A solution contain 8 g of a carbohydrate in 100 g of water has a density 1.025 g/mL and an osmotic pressure of 5 atm at `27^(@) C`. What is the molar mass of the carbohydrate?

Options

A. 387
B. 374
C. 3740
D. None of these

Answer

To find the molar mass of the carbohydrate in the solution, we will follow these steps:

### Step 1: Identify the given data
- Mass of the carbohydrate (solute), W = 8 g
- Mass of water (solvent) = 100 g
- Density of the solution = 1.025 g/mL
- Osmotic pressure (π) = 5 atm
- Temperature (T) = 27°C = 300 K (after converting to Kelvin)

### Step 2: Calculate the total mass of the solution
The total mass of the solution (m_solution) is the sum of the mass of the solute and the mass of the solvent:
\[ m_{\text{solution}} = W + \text{mass of water} = 8 \, \text{g} + 100 \, \text{g} = 108 \, \text{g} \]

### Step 3: Calculate the volume of the solution
Using the density formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
We can rearrange this to find the volume (V):
\[ V = \frac{m_{\text{solution}}}{\text{Density}} = \frac{108 \, \text{g}}{1.025 \, \text{g/mL}} \]
Calculating this gives:
\[ V \approx 105.37 \, \text{mL} = 0.10537 \, \text{L} \]

### Step 4: Use the osmotic pressure formula
The osmotic pressure formula is given by:
\[ \pi = i \cdot C \cdot R \cdot T \]
For a non-electrolyte like a carbohydrate, \( i = 1 \), so:
\[ \pi = C \cdot R \cdot T \]
Where \( C \) is the concentration in moles per liter (mol/L). Rearranging gives:
\[ C = \frac{\pi}{R \cdot T} \]

### Step 5: Calculate the concentration (C)
Substituting the known values:
- \( \pi = 5 \, \text{atm} \)
- \( R = 0.0821 \, \text{L atm/(K mol)} \)
- \( T = 300 \, \text{K} \)

Calculating \( C \):
\[ C = \frac{5}{0.0821 \cdot 300} \]
\[ C \approx 0.202 \, \text{mol/L} \]

### Step 6: Calculate the number of moles of carbohydrate
Using the concentration and the volume of the solution:
\[ n = C \cdot V = 0.202 \, \text{mol/L} \cdot 0.10537 \, \text{L} \]
Calculating this gives:
\[ n \approx 0.0213 \, \text{mol} \]

### Step 7: Calculate the molar mass (M)
The molar mass is calculated using the formula:
\[ M = \frac{W}{n} \]
Substituting the values:
\[ M = \frac{8 \, \text{g}}{0.0213 \, \text{mol}} \]
Calculating this gives:
\[ M \approx 375.93 \, \text{g/mol} \]

### Conclusion
The molar mass of the carbohydrate is approximately **374 g/mol**.

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