief review of fundamental thermodynamics and free energy:

In previous courses, you have been introduced to the concepts of enthalpy (H) and entropy (S). Enthalpy is a measure of the potential to take up or give off energy in the form of heat, and entropy is a measure of disorder. Both properties are directly related to Gibbs free energy according to the following equation, in which $T$ is the temperature in Kelvin:

$G = H - T \cdot S$

Enthalpy and entropy, like Gibbs free energy, describe properties of a system at a given state. Like Gibbs free energy, the absolute amount of entropy or enthalpy in a system is not directly measurable; instead, we typically discuss changes in these values as a system moves from one state to another:

$Δ G = Δ H - T \cdot Δ S$

If a process is spontaneous, then $Δ G < 0$ and if it is non-spontaneous, then $Δ G > 0$ .

PROBLEM 1

What is true of the enthalpy of mixing, $Δ H_{m,}$ for an ideal solution?

$Δ H_{m} > 0$
$Δ H_{m} < 0$
$Δ H_{m} = 0$

Click here for the answer to Problem 1.

ANSWER 1

C. $Δ H_{m} = 0$

All intermolecular interactions are equivalent, thus there can be no change in enthalpy upon mixing.

PROBLEM 2

What is true of the entropy of mixing, $Δ S_{m,}$ for an ideal solution?

$Δ S_{m} > 0$
$Δ S_{m} < 0$
$Δ S_{m} = 0$

Click here for the answer to Problem 2.

ANSWER 2

A. $Δ S_{m} > 0$

Entropy is a measure of disorder. If we go from two pure substances (solvent and solute) and mix them together, there will be more disorder, as now we have to account for all of the different arrangements of the solvent and solute in space.

PROBLEM 3

What is true of the Gibbs free energy of mixing, $Δ G_{m,}$ for an ideal solution?

$Δ G_{m} > 0$
$Δ G_{m} < 0$
$Δ G_{m} = 0$

Click here for the answer to Problem 3.

ANSWER 3

B. $Δ G_{m} < 0$

If $Δ H_{m} = 0$ and $Δ S_{m} > 0$ , then by using $Δ G = Δ H - T \cdot Δ S$ , we know $Δ G$ must be negative and mixing for an ideal solution will always be spontaneous.

So we know that entropy increases in an ideal solution, and this is what drives the spontaneous mixing. But what is the entropy exactly? In our simple model, imagine an array like that pictured (figure 10.2) in which you place the solvent and solute molecules in various spaces. How many different distinguishable arrangements of the solvent and solute molecules in this array can there be?

Enter image and alt text here. No sizes!

Figure 10.2: Simple model of an ideal solution

Source: Lauren Zarzar

Entropy is related to the “ $t o t a l # a r r a n g e m e n t s$ ”, where $N_{1}$ and $N_{2}$ are the numbers of solvent and solute molecules:

$t o t a l # a r r a n g e m e n t s = Ω = \frac{(n_{1} + n_{2})!}{N_{1}! N_{2}!}$

Entropy is related to $Ω$ through the Boltzmann equation, where $S$ is entropy and $k$ is the Boltzmann constant:

$S = k \ln Ω$

If we substitute the expression for $Ω$ into the Boltzmann equation, we get $Δ S_{m}$ :

$Δ S_{m} = R [n_{1} \ln X_{1} + n_{2} \ln X_{2}]$

…where have now switched from using “number of molecules” $N_{1}$ and $N_{2}$ to using mole fractions $X_{1}$ and $X_{2}$ (i.e. $X_{1} = N_{1} / (N_{1} + N_{2})$ and $X_{2} = N_{2} / (N_{1} + N_{2})$ ). Notice the switch from $k$ to $R$ (the gas constant) as well; the use of $k$ vs $R$ depends on whether you are dealing with individual molecules or moles of molecules, because $R = k N_{A}$ where $N_{A}$ is Avogadro’s number. (Usually we deal with moles, and we have introduced mole fractions into our expression for entropy, thus we are using $R$ ).

Because we know that $Δ H_{m} = 0$ in an ideal solution, we can also write an expression for the change in free energy:

$Δ G_{m} = R T [n_{1} \ln X_{1} + n_{2} \ln X_{2}]$

Now we have a good understanding of the thermodynamics of ideal solutions. Do polymer solutions behave like ideal solutions? Usually, they do not! In that case, what’s wrong with the theory of ideal solutions? There are two major flaws in the assumptions made for an ideal solution: 1) the assumption that all solvent and solute molecules are the same size is very wrong, especially in the case of polymer solutions because polymers are very large compared to solvent; 2) intermolecular interactions do occur, and they are usually different between solvent-solvent, solute-solute, and solute-solvent. After all, what if you choose a “bad” solvent for the polymer, like trying to dissolve polyethylene (a hydrophobic polymer) in water? Well, we know that won’t work. So intermolecular interactions are definitely important in determining whether two components will mix!