Cellobiose is a sugar obtained by degradation of cellulose. If 200.0 mL of an aqueous solution containing 1.500 g of cellobiose at 25.0 °C gives rise to an osmotic pressure of 407.2 mm Hg, what is the molecular weight of cellobiose?

Verified step by step guidance

1. Step 1: Convert the osmotic pressure from mm Hg to atm using the conversion factor 1 atm = 760 mm Hg.
2. Step 2: Use the formula for osmotic pressure \( \Pi = iMRT \), where \( \Pi \) is the osmotic pressure in atm, \( i \) is the van't Hoff factor (assumed to be 1 for non-electrolytes), \( M \) is the molarity, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is the temperature in Kelvin.
3. Step 3: Convert the temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature.
4. Step 4: Rearrange the osmotic pressure formula to solve for molarity \( M \): \( M = \frac{\Pi}{iRT} \).
5. Step 5: Calculate the number of moles of cellobiose using the molarity and the volume of the solution in liters, then use the mass of cellobiose to find the molecular weight by dividing the mass by the number of moles.

Key Concepts

Osmotic Pressure

Osmotic pressure is the pressure required to prevent the flow of solvent into a solution through a semipermeable membrane. It is directly proportional to the concentration of solute particles in the solution, as described by the formula π = iCRT, where π is osmotic pressure, i is the van 't Hoff factor, C is the molar concentration, R is the ideal gas constant, and T is the temperature in Kelvin.

Molecular Weight

Molecular weight, or molar mass, is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It can be calculated by dividing the mass of the substance by the number of moles present. In the context of osmotic pressure, knowing the molecular weight allows for the determination of the number of moles of solute in a given mass, which is essential for calculating concentration.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. While it primarily applies to gases, it can be adapted for solutions in terms of osmotic pressure. Understanding this law helps in manipulating the variables to find unknowns, such as the molecular weight of solutes in solution when osmotic pressure is known.