In the molten polymer tube model, how do I work out the volume of a tube?

[jeebs]

I've got this problem where I've got a molten polymer (polyisoprene) in a region where I can imagine a cubic lattice. I am told that it has a density of 910 kg/m^3 and an entanglement molecular weight of 3.7 kg/mol.

earlier in the problem I have to figure out the molecular weight of an isoprene monomer, which is 68 (relative molecular mass) or 1.14x10^-25kg (normal mass). Also, I had to imagine each monomer sitting on a cubic lattice point, and calculate the lattice spacing. Using the density and the monomer mass, and taking a 1m x 1m x 1m volume of this molten stuff, I worked out the total number of monomers within this volume. I then cube-rooted that to find out the number of monomers stretching along one side of this imaginary cube of side length 1m, and divided 1m by this number to get the distance spanned by 1 monomer, which is the lattice spacing.

Now onto where I get stuck. I need to find the number of these cubic lattice points that would be found within the tube corresponding to one molten polyisoprene chain of mass 74kg/mol.

So, I've started by saying that this molecule will have a mass of 74kg / 6.02x10^23 = 1.23x10^-22 kg, because I know the mass of 1 mole of this molecule, so I divided by the avogadro number to get the mass of the chain.
Using the mass per monomer, I found that the chain has N = 1.23x10^-22 kg / 1.14x10^-25 kg = 1078 monomers in it.
Also, apparently molten polymer chains have a "random walk" shape, where the average end to end distance is given by L = a(N^0.5) where a is the length of the monomer or the lattice spacing.

I'm really not sure where to go from here though, because I need to work out the volume that this chain occupies. I could then use the number density of cubic lattice points since I know their separation and I would have my answer. Finding the tube volume is proving difficult though.

Has anyone got any suggestions?
Could it be a valid suggestion to assume the tube has a circular cross-sectional diameter equal to the monomer length? If so, that would solve this problem...
Thanks.

 

[AndySh]

I have the same problem! Can anyone help?

 

[Chestermiller]

I have the same problem! Can anyone help?

If it's laid out in a cubic lattice, you should be able to figure out the equivalent cross sectional area of a monomer unit geometrically.

Chet

 

[AndySh]

https://mail.google.com/mail/u/0/?ui=2&ik=f008be4e61&view=fimg&th=14a0207ebab747d7&attid=0.1&disp=inline&realattid=85ef658de2583b00_0.1&safe=1&attbid=ANGjdJ83mELN7FV0fVAYLCBHdeTOkCTXSvgcQ79kFceQ0d9T8HFoWNkFEN8LaY_2zoxcAJ4Kc0Hkqh0QD1HDIKNnh7mBxVwNlbm9H7XqMFIFaWwgg3P009KNMKTWs4Q&ats=1417373297883&rm=14a0207ebab747d7&zw&sz=w1342-h480 Hi Chet, does this look like what you suggested? If I go about the method already stated whereby I find the volume of the tube as a fraction of the volume of the lattice, I get an answer about 10x less than the number of monomers on the respective chain which just doesn't seem right. The formula I used for the volume is pi*a^2 (circle area) * a*sqrt(N) (random walk length). The fact that the random walk length is shorter than when the tube is stretched out doesn't quite make sense to me either. Thanks.

 

[AndySh]

a has already been determined

 

[Chestermiller]

https://mail.google.com/mail/u/0/?ui=2&ik=f008be4e61&view=fimg&th=14a0207ebab747d7&attid=0.1&disp=inline&realattid=85ef658de2583b00_0.1&safe=1&attbid=ANGjdJ83mELN7FV0fVAYLCBHdeTOkCTXSvgcQ79kFceQ0d9T8HFoWNkFEN8LaY_2zoxcAJ4Kc0Hkqh0QD1HDIKNnh7mBxVwNlbm9H7XqMFIFaWwgg3P009KNMKTWs4Q&ats=1417373297883&rm=14a0207ebab747d7&zw&sz=w1342-h480 Hi Chet, does this look like what you suggested? If I go about the method already stated whereby I find the volume of the tube as a fraction of the volume of the lattice, I get an answer about 10x less than the number of monomers on the respective chain which just doesn't seem right. The formula I used for the volume is pi*a^2 (circle area) * a*sqrt(N) (random walk length). The fact that the random walk length is shorter than when the tube is stretched out doesn't quite make sense to me either. Thanks.

To get the volume occupied by the chain, you should be using the stretched out length. Also, if it's on a cubic lattice, I think the cross sectional area assigned to a chain should be the area of a cube wall s². I take it that you are assuming that the entire lattice is filled.

Chet

 

[AndySh]

Thanks that helped me out! I found that the cross sectional area should be pi*(R/2)^2 where R is the diameter of the tube given by a*sqrt(N) where N in this case is the number of monomers associated with a chain at the entanglement length.