The osmotic pressure of the red blood cell is 5.94 bar at 25 °C, how much (in terms of volume) of the 12% sucrose solution do you need to make 1 L of isotonic (isosmotic) solution for the red blood cell?

To prepare an isotonic (isosmotic) solution for the red blood cell, we need to match the osmotic pressure of the red blood cell solution with that of the sucrose solution. Here’s a step-by-step solution:

### Step 1: Understand the Osmotic Pressure Formula
The osmotic pressure (\(\Pi\)) for a solution is given by the formula:
\[
\Pi = i \cdot C \cdot R \cdot T
\]
where:
- \( i \) is the van't Hoff factor (1 for sucrose, as it does not dissociate in solution),
- \( C \) is the molarity of the sucrose solution,
- \( R \) is the gas constant, \( R = 0.0831 \, \text{L bar K}^{-1} \text{mol}^{-1} \),
- \( T \) is the temperature in Kelvin (25 °C = 298 K).

### Step 2: Set Osmotic Pressure to be Equal
Since we want the osmotic pressure of the sucrose solution to match that of the red blood cell, we set:
\[
5.94 \, \text{bar} = C \cdot 0.0831 \cdot 298
\]

### Step 3: Solve for the Required Molarity (\(C\))
Rearranging the formula to solve for \(C\):
\[
C = \frac{5.94}{0.0831 \cdot 298}
\]
Calculating this gives:
\[
C \approx 0.24 \, \text{mol/L}
\]

### Step 4: Determine the Required Volume of 12% Sucrose Solution
A 12% sucrose solution means there are 12 grams of sucrose per 100 mL of solution. Sucrose has a molar mass of 342 g/mol, so the molarity of the 12% sucrose solution is:
\[
\text{Molarity} = \frac{12 \, \text{g}}{100 \, \text{mL}} \times \frac{1000 \, \text{mL}}{1 \, \text{L}} \times \frac{1}{342 \, \text{g/mol}}
\]
Calculating this, we get:
\[
\text{Molarity} \approx 0.35 \, \text{mol/L}
\]

### Step 5: Use Dilution Formula to Find Volume Needed
Since we need a 0.24 M solution from a 0.35 M stock solution, we apply the dilution formula:
\[
C_1 V_1 = C_2 V_2
\]
where:
- \( C_1 = 0.35 \, \text{mol/L} \) (initial concentration),
- \( V_1 \) is the volume of the 12% solution we need to find,
- \( C_2 = 0.24 \, \text{mol/L} \) (desired concentration),
- \( V_2 = 1 \, \text{L} \).

Rearrange to solve for \( V_1 \):
\[
V_1 = \frac{C_2 \cdot V_2}{C_1} = \frac{0.24 \cdot 1}{0.35} \approx 0.686 \, \text{L}
\]

### Answer:
To prepare 1 L of an isotonic solution with the red blood cell, you need approximately **0.686 L (or 686 mL) of the 12% sucrose solution**.

Would you like further details or have any questions?

### Additional Questions:
1. How would the volume change if the temperature were different?
2. What effect does the van’t Hoff factor have on different solutes?
3. How would you adjust the solution if using a different solute with similar molarity?
4. Why does sucrose have a van’t Hoff factor of 1?
5. How does osmotic pressure relate to cell membrane permeability?

#### Tip:
When preparing solutions for biological applications, consider both osmotic pressure and pH for compatibility with cell environments.

Learn how to calculate the volume of a 12% sucrose solution needed to prepare 1 L of an isotonic solution for red blood cells. This problem involves osmotic pressure, molarity calculations, and dilution formulas, with step-by-step solutions. Suitable for Grades 11-12.

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