A solution of a polymer containing 5g `dm^(-3)` was found to give an osmotic pressure of 600 `Nm^(-2)` (4.5 mm Hg) at `15^(@)C`. Calculate the molecular mass of the polymer.

Explanation

To calculate the molecular mass of the polymer from the given data, we can follow these steps:

### Step 1: Convert the given values to appropriate units
- The concentration of the polymer solution is given as 5 g/dm³. This can be converted to g/L, which is the same value: 5 g/L.
- The osmotic pressure (π) is given as 600 N/m². We can also convert this to atm for consistency with the gas constant. Since 1 atm = 101325 N/m², we can convert:
\[ \pi = \frac{600 \, \text{N/m}^2}{101325 \, \text{N/m}^2/\text{atm}} \approx 0.00593 \, \text{atm} \]
- The temperature (T) is given as 15°C, which can be converted to Kelvin:
\[ T = 15 + 273 = 288 \, \text{K} \]

### Step 2: Use the osmotic pressure formula
The formula for osmotic pressure is given by:
\[ \pi = CRT \]
Where:
- π = osmotic pressure
- C = concentration in mol/L
- R = ideal gas constant (0.0821 L·atm/(K·mol))
- T = temperature in Kelvin

### Step 3: Rearrange the formula to find the concentration (C)
We can rearrange the formula to find C:
\[ C = \frac{\pi}{RT} \]

### Step 4: Substitute the values into the equation
Substituting the values we have:
\[ C = \frac{0.00593 \, \text{atm}}{0.0821 \, \text{L·atm/(K·mol)} \times 288 \, \text{K}} \approx \frac{0.00593}{23.62968} \approx 0.000251 \, \text{mol/L} \]

### Step 5: Calculate the number of moles of polymer
Since we know the concentration (C) and the volume of the solution (1 L), we can calculate the number of moles (n):
\[ n = C \times V = 0.000251 \, \text{mol/L} \times 1 \, \text{L} = 0.000251 \, \text{mol} \]

### Step 6: Calculate the molar mass (M) of the polymer
The molar mass can be calculated using the formula:
\[ M = \frac{\text{mass}}{\text{moles}} = \frac{5 \, \text{g}}{0.000251 \, \text{mol}} \approx 19967.3 \, \text{g/mol} \]

### Final Answer
The molecular mass of the polymer is approximately **19967.3 g/mol**.

---