A solution prepared by dissolving 8.95 mg of a gene fragment in 35.0 mL
of water has an osmotic pressure of 0.335 torr at `25^(@)C`.
Assuming that the gene fragment is a non-electrolyte, calculate its molar mass.

Explanation

To calculate the molar mass of the gene fragment using the given osmotic pressure, we can follow these steps:

### Step 1: Convert the mass of the gene fragment to grams
The mass of the gene fragment is given as 8.95 mg. We need to convert this to grams:
\[ \text{Mass in grams} = 8.95 \, \text{mg} \times \frac{1 \, \text{g}}{1000 \, \text{mg}} = 0.00895 \, \text{g} \]

### Step 2: Convert the volume of the solution to liters
The volume of the solution is given as 35.0 mL. We need to convert this to liters:
\[ \text{Volume in liters} = 35.0 \, \text{mL} \times \frac{1 \, \text{L}}{1000 \, \text{mL}} = 0.0350 \, \text{L} \]

### Step 3: Use the osmotic pressure formula
The formula for osmotic pressure (\(\Pi\)) is given by:
\[ \Pi = C \cdot R \cdot T \]
Where:
- \(\Pi\) = osmotic pressure (in atm)
- \(C\) = molarity (moles of solute per liter of solution)
- \(R\) = ideal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature in Kelvin (25°C = 298 K)

First, we need to convert the osmotic pressure from torr to atm:
\[ \Pi = 0.335 \, \text{torr} \times \frac{1 \, \text{atm}}{760 \, \text{torr}} = 0.00044079 \, \text{atm} \]

### Step 4: Rearrange the formula to find molarity (C)
Rearranging the formula gives us:
\[ C = \frac{\Pi}{R \cdot T} \]
Substituting the values:
\[ C = \frac{0.00044079 \, \text{atm}}{0.0821 \, \text{L·atm/(K·mol)} \cdot 298 \, \text{K}} \approx 0.000018 \, \text{mol/L} \]

### Step 5: Calculate the number of moles of the gene fragment
Using the molarity and the volume of the solution:
\[ \text{Moles} = C \cdot \text{Volume in L} = 0.000018 \, \text{mol/L} \times 0.0350 \, \text{L} \approx 0.00000063 \, \text{mol} \]

### Step 6: Calculate the molar mass
The molar mass (M) can be calculated using the formula:
\[ M = \frac{\text{mass in grams}}{\text{number of moles}} \]
Substituting the values:
\[ M = \frac{0.00895 \, \text{g}}{0.00000063 \, \text{mol}} \approx 14193 \, \text{g/mol} \]

### Final Answer
The molar mass of the gene fragment is approximately **14193 g/mol**.

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