Problem 81
Question
The number average molecular mass and mass average molecular mass of a polymer are respectively 30,000 and \(40,000 .\) The poly dispersity index of the polymer is (a) \(<1\) (b) \(>1\) (c) 1 (d) 0
Step-by-Step Solution
Verified Answer
The poly dispersity index of the polymer is (b) >1.
1Step 1: Understanding the Dispersity Index
The Polydispersity Index (PDI) is a measure of the distribution of molecular mass in a given polymer sample. It is calculated as the ratio of the mass average molecular mass (\(M_w\)) to the number average molecular mass (\(M_n\)). The formula for PDI is: \[ \text{PDI} = \frac{M_w}{M_n} \]
2Step 2: Calculating the Polydispersity Index
Given that \(M_n = 30,000\) and \(M_w = 40,000\), substitute these values into the PDI formula to calculate: \[ \text{PDI} = \frac{40,000}{30,000} = \frac{4}{3} = 1.333\]
3Step 3: Analyzing the PDI Value
Since the calculated PDI is \(1.333\), which is greater than 1, we can determine that the polydispersity index indicates a wide distribution of molecular weights.
4Step 4: Choosing the Correct Option
From the options provided, the correct answer is the one that states the PDI is greater than 1. Therefore, the correct option is (b) \(>1\).
Key Concepts
Molecular Mass DistributionMass Average Molecular MassNumber Average Molecular Mass
Molecular Mass Distribution
Molecular mass distribution in polymers refers to the range of molecular weights present in a polymer sample. Imagine it like a classroom full of students where each student represents a different chain length of the polymer. Some chains are long and others are short. The distribution gives us an idea of whether most chains are similar in size or if there's a wide variety.
A narrow distribution means most of the molecules have similar sizes, much like a class where students are of almost the same height. A wide distribution, however, indicates diversity in molecular sizes, like a class with a mix of short and tall students.
Polydispersity Index (PDI) comes into play here. It is a numerical representation of this distribution. A PDI of 1 means all polymer chains are equal in size, which is exceedingly rare. More commonly, polymers have a PDI greater than 1, suggesting variation in chain lengths. This variability affects the physical properties of the polymer, such as strength, toughness and flexibility.
A narrow distribution means most of the molecules have similar sizes, much like a class where students are of almost the same height. A wide distribution, however, indicates diversity in molecular sizes, like a class with a mix of short and tall students.
Polydispersity Index (PDI) comes into play here. It is a numerical representation of this distribution. A PDI of 1 means all polymer chains are equal in size, which is exceedingly rare. More commonly, polymers have a PDI greater than 1, suggesting variation in chain lengths. This variability affects the physical properties of the polymer, such as strength, toughness and flexibility.
Mass Average Molecular Mass
The mass average molecular mass, noted as \( M_w \), focuses on the weight contribution of each polymer molecule in a sample. It's like weighing students in a class based on their contributions in group activities. More active students (heavier molecules) influence the result more than the quieter ones.
To calculate \( M_w \), you multiply each molecular weight by its concentration and then divide by the total concentration of all molecules. This calculation gives you an average that accounts for the weightier presence of heavier molecules compared to lighter ones.
A higher \( M_w \) indicates that the sample contains very heavy molecules, which could influence the polymer's properties significantly, such as defining the melting and mechanical behavior. It's an essential measurement, especially for industries interested in creating strong and durable plastics.
To calculate \( M_w \), you multiply each molecular weight by its concentration and then divide by the total concentration of all molecules. This calculation gives you an average that accounts for the weightier presence of heavier molecules compared to lighter ones.
A higher \( M_w \) indicates that the sample contains very heavy molecules, which could influence the polymer's properties significantly, such as defining the melting and mechanical behavior. It's an essential measurement, especially for industries interested in creating strong and durable plastics.
Number Average Molecular Mass
The number average molecular mass, represented by \( M_n \), is slightly different from \( M_w \), as it treats each molecule as holding the same importance, regardless of its size. You could think of it like counting the number of students in a class without considering each student's skill level in a particular subject.
To find \( M_n \), you add up the molecular weights of all polymer molecules and divide by the total number of molecules. It's a simpler average compared to \( M_w \), focusing on how many of each type of molecule is present rather than their overall weight.
This measurement is crucial when a uniform product quality is desired since it describes the overall molecular composition in terms of the number of molecules and provides insights into the synthesis process. Understanding \( M_n \) helps in controlling the polymerization process to achieve desired characteristics for specific applications.
To find \( M_n \), you add up the molecular weights of all polymer molecules and divide by the total number of molecules. It's a simpler average compared to \( M_w \), focusing on how many of each type of molecule is present rather than their overall weight.
This measurement is crucial when a uniform product quality is desired since it describes the overall molecular composition in terms of the number of molecules and provides insights into the synthesis process. Understanding \( M_n \) helps in controlling the polymerization process to achieve desired characteristics for specific applications.
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