A solution containing 500 g of a protein per litre is isotonic with a solution containing 3.42 g of sucrose per litre . The molar mass of protein is ________.`g mol ^(-1)`

Options

A. 500 g/mole
B. 14600 g/mole
C. 34200 g/mole
D. 50000 g/mole

Answer

To find the molar mass of the protein in the given isotonic solutions, we can follow these steps:

### Step 1: Understand the concept of isotonic solutions
Isotonic solutions have the same osmotic pressure. This means that the osmotic pressure of the protein solution is equal to the osmotic pressure of the sucrose solution.

### Step 2: Write the formula for osmotic pressure
The osmotic pressure (π) can be calculated using the formula:
\[ \pi = C \cdot R \cdot T \]
Where:
- \( C \) = concentration (in moles per liter)
- \( R \) = universal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin

### Step 3: Set up the equation for the two solutions
Since the two solutions are isotonic, we can equate their osmotic pressures:
\[ \pi_{\text{protein}} = \pi_{\text{sucrose}} \]
This gives us:
\[ C_{\text{protein}} \cdot R \cdot T = C_{\text{sucrose}} \cdot R \cdot T \]
Since \( R \) and \( T \) are constants, they can be canceled out:
\[ C_{\text{protein}} = C_{\text{sucrose}} \]

### Step 4: Calculate the concentration of sucrose
The concentration of sucrose can be calculated using its mass and molar mass:
- Mass of sucrose = 3.42 g
- Molar mass of sucrose (C₁₂H₂₂O₁₁) = 342 g/mol

Now, we can find the number of moles of sucrose:
\[ \text{Number of moles of sucrose} = \frac{\text{mass}}{\text{molar mass}} = \frac{3.42 \text{ g}}{342 \text{ g/mol}} = 0.01 \text{ mol} \]
Since this is for 1 liter, the concentration \( C_{\text{sucrose}} \) is:
\[ C_{\text{sucrose}} = 0.01 \text{ mol/L} \]

### Step 5: Calculate the concentration of protein
Since the concentrations are equal:
\[ C_{\text{protein}} = 0.01 \text{ mol/L} \]
The concentration of protein can also be expressed in terms of its mass and molar mass:
\[ C_{\text{protein}} = \frac{\text{mass of protein}}{\text{molar mass of protein} \times \text{volume}} = \frac{500 \text{ g}}{M \times 1 \text{ L}} \]
Where \( M \) is the molar mass of the protein.

### Step 6: Set the concentrations equal
Now we can set the two concentrations equal to each other:
\[ \frac{500 \text{ g}}{M} = 0.01 \text{ mol/L} \]

### Step 7: Solve for the molar mass of protein
Rearranging the equation gives:
\[ M = \frac{500 \text{ g}}{0.01 \text{ mol}} = 50000 \text{ g/mol} \]

### Final Answer
The molar mass of the protein is **50000 g/mol**.

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