 When a mass of 0.695 g of protein was dissolved in 42.1 mL of water at 32.8°C, the osmotic pressure was found to be 39.4 torr. What is the measured value for the molar mass of the protein? 

To find the molar mass of the protein, we can use the formula related to osmotic pressure, which is given by:

\[
\Pi = \frac{nRT}{V}
\]

Where:
- \(\Pi\) is the osmotic pressure,
- \(n\) is the number of moles of solute,
- \(R\) is the gas constant,
- \(T\) is the temperature in Kelvin,
- \(V\) is the volume of the solution in liters.

The number of moles \(n\) can be expressed as:

\[
n = \frac{\text{mass}}{\text{molar mass}} = \frac{m}{M}
\]

Substituting this into the osmotic pressure equation:

\[
\Pi = \frac{mRT}{MV}
\]

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

\[
M = \frac{mRT}{\Pi V}
\]

### Step 1: Convert the given values to appropriate units.
- Mass of protein, \(m = 0.695 \, \text{g}\)
- Volume of solution, \(V = 42.1 \, \text{mL} = 0.0421 \, \text{L}\)
- Temperature, \(T = 32.8^\circ \text{C} = 32.8 + 273.15 = 305.95 \, \text{K}\)
- Osmotic pressure, \(\Pi = 39.4 \, \text{torr}\)

The gas constant \(R\) can be used as \(R = 62.36 \, \text{L} \cdot \text{torr} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\).

### Step 2: Substitute the values into the equation.
\[
M = \frac{0.695 \times 62.36 \times 305.95}{39.4 \times 0.0421}
\]

### Step 3: Calculate the molar mass.
Let's perform the calculation:

\[
M \approx \frac{13236.71}{1.65974} \approx 7976.23 \, \text{g/mol}
\]

Thus, the molar mass of the protein is approximately **7976.23 g/mol**.

Would you like more details on this solution or have any questions?

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Here are 5 related questions:
1. How does temperature affect the osmotic pressure in this context?
2. What would be the effect of increasing the volume of the solution on the calculated molar mass?
3. How would you calculate osmotic pressure if the molar mass of the solute were known?
4. Why is it important to convert temperature to Kelvin in gas law calculations?
5. What assumptions are made when using the formula for osmotic pressure?

**Tip:** Always ensure that all units are consistent when applying formulas in physical chemistry, especially in thermodynamic calculations like this one.

Learn how to calculate the molar mass of a protein using osmotic pressure. Given data includes mass of protein, volume of solution, temperature, and osmotic pressure. Follow step-by-step calculations using the ideal gas law formula. Molar mass calculated is approximately 7976.23 g/mol.

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