To solve the problem, we need to calculate the height of the solution that will represent the osmotic pressure generated by dissolving a polymer in the solution. Here’s a step-by-step breakdown of the solution:
### Step 1: Calculate the number of moles of the polymer
Given:
- Mass of polymer = 5 g
- Molecular mass of polymer = 50 kg/mol = 50000 g/mol
Using the formula for moles:
\[
\text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{5 \, \text{g}}{50000 \, \text{g/mol}} = 0.0001 \, \text{mol}
\]
### Step 2: Calculate the concentration of the solution
The volume of the solution is given as 1 dm³, which is equivalent to 0.001 m³. The concentration (C) in moles per cubic meter (mol/m³) is calculated as:
\[
C = \frac{\text{Number of moles}}{\text{Volume in m}^3} = \frac{0.0001 \, \text{mol}}{0.001 \, \text{m}^3} = 0.1 \, \text{mol/m}^3
\]
### Step 3: Calculate the osmotic pressure
The formula for osmotic pressure (\(\Pi\)) is given by:
\[
\Pi = C \cdot R \cdot T
\]
Where:
- \(R = 8.314 \, \text{J/(mol K)}\)
- \(T = 300 \, \text{K}\)
Substituting the values:
\[
\Pi = 0.1 \, \text{mol/m}^3 \cdot 8.314 \, \text{J/(mol K)} \cdot 300 \, \text{K} = 249.42 \, \text{Pa} \, (\text{or N/m}^2)
\]
### Step 4: Convert the density of the solution
The density of the solution is given as 0.96 kg/dm³. To convert this to kg/m³:
\[
\text{Density} = 0.96 \, \text{kg/dm}^3 = 0.96 \times 1000 \, \text{kg/m}^3 = 960 \, \text{kg/m}^3
\]
### Step 5: Use the hydrostatic pressure formula
The hydrostatic pressure formula is given by:
\[
\Pi = \rho g h
\]
Where:
- \(\rho = 960 \, \text{kg/m}^3\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(h = ?\)
Rearranging for \(h\):
\[
h = \frac{\Pi}{\rho g} = \frac{249.42 \, \text{Pa}}{960 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2}
\]
Calculating \(h\):
\[
h = \frac{249.42}{960 \cdot 9.8} \approx 0.02652 \, \text{m}
\]
### Step 6: Convert height to millimeters
To convert meters to millimeters:
\[
h = 0.02652 \, \text{m} \times 1000 = 26.52 \, \text{mm}
\]
### Final Answer
The height of the solution that will represent this pressure is approximately **26.52 mm**.
