Lecture 10 Flashcards
what causes greater deviation from Raoult’s law as well as osmotic pressure for polymer solution?
Raoults law:
1. lower polymer concentration
2. high molar mass
Osmotic pressure:
1. low polymer concentration
How is the lattice approach used for polymer solutions?
The lattice approach for polymer solutions is very similar to the lattice approach of the regular solutions
On this lattice, Np homopolymers (polymers made with the same monomer) are placed. Each chain contains M identical segments, where each segment occupies exactly 1 site on the lattice. In total, the polymer chains occupy MNp lattice sites. furthermore, N = Ns + MNp, since each solvent molecule occupies exactly 1 lattice site.
This is illustrated on notes.
What is the derivation of mixing entropy?
Inorder to derive mixing entropy we need to first understand the steps to get there:
Suppose that there are already i polymer chains placed on an empty lattice. This means that there are N - Mi lattice sites still available. When we then add the first monomer of a new polymer to the lattice (polymer i +1), it can be placed on N - Mi lattice sites.
Since the polymers are covalently bonded we know that the second monomer can only be placed in the neighboring position of the first monomer, but it is unknown to us whether or not that position is occupied. To get past that we use the probability of the neighboring site being empty N-iM/N. This is called the mean-field approximation. From this, we know that the second moner can be placed on z(N-iM)/N sites.
Furthermore, there are (z-1)(N-iM)/N sites available for the third till Mth monomer. The Z-1 is due to one of the neighboring sites of the monomer being occupied by the monomer before it.
The rest of the very long and complicated derivation is in the notes :).
What is volume fraction? How are they related to mole fractions?
The volume fractions indicate which part of the total volume N*v (where v is the volume of each site) is occupied by the different molecules.
They are related to mole fraction via:
xs = M(ϕs)/ 1 + (m-1)ϕs, and xp = (ϕs)/ 1 + (m-1)ϕp.
Where for a mixture of monomeric types of substances (M=1 for both y3ani no chains), volume fraction equals mole fraction and the equation of mixing entropy becomes the one derived for ideal solutions. Since we are talking about polymer (M>1) ofc this is not the case, but can happen in very dilute solutions where ϕp approaches 0.
What is the expression given for mixing enthalpies of polymer solutions?
Before derivation let’s set some foundation:
- In this lattice model we assume constant volume, therefore it is not possible to describe the small volume changes, hence mH = mU and mG = mA
- when calculating mH we consider only the neighboring interactions, and we assume random mixing. This means that the solvent molecules are surrounded with z * ϕp polymer segment, and with z(ϕs) = z(1- ϕp) solvent molecules, and the polymer segments are surrounded with z* ϕp polymer segments, and with z(ϕs) = z(1- ϕs) solvent segments.
- This model doesn’t account for the fact that a polymer segment in between the two end segments is always covalently linked to two other segments, but this doesn’t have too much influence on the final results.)
The rest of the derivation is found in notes on page 3
What is the derivation for Gibbs energy of mixing for polymer solutions?
Found in notes on page 3
What is the derivation for the chemical potential of the solvent and polymer molecules?
Found in notes on page 4
What is the derivation for the vapour pressure and its series expansion?
Found in notes on page 4.
What is the derivation for the osmotic pressure of polymer solutions?
Found in notes on pages 5-6
How do we use the viral expansion of osmotic pressure to define the ideality of the polymer and the quality of the solvent?
Looking at the viral expansion of osmotic pressure it is noted that at kay = 1/2 the second viral coefficient disappears, the temp that this occurs at is termed the θ or flory temp. At this temperature the polymer behaves ideally for a broad range of concentrations.
for T > θ, kay < 1/2 we will have a good solvent
for T = θ, kay = 1/2 we will have θ solvent
for T < θ, kay > 1/2 we will have poor solevnt.
How can we obtain the second virial coefficient of the osmotic pressure and the molecular weight of the polymer using osmotic pressure measurements?
Very simply plot an osmotic pressure/ Pp against Pp where your slope = A2 and the reciprocal of the intercept gives you the Mw
How does the mH determine solubility?
The enthalpy of mixing (mH) depends on the specific molecular properties of the solvent molecules and polymer segments, therefore the value of mH determines if the polymer dissolved in the solvent or not, Where the lowest (possible) mH values, and hence the highest solubility is expected in a situation where the chemical composition and the structure of the solvent is similar to the polymer
Also, note the mG is always negative while mH is typically positive (idc how this fits but seemed important it’s 3 am I’m sorry I love you) ;)
What are the conditions required for phase separation to occur in polymer solutions?
the chemical potential of both the solvent and polymer need to be equal in both phases:
μs(ϕp’) = μs(ϕp”), μp(ϕp’) = μp(ϕp”)
and since we have related the chemical potential to osmotic pressure it follows that:
Osmotic pressure(ϕp’) =Osmotic pressure(ϕp”)
How do we know if phase separation occurs in polymer solutions and how can we find the said volume fraction of the coexisting phases?
The only way to find the volume fraction of the coexistence curves and also to find out if phase separation occurs is by graphically plotting an mG / NkbT against ϕp, where if a common tangent can be drawn between two points on the graph then phase separation occurs( since this shows equal chemical potentials).
Then we can find the volume fraction of the coexisting curves from the graph at two points the common tangent crosses!
What is the derivation for the critical (demixing) volume fraction of the polymer and kay?
found in notes on page 7
