Sure, let's walk through the step-by-step solution to calculate the molar mass of the protein given the problem's parameters.
### Given Data:
1. [tex]\( \text{Mass of protein} (m) = 3.5 \)[/tex] g
2. [tex]\( \text{Volume of solution} (V) = 0.05 \)[/tex] L
3. [tex]\( \text{Osmotic pressure} (\pi) = 0.035 \)[/tex] atm
4. [tex]\( \text{Temperature} (T) = 310 \)[/tex] K
5. [tex]\( \text{Gas constant} (R) = 0.0821 \)[/tex] L atm K[tex]\(^{-1}\)[/tex] mol[tex]\(^{-1}\)[/tex]
### Step-by-Step Calculation:
#### Step 1: Calculate the number of moles ([tex]\( n \)[/tex]) of the protein
Using the formula for osmotic pressure:
[tex]\[ \pi = \frac{n}{V} RT \][/tex]
Rearrange the formula to solve for the number of moles ([tex]\( n \)[/tex]):
[tex]\[ n = \frac{\pi V}{RT} \][/tex]
Plug in the given values:
[tex]\[ n = \frac{0.035 \text{ atm} \times 0.05 \text{ L}}{0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \times 310 \text{ K}} \][/tex]
Calculate:
[tex]\[ n = \frac{0.00175 \text{ atm L}}{25.451 \text{ L atm K}^{-1} \text{ mol}^{-1}} \][/tex]
[tex]\[ n \approx 6.875957722682803 \times 10^{-5} \text{ mol} \][/tex]
#### Step 2: Calculate the molar mass ([tex]\( M \)[/tex]) of the protein
The molar mass ([tex]\( M \)[/tex]) is the mass of the solute ([tex]\( m \)[/tex]) divided by the number of moles ([tex]\( n \)[/tex]):
[tex]\[ M = \frac{m}{n} \][/tex]
Plug in the given values:
[tex]\[ M = \frac{3.5 \text{ g}}{6.875957722682803 \times 10^{-5} \text{ mol}} \][/tex]
Calculate:
[tex]\[ M \approx 50901.99999999999 \text{ g/mol} \][/tex]
### Conclusion:
The molar mass of the protein is approximately 50,902 g/mol.
