Solution Analysis:

To determine the molecular weight of the solute, we can use the formula for osmotic pressure:

Osmotic pressure (π) = (n/V)RT

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

Step 1: Convert the given data to the appropriate units:
- The solute mass is given as 20g.
- The volume of the solvent is given as 500ml, which needs to be converted to liters. 1 liter = 1000ml, so 500ml = 0.5L.
- The temperature is given as 15°C, which needs to be converted to Kelvin. Kelvin = °C + 273, so 15°C = 288K.

Step 2: Calculate the number of moles of solute:
Use the formula: moles = mass / molar mass

To find the molar mass, we need to use the formula: molar mass = mass / moles

Let's assume the molecular weight of the solute is "x" g/mol.

20g / x g/mol = moles

Step 3: Calculate the osmotic pressure:
Plug the values into the osmotic pressure formula:

π = (n/V)RT

π = (moles / 0.5L) * (0.0821 L·atm/(mol·K)) * 288K

Step 4: Solve for x (molecular weight):
Now we can rearrange the formula to solve for x (molecular weight):

π = (20g / x g/mol) / 0.5L * (0.0821 L·atm/(mol·K)) * 288K

Multiply both sides by x:

π * x = 20g / 0.5L * (0.0821 L·atm/(mol·K)) * 288K

Divide both sides by π:

x = (20g / 0.5L * (0.0821 L·atm/(mol·K)) * 288K) / π

Step 5: Calculate the molecular weight of the solute:
Plug in the given osmotic pressure value of 600mm Hg (which is equivalent to 0.789 atm) into the equation:

x = (20g / 0.5L * (0.0821 L·atm/(mol·K)) * 288K) / 0.789 atm

Simplify the equation:

x = (20g / 0.5L * 0.0821 L·atm/(mol·K) * 288K) / 0.789

x ≈ 173.6 g/mol

Conclusion:
The molecular weight of the solute is approximately 173.6 g/mol.

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