A polymeric sample in which`30%`molecules have a molecules mass `20,000`,40%, have `30,000` and the rest `30%` have `60,000`.The`(Mbar(n))`and `(Mbar(w))`of this sample was:

Options

A. `36,000,43,333`
B. `43,333,36000`
C. `72,000,86,666`
D. `86,666,72000`

Answer

To solve the problem of finding the number average molecular mass (\( \bar{M}_n \)) and the weight average molecular mass (\( \bar{M}_w \)) of the given polymeric sample, we can follow these steps:

### Step 1: Define the given data
We have the following molecular masses and their respective percentages:
- 30% of molecules have a molecular mass of 20,000
- 40% of molecules have a molecular mass of 30,000
- 30% of molecules have a molecular mass of 60,000

### Step 2: Calculate the number average molecular mass (\( \bar{M}_n \))
The formula for number average molecular mass is given by:
\[ \bar{M}_n = \frac{\sum (N_i \cdot M_i)}{\sum N_i} \]
Where:
- \( N_i \) = percentage of molecules (considered as number of molecules)
- \( M_i \) = molecular mass of the respective molecules

Substituting the values:
\[ \bar{M}_n = \frac{(30 \cdot 20000) + (40 \cdot 30000) + (30 \cdot 60000)}{30 + 40 + 30} \]

Calculating the numerator:
\[ = (600000) + (1200000) + (1800000) = 3600000 \]

Calculating the denominator:
\[ = 30 + 40 + 30 = 100 \]

Now, substituting back into the formula:
\[ \bar{M}_n = \frac{3600000}{100} = 36000 \]

### Step 3: Calculate the weight average molecular mass (\( \bar{M}_w \))
The formula for weight average molecular mass is given by:
\[ \bar{M}_w = \frac{\sum (N_i \cdot M_i^2)}{\sum (N_i \cdot M_i)} \]

Calculating the numerator:
\[ \bar{M}_w = \frac{(30 \cdot 20000^2) + (40 \cdot 30000^2) + (30 \cdot 60000^2)}{(30 \cdot 20000) + (40 \cdot 30000) + (30 \cdot 60000)} \]

Calculating each term:
- \( 30 \cdot 20000^2 = 30 \cdot 400000000 = 12000000000 \)
- \( 40 \cdot 30000^2 = 40 \cdot 900000000 = 36000000000 \)
- \( 30 \cdot 60000^2 = 30 \cdot 3600000000 = 108000000000 \)

Adding these:
\[ = 12000000000 + 36000000000 + 108000000000 = 216000000000 \]

Calculating the denominator (already calculated):
\[ = 3600000 \]

Now substituting back into the formula:
\[ \bar{M}_w = \frac{216000000000}{3600000} = 60000 \]

### Final Results
Thus, we have:
- \( \bar{M}_n = 36000 \)
- \( \bar{M}_w = 60000 \)

### Summary
The number average molecular mass (\( \bar{M}_n \)) is 36,000 and the weight average molecular mass (\( \bar{M}_w \)) is 60,000.