Volume Fraction of a Polymer

[yamborghini]

I have a tutorial question that I'm not too sure of.

The question states:

Calculate the volume fraction Φ for a polymer solution of concentration C (given in g/L) and hydrodynamic radius R?

The answer was given as
https://s.yimg.com/hd/answers/i/f9b738ccf1bf42f38089e06bf1e8035f_A.jpeg?a=answers&mr=0&x=1433625143&s=43c9115d1b188fb3b3e983b9eb00f179

Could someone explain how V total was gotten?

Am I right in assuming the Volume of Polymer would be the volume of a the sphere multiplied by the number of polymers?

[Could this be a potential error in the solutions? In my notebook I have (C / NM)*sphere Volume]

 

[_coyote_]

The volume fraction is the ratio of the volume of the polymer to the total volume of the solution. To calculate this, you will need the concentration of the polymer in the solution (C) and the hydrodynamic radius of the polymer (R). The volume of a single polymer molecule is the volume of a sphere with radius R, which is 4/3πR^3. The number of polymer molecules in the solution is then C/(N_A M), where N_A is Avogadro's constant (6.022 x 10^23 mol-1) and M is the molecular weight of the polymer. Therefore, the volume of the polymer in the solution is: V_polymer = 4/3πR^3 * C/(N_A M)The total volume of the solution is the same as the volume of the solvent (usually water). This can be calculated using the volume of a sphere with radius R, which is 4/3πR^3. Therefore, the total volume of the solution is: V_total = 4/3πR^3The volume fraction is then the ratio of the volume of the polymer to the total volume of the solution, which is V_polymer/V_total.Therefore, the volume fraction is:Φ = V_polymer/V_total = C/(N_A M)