A polymer sample has the following composition: 50 molecules of molecular mass 1000 g/mol, 100 molecules of molecular mass 2000 g/mol, and 150 molecules of molecular mass 3000 g/mo... A polymer sample has the following composition: 50 molecules of molecular mass 1000 g/mol, 100 molecules of molecular mass 2000 g/mol, and 150 molecules of molecular mass 3000 g/mol. Calculate the number and weight average molecular weight and the polydispersity index.
Understand the Problem
The question is asking to calculate the number average molecular weight, weight average molecular weight, and the polydispersity index for a polymer sample with given molecular masses and quantities. This involves using formulas from polymer chemistry related to molecular weights and distributions.
Answer
$M_n \approx 2333.33 \, \text{g/mol}, \, M_w \approx 2707.14 \, \text{g/mol}, \, PDI \approx 1.16$
Answer for screen readers
The number average molecular weight ($M_n$) is approximately $2333.33 , \text{g/mol}$, the weight average molecular weight ($M_w$) is approximately $2707.14 , \text{g/mol}$, and the polydispersity index (PDI) is approximately $1.16$.
Steps to Solve
-
Calculate Total Number of Molecules First, we need to find the total number of molecules in the sample. [ N = 50 + 100 + 150 = 300 ]
-
Calculate Number Average Molecular Weight ($M_n$) The number average molecular weight is given by the formula: [ M_n = \frac{\sum (N_i M_i)}{N} ] Where (N_i) is the number of molecules and (M_i) is the molecular mass. Calculating: [ M_n = \frac{(50 \times 1000) + (100 \times 2000) + (150 \times 3000)}{300} ] [ M_n = \frac{(50000 + 200000 + 450000)}{300} ] [ M_n = \frac{700000}{300} = 2333.33 , \text{g/mol} ]
-
Calculate Weight Average Molecular Weight ($M_w$) The weight average molecular weight is calculated using the formula: [ M_w = \frac{\sum (N_i M_i^2)}{\sum (N_i M_i)} ] Calculating: [ M_w = \frac{(50 \times 1000^2) + (100 \times 2000^2) + (150 \times 3000^2)}{(50 \times 1000) + (100 \times 2000) + (150 \times 3000)} ] Calculating the numerator: [ M_w = \frac{(50 \times 1000000) + (100 \times 4000000) + (150 \times 9000000)}{700000} ] [ = \frac{(50000000 + 400000000 + 1350000000)}{700000} = \frac{1895000000}{700000} = 2707.14 , \text{g/mol} ]
-
Calculate Polydispersity Index (PDI) The polydispersity index is calculated using: [ PDI = \frac{M_w}{M_n} ] Calculating: [ PDI = \frac{2707.14}{2333.33} \approx 1.16 ]
The number average molecular weight ($M_n$) is approximately $2333.33 , \text{g/mol}$, the weight average molecular weight ($M_w$) is approximately $2707.14 , \text{g/mol}$, and the polydispersity index (PDI) is approximately $1.16$.
More Information
The number average molecular weight gives an average based on the number of molecules, whereas the weight average accounts for the weight contribution of each molecule, reflecting the distribution of molecular sizes. The polydispersity index indicates the diversity of molecular weights in the sample; values closer to 1 suggest a uniform distribution.
Tips
- Forgetting to sum the total number of molecules before calculating averages.
- Miscalculating the individual terms in either $M_n$ or $M_w$ calculations.
- Confusing the definitions of $M_n$ and $M_w$, which may lead to incorrect interpretation of results.
AI-generated content may contain errors. Please verify critical information
