200 mL of an aqueous solution of a protein contains its 1.26g. The osmotic pressure of this solution at 300K is found to be `2.57 xx 10^(-3)` bar. The molar mass of protein will be `(R = 0.083 L bar mol^(-1)K^(-1))`

Options

A. `61038 g mol^(–1)`
B. `51022 g mol^(–1)`
C. `122044 g mol^(–1)`
D. `31011 g mol^(–1)`

Explanation

To find the molar mass of the protein from the given data, we can use the formula for osmotic pressure:

\[ \pi = \frac{n}{V} RT \]

Where:
- \(\pi\) = osmotic pressure
- \(n\) = number of moles of solute
- \(V\) = volume of solution in liters
- \(R\) = ideal gas constant
- \(T\) = temperature in Kelvin

We can also express the number of moles \(n\) in terms of mass and molar mass:

\[ n = \frac{W}{M} \]

Where:
- \(W\) = mass of the solute (protein)
- \(M\) = molar mass of the solute

Substituting this into the osmotic pressure equation gives:

\[ \pi = \frac{W}{MV} RT \]

Rearranging this to solve for molar mass \(M\):

\[ M = \frac{WRT}{\pi V} \]

### Step-by-Step Calculation:

1. **Identify the given values:**
- Weight of protein, \(W = 1.26 \, \text{g}\)
- Osmotic pressure, \(\pi = 2.57 \times 10^{-3} \, \text{bar}\)
- Volume, \(V = 200 \, \text{mL} = 200 \times 10^{-3} \, \text{L} = 0.200 \, \text{L}\)
- Temperature, \(T = 300 \, \text{K}\)
- Ideal gas constant, \(R = 0.083 \, \text{L bar mol}^{-1} \text{K}^{-1}\)

2. **Substitute the values into the molar mass formula:**

\[ M = \frac{(1.26 \, \text{g}) \times (0.083 \, \text{L bar mol}^{-1} \text{K}^{-1}) \times (300 \, \text{K})}{(2.57 \times 10^{-3} \, \text{bar}) \times (0.200 \, \text{L})} \]

3. **Calculate the numerator:**

\[ 1.26 \times 0.083 \times 300 = 31.398 \, \text{g L bar mol}^{-1} \]

4. **Calculate the denominator:**

\[ 2.57 \times 10^{-3} \times 0.200 = 5.14 \times 10^{-4} \, \text{bar L} \]

5. **Now divide the numerator by the denominator:**

\[ M = \frac{31.398}{5.14 \times 10^{-4}} \approx 61038.5 \, \text{g mol}^{-1} \]

6. **Final result:**

The molar mass of the protein is approximately \(61038 \, \text{g mol}^{-1}\).

### Conclusion:

The molar mass of the protein is \(61038 \, \text{g mol}^{-1}\).

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