Insulin is dissolved in suitable solvent and the osmotic pressure `(pi)` of solution of various concentration `(g//cm^(3))` C is measured at `27^(@)C`. The slope of plot of `pi` against C is found to be `4.1 xx 10^(-3)`. The molecular mass of inulin is:

Options

A. `3xx10^3`
B. `6xx10^6`
C. `3xx10^6`
D. `6xx10^3`

Answer

To find the molecular mass of insulin based on the given information, we can follow these steps:

### Step 1: Understand the relationship between osmotic pressure, concentration, and molecular mass.
The formula for osmotic pressure (\( \pi \)) is given by:
\[ \pi = \frac{n}{V}RT \]
where:
- \( n \) = number of moles of solute
- \( V \) = volume of solution in liters
- \( R \) = universal gas constant (0.082 L·atm/(K·mol))
- \( T \) = temperature in Kelvin

### Step 2: Relate concentration to osmotic pressure.
The number of moles (\( n \)) can also be expressed in terms of mass (\( m \)) and molecular mass (\( M \)):
\[ n = \frac{m}{M} \]
Thus, we can rewrite the osmotic pressure equation as:
\[ \pi = \frac{m}{MV}RT \]
The concentration \( C \) (in g/cm³) can be expressed as:
\[ C = \frac{m}{V} \]
Substituting \( m = CV \) into the osmotic pressure equation gives:
\[ \pi = \frac{CV}{MV}RT \implies \pi = \frac{CR}{M}T \]

### Step 3: Rearranging the equation to find molecular mass.
From the above equation, we can rearrange to find the molecular mass \( M \):
\[ M = \frac{CR}{\pi/T} \]
This can be simplified to:
\[ M = \frac{CRT}{\pi} \]

### Step 4: Substitute the values into the equation.
Given:
- Slope of the plot \( \frac{\pi}{C} = 4.1 \times 10^{-3} \)
- \( R = 0.082 \, \text{L·atm/(K·mol)} \)
- Temperature \( T = 27^\circ C = 300 \, \text{K} \)

We can express \( \pi \) in terms of \( C \):
\[ \pi = 4.1 \times 10^{-3} C \]

### Step 5: Substitute \( \pi \) into the molecular mass equation.
Substituting \( \pi \) into the equation for \( M \):
\[ M = \frac{C \cdot 0.082 \cdot 300}{4.1 \times 10^{-3} C} \]
The \( C \) cancels out:
\[ M = \frac{0.082 \cdot 300}{4.1 \times 10^{-3}} \]

### Step 6: Calculate the molecular mass.
Calculating the right side:
\[ M = \frac{24.6}{4.1 \times 10^{-3}} = 6000 \, \text{g/mol} \]

### Conclusion
The molecular mass of insulin is approximately \( 6000 \, \text{g/mol} \).