Art what temperature `(s)` a `5%` solution `(w//V)` of glucose will develop an osmotic pressure of `7 atm`?

Options

A. `273 K`
B. `306.94 K`
C. `33.94^(@)C`
D. `33.94 K`

Answer

To find the temperature at which a 5% (w/V) solution of glucose develops an osmotic pressure of 7 atm, we can use the formula for osmotic pressure:

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

Where:
- \(\pi\) = osmotic pressure (in atm)
- \(n\) = number of moles of solute
- \(R\) = ideal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature in Kelvin
- \(V\) = volume of the solution in liters

### Step 1: Determine the values
1. **Osmotic Pressure (\(\pi\))**: Given as 7 atm.
2. **Weight of Glucose**: A 5% (w/V) solution means there are 5 grams of glucose in 100 mL of solution.
3. **Volume (V)**: Convert 100 mL to liters:
\[ V = 100 \text{ mL} = 0.1 \text{ L} \]
4. **Molecular Weight of Glucose**: The molecular weight of glucose (C6H12O6) is 180 g/mol.

### Step 2: Calculate the number of moles of glucose (n)
Using the formula:
\[ n = \frac{\text{mass of solute (g)}}{\text{molar mass (g/mol)}} \]
Substituting the values:
\[ n = \frac{5 \text{ g}}{180 \text{ g/mol}} = \frac{5}{180} \approx 0.02778 \text{ mol} \]

### Step 3: Substitute the values into the osmotic pressure formula
Rearranging the formula to solve for \(T\):
\[ T = \frac{\pi V}{nR} \]
Substituting the known values:
\[ T = \frac{7 \text{ atm} \times 0.1 \text{ L}}{0.02778 \text{ mol} \times 0.0821 \text{ L·atm/(K·mol)}} \]

### Step 4: Calculate the temperature
Calculating the numerator:
\[ 7 \times 0.1 = 0.7 \]
Calculating the denominator:
\[ 0.02778 \times 0.0821 \approx 0.002283 \]
Now substituting back into the equation:
\[ T = \frac{0.7}{0.002283} \approx 306.94 \text{ K} \]

### Step 5: Convert Kelvin to Celsius
To convert from Kelvin to Celsius:
\[ T(°C) = T(K) - 273 \]
Substituting the value:
\[ T(°C) = 306.94 - 273 \approx 33.94 °C \]

### Final Answer
The temperature at which a 5% solution of glucose will develop an osmotic pressure of 7 atm is approximately **306.94 K** or **33.94 °C**.

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