Question

A solution of \(6.85 \mathrm{~g}\) of a carbohydrate in \(100.0 \mathrm{~g}\) of water has a density of \(1.024 \mathrm{~g} / \mathrm{mL}\) and an osmotic pressure of 4.61 atm at \(20.0^{\circ} \mathrm{C}\). Calculate the molar mass of the carbohydrate.

Short answer

The molar mass of the carbohydrate is approximately 342 g/mol.

Step-by-step solution

Understand the formula for osmotic pressure

Osmotic pressure (\( \Pi \)) is given by the formula:\[\Pi = iMRT\]where \(i\) is the Van't Hoff factor, \(M\) is molarity in moles per liter, \(R\) is the ideal gas constant \(0.0821\, \text{L atm mol}^{-1} \text{K}^{-1}\), and \(T\) is the temperature in Kelvin.For carbohydrates, \(i = 1\) because they do not ionize in solution.

Convert temperature to Kelvin

Convert the given temperature from Celsius to Kelvin by adding 273.15.\[T = 20.0 + 273.15 = 293.15 \, \text{K}\]

Calculate molarity of the solution

First, calculate the volume of the solution using its mass and density. The total mass of the solution is \(6.85 + 100.0 = 106.85\, \text{g}\). Using the formula for density \(\rho = \text{mass/volume}\), solve for volume:\[\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{106.85\, \text{g}}{1.024\, \text{g/mL}} = 104.32\, \text{mL} = 0.10432\, \text{L}\]Molarity (\(M\)) is the number of moles of solute per liter of solution, calculated as follows:\[M = \frac{x}{0.10432} \] where \(x\) is moles of the carbohydrate.

Relate osmotic pressure to molarity and calculate moles

Given \(\Pi = 4.61 \text{ atm}\), use the formula from step 1:\[4.61 = 1 \cdot M \cdot 0.0821 \cdot 293.15\]Solving for \(M\),\[M = \frac{4.61}{0.0821 \times 293.15} = 0.192\, \text{mol/L}\]The moles of carbohydrate (\(x\)) can be calculated using the molarity:\[ x = 0.192 \times 0.10432 = 0.02003 \text{ mol} \]

Calculate molar mass

Molar mass is given by the mass of the solute divided by the number of moles of the solute:\[\text{Molar mass} = \frac{6.85\, \text{g}}{0.02003\, \text{mol}} \approx 342\, \text{g/mol}\]

Key concepts

Osmotic Pressure

Osmotic pressure is a key concept in chemistry that deals with the pressure required to stop osmosis, which is the movement of a solvent across a semipermeable membrane. It is especially useful for determining molecular weights of solutes, such as carbohydrates, dissolved in a solvent.
The formula for osmotic pressure (\( \Pi \)) is:\[\Pi = iMRT\]where:

• \(i\) is the Van't Hoff factor, which equals 1 for non-ionizing solutes like carbohydrates.
• \(M\) is the molarity of the solution.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

Understanding osmotic pressure helps in calculating the concentration of the solution, giving insights into the properties of the solute in different conditions.

Ideal Gas Constant

The ideal gas constant \(R\) is a fundamental constant used in numerous equations, connecting various physical properties of substances. It plays a crucial role in the ideal gas law and formulas related to solutions. For osmotic pressure calculations, \(R\) is expressed in units of \(0.0821 \, \text{L atm mol}^{-1} \text{K}^{-1}\).
Choosing the right units for \(R\) ensures compatibility with the other measurements in the equation, like temperature in Kelvin, pressure in atm, and volume in liters. This consistency is vital for accurate measurements and calculations in chemistry and physics.

Molarity Calculation

Molarity (\(M\)) is a measure of the concentration of a solute in a solution, represented in moles of solute per liter of solution. It is calculated by dividing the number of moles of a solute (\(x\)) by the volume of the solution in liters.
In the provided exercise, the calculation of molarity involves not just the solute but understanding the total mass and resulting volume of the solution:

• Calculate the total mass by adding solute and solvent masses.
• Convert this mass into volume using the solution's density.
• Determine molarity using the formula \(M = \frac{x}{{\text{volume in Liters}}}\).

This multiple-step approach ensures an accurate understanding of the concentration, crucial for further calculations like molar mass.

Density and Volume Relationship

Density is an important property that connects the mass and volume of a substance. For solutions, density (\(\rho\)) is used to find the volume of a given mass using the formula:\[\text{Volume} = \frac{\text{Mass}}{\text{Density}}\]This relationship is especially important when converting mass measurements into volume, as in the case of our carbohydrate solution.
The total mass of the solution is determined and then divided by its density to achieve precise volume calculations. This step is essential for converting measurements to the appropriate units for further calculations, such as molarity or osmotic pressure.

Temperature Conversion

Converting temperature from Celsius to Kelvin is essential when dealing with scientific equations such as the one for osmotic pressure. The conversion is straightforward: simply add 273.15 to the Celsius temperature.
For the exercise at hand, \(20.0^{\circ} \text{C}\) was converted to:

• \(20.0 + 273.15 = 293.15 \, \text{K}\)

The Kelvin scale is preferred in scientific calculations because it starts from absolute zero, providing consistency and accuracy for formulas involving thermal dynamics and physical chemistry.