Question

Suppose a semipermeable membrane separates two solutions. One solution has \(0.72 \mathrm{~g}\) glucose to \(1.0 \mathrm{~L}\) of water; the other has \(0.117 \mathrm{~g} \mathrm{NaCl}\) to \(1.0 \mathrm{~L}\) of water. Given that glucose has a molecular weight of 180 and \(\mathrm{NaCl}\) has a molecular weight of \(58.5\), perform the following calculations. Calculate the molality and osmolality of each solution (see fig. 6.10).

Short answer

The molality of the glucose solution is \(4.0 \times 10^{-3}\) mol/kg and its osmolality is \(4.0 \times 10^{-3}\) Osm/kg. Meanwhile, the molality of the NaCl solution is \(2.0 \times 10^{-3}\) mol/kg and its osmolality is \(4.0 \times 10^{-3}\) Osm/kg.

Step-by-step solution

Calculate moles of solute

First, we need to find the moles of glucose and NaCl in the solutions. The moles of solute can be calculated using the formula: moles = mass (g) / molecular weight For glucose: moles = \(0.72 \mathrm{~g} \) / 180 moles of glucose = \(4.0 \times 10^{-3}\) moles For NaCl: moles = \(0.117 \mathrm{~g}\) / 58.5 moles of NaCl = \(2.0 \times 10^{-3}\) moles

Calculate molality

Molality (m) is defined as the number of moles of solute per kilogram of solvent. In this case, the solvent is water. Since 1.0 L of water has a mass of approximately 1.0 kg, we can use the moles of solute calculated in the previous step to find the molality of each solution. For glucose: molality = moles of glucose / mass of water (in kg) molality = \((4.0 \times 10^{-3}) \mathrm{~moles}\) / 1.0 \(\mathrm{kg}\) molality of glucose solution = \(4.0 \times 10^{-3}\) mol/kg For NaCl: molality = moles of NaCl / mass of water (in kg) molality = \((2.0 \times 10^{-3}) \mathrm{~moles}\) / 1.0 \(\mathrm{kg}\) molality of NaCl solution = \(2.0 \times 10^{-3}\) mol/kg

Calculate osmolality

Osmolality (Osm) is defined as the number of osmoles of solute particles per kilogram of solvent. In this case, glucose does not dissociate in water, so the osmolality is the same as molality. But NaCl dissociates into two ions (Na+ and Cl-) in water, so the osmolality will be twice the molality for the NaCl solution. For glucose: osmolality = molality osmolality = \(4.0 \times 10^{-3}\) Osm/kg For NaCl: osmolality = 2 * molality osmolality = 2 * \((2.0 \times 10^{-3}) \mathrm{~Osm/kg}\) osmolality = \(4.0 \times 10^{-3}\) Osm/kg The molality of the glucose solution is \(4.0 \times 10^{-3}\) mol/kg, and its osmolality is \(4.0 \times 10^{-3}\) Osm/kg. The molality of the NaCl solution is \(2.0 \times 10^{-3}\) mol/kg, and its osmolality is \(4.0 \times 10^{-3}\) Osm/kg.

Key concepts

Molality

Understanding molality is vital for accurate solution concentration calculations. It is a measure of the amount of a solute that's mixed into a given mass of solvent. The key difference between molality and other concentration measures, like molarity, is that molality is not affected by temperature changes. This is because molality is based on the mass of the solvent, not its volume, and mass doesn't change with temperature.

When calculating molality, the formula \( m = \frac{{\text{{moles of solute}}}}{{\text{{mass of solvent (in kg)}}}} \) is used. If we consider water as the solvent, with a density close to 1 kg/L, then for a liter solution, the mass of the solvent is approximately 1 kg. The exercise provided demonstrates this calculation with glucose and NaCl dissolved in water.

Semipermeable Membrane

A semipermeable membrane is a selective barrier; it allows certain particles to pass through while blocking others. This property is essential in biological systems and technology such as dialysis machines. In the educational scenario, a semipermeable membrane separates two solutions of different concentrations.

These membranes are the basis for osmosis, where water moves from an area with a lower solute concentration (hypoosmotic solution) to one with a higher solute concentration (hyperosmotic solution) to achieve equilibrium. An understanding of how semipermeable membranes function helps explain why certain particles, like the ions of NaCl, can affect the movement of water between solutions.

Solution Concentration

Solution concentration is a quantitative way of expressing the amount of solute dissolved in a given quantity of solvent. It's important in many fields, such as chemistry, medicine, and environmental science. Concentrations can be expressed in various ways, including molarity, molality, and percent composition. Each expression has its own usage depending on the situation.

For the mathematically-inclined, solution concentration affects properties like boiling point elevation and freezing point depression. In biological contexts, it's crucial for maintaining proper osmotic balance in cells. For example, an incorrect electrolyte concentration can disrupt cellular processes, demonstrating the significance of correct solution concentration calculations.

Solute Dissociation

Solute dissociation refers to the breaking apart of a compound into its constituent ions when it dissolves in a solvent. This is a critical aspect to consider when calculating osmolality. Keep in mind that not all solutes dissociate; glucose, for example, remains whole in aqueous solutions, while NaCl separates into Na+ and Cl- ions.

When calculating osmolality, solute dissociation is vital because it changes the number of particles in the solution. In the case of NaCl, you must multiply the molality by two (since it breaks apart into two ions) to get the true osmolality. This affects the freezing point depression and boiling point elevation of solutions, as well as the osmotic balance across membranes.