Write the significance of poly dispersity index; also calculate the number-average molecular weight, weight-average molecular weight and degree of polymerization for the polypropylene by using given data. Molecular Weight Range (g/mol) Xi Wi 8,000-18,000 0.05 0.02 Write the significance of poly 18,000-26,000 0.16 0.10 26,000-34,000 0.24 0.20 34,000-42,000 0.28 0.30 42,000-50,000 0.20 0.27 50,000-58,000 0.07 0.11
The Correct Answer and Explanation is :
Significance of Polydispersity Index (PDI)
The Polydispersity Index (PDI) is a key parameter in polymer science that provides insight into the distribution of molecular weights in a polymer sample. It is defined as the ratio of the weight-average molecular weight ((M_w)) to the number-average molecular weight ((M_n)):
[
PDI = \frac{M_w}{M_n}
]
A PDI value of 1 indicates that the polymer sample is monodisperse, meaning all polymer chains have the same molecular weight. However, in most real-world scenarios, polymer samples are polydisperse, meaning the chains vary in size. Higher PDI values indicate a broader distribution of molecular weights. PDI is important because it influences the physical properties of the polymer, such as its viscosity, tensile strength, and processing behavior. For example, a narrow PDI (close to 1) typically yields polymers with more predictable and uniform properties, while a broader PDI (greater than 1) can lead to more varied material characteristics.
Given Data and Calculation
The data for polypropylene is provided in the form of molecular weight ranges and the corresponding (X_i) (number fraction) and (W_i) (weight fraction).
Data Table:
| Molecular Weight Range (g/mol) | (X_i) (number fraction) | (W_i) (weight fraction) |
|---|---|---|
| 8,000 – 18,000 | 0.05 | 0.02 |
| 18,000 – 26,000 | 0.16 | 0.10 |
| 26,000 – 34,000 | 0.24 | 0.20 |
| 34,000 – 42,000 | 0.28 | 0.30 |
| 42,000 – 50,000 | 0.20 | 0.27 |
| 50,000 – 58,000 | 0.07 | 0.11 |
Step 1: Number-Average Molecular Weight ((M_n))
The number-average molecular weight is calculated by the sum of the products of the number fraction (X_i) and the average molecular weight of each range (M_i), divided by the total number fraction.
[
M_n = \frac{\sum (X_i M_i)}{\sum X_i}
]
Where (M_i) is the average of the molecular weight range for each interval.
[
M_i = \frac{(M_{\text{low}} + M_{\text{high}})}{2}
]
Step 2: Weight-Average Molecular Weight ((M_w))
The weight-average molecular weight is calculated by the sum of the products of the weight fraction (W_i) and the average molecular weight of each range (M_i), divided by the total weight fraction.
[
M_w = \frac{\sum (W_i M_i)}{\sum W_i}
]
Step 3: Degree of Polymerization (DP)
The degree of polymerization is the ratio of the molecular weight of the polymer chains to the molecular weight of the monomer unit. For polypropylene, the monomer unit is the propylene monomer with a molecular weight of 42 g/mol.
[
DP = \frac{M_n}{M_{\text{monomer}}}
]
Now, let’s perform the calculations.
Calculation of (M_n)
- For each range, calculate the average molecular weight (M_i):
[
M_1 = \frac{8000 + 18000}{2} = 13,000
]
[
M_2 = \frac{18000 + 26000}{2} = 22,000
]
[
M_3 = \frac{26000 + 34000}{2} = 30,000
]
[
M_4 = \frac{34000 + 42000}{2} = 38,000
]
[
M_5 = \frac{42000 + 50000}{2} = 46,000
]
[
M_6 = \frac{50000 + 58000}{2} = 54,000
]
- Multiply (X_i) by (M_i) for each range and sum the results:
[
\sum (X_i M_i) = (0.05 \times 13000) + (0.16 \times 22000) + (0.24 \times 30000) + (0.28 \times 38000) + (0.20 \times 46000) + (0.07 \times 54000)
]
[
\sum X_i = 0.05 + 0.16 + 0.24 + 0.28 + 0.20 + 0.07 = 1
]
[
M_n = \frac{(0.05 \times 13000) + (0.16 \times 22000) + (0.24 \times 30000) + (0.28 \times 38000) + (0.20 \times 46000) + (0.07 \times 54000)}{1}
]
Calculation of (M_w)
- Multiply (W_i) by (M_i) for each range and sum the results:
[
\sum (W_i M_i) = (0.02 \times 13000) + (0.10 \times 22000) + (0.20 \times 30000) + (0.30 \times 38000) + (0.27 \times 46000) + (0.11 \times 54000)
]
[
\sum W_i = 0.02 + 0.10 + 0.20 + 0.30 + 0.27 + 0.11 = 1
]
[
M_w = \frac{(0.02 \times 13000) + (0.10 \times 22000) + (0.20 \times 30000) + (0.30 \times 38000) + (0.27 \times 46000) + (0.11 \times 54000)}{1}
]
Calculation of DP
[
DP = \frac{M_n}{42}
]
Now, I will calculate the values for (M_n), (M_w), and DP.
The calculations for the number-average molecular weight ((M_n)), weight-average molecular weight ((M_w)), and degree of polymerization (DP) are as follows:
- Number-Average Molecular Weight ((M_n)): 34,990 g/mol
- Weight-Average Molecular Weight ((M_w)): 38,220 g/mol
- Degree of Polymerization (DP): 833.1
Explanation
- (M_n) is calculated by averaging the molecular weights based on the number fraction of polymer chains in each molecular weight range. A higher (M_n) suggests a sample with a greater average chain length.
- (M_w) is a more weight-sensitive average that gives greater weight to the larger chains, making it higher than (M_n). The weight fraction (W_i) emphasizes the contribution of heavier molecules to the average molecular weight.
- Degree of Polymerization (DP) represents the average number of monomer units in a polymer chain. A higher DP indicates longer chains, which typically leads to better mechanical properties and strength in the polymer material.
Polydispersity Index (PDI)
Finally, the Polydispersity Index (PDI) can be calculated as the ratio of (M_w) to (M_n):
[
PDI = \frac{M_w}{M_n} = \frac{38,220}{34,990} \approx 1.09
]
This PDI value of 1.09 indicates a relatively narrow distribution of molecular weights, meaning the polymer sample is fairly uniform, but still exhibits some variance in chain lengths.