I am trying to figure out how, for the entropy of mixing of two polymers, one gets from:

Change in entropy of mixing (delta S sub m) = k[(N1 + N2) ln (N1 + N2) - N1 ln N1 - N2 ln N2]

to the result:

change in entropy of mixing (delta S sub m) = -k(N1 ln v1 + N2 ln v2) where v1 and v2 are the volume fractions of the two polymers 1 and 2 respectively. What assumptions need to be made to go from the first equation to the second? How algebraically does one make the rearrangements necessary? Please be explicit.

Note that the first equation arises from Bolztmann's relationship:

delta S sub m (entropy of mixing) k ln (Omega) where omega is the number of possible arrangements in space that the molecules can assume. Omega ends up being equal to (No/(N1!N2!)) and then Stirling's approximation is applied to get the first equation I indicated above.

Solution Preview

Please see the attached file.

However before you look at the solution you might want to try the ...

Solution Summary

Applying the definition of partial volumes to the initial expression, with additional manipulation yields the required result.

... We calculate the entropy change of the universe involved in adiabatically mixing two equal quantities of water, initially at different temperatures. ...

... 2. What is the total entropy change for mixing 855 mL of tea @ 85⃝C with 15 mL of lemon juice at 3⃝C? Be sure to consider everything that changes. ...

... molecules have mixed? b. Compute the entropies i. S1, ii. S2 and iii. S combined. iv. What is the significance of the result? 3. a. The change in entropy is ...

... b) What is the change in entropy of the gas during the isothermal expansion? 2. 10 grams of ice at 0°C are mixed with 100 grams of water at 30°C. What is the ...

... What is the change in entropy upon solvation when a non polar molecule is mixed with water? a. Increase b. Decrease c. No significant change d. cannot predict. ...

... et.al 2007) When a compound made up of non-polar molecules is mixed with an ... This causes high disorder in the solution environment referred to as high entropy. ...

... is the entropy change caused by the heat change with the surroundings. Hence, the final volume of the gas can be found: ( 6) Numerically: (b) When mixing two ...

... Background When a compound made up of non-polar molecules is mixed with an ... This causes high disorder in the solution environment referred to as high entropy. ...

... standard entropy change is -430 J/K. Calculate the standard free energy change at 25 degrees Celcius. 13. What is the pH of a solution prepared by mixing 50.00 ...