Problem 74 - Chemistry An Atoms First Approach

Question

Erythrocytes are red blood cells containing hemoglobin. In a saline solution they shrivel when the salt concentration is high and swell when the salt concentration is low. In a \(25^{\circ} \mathrm{C}\) aqueous solution of \(\mathrm{NaCl}\), whose freezing point is \(-0.406^{\circ} \mathrm{C},\) erythrocytes neither swell nor shrink. If we want to calculate the osmotic pressure of the solution inside the erythrocytes under these conditions, what do we need to assume? Why? Estimate how good (or poor) of an assumption this is. Make this assumption and calculate the osmotic pressure of the solution inside the erythrocytes.

Step-by-Step Solution

Answer

To calculate the osmotic pressure of the solution inside the erythrocytes, we need to assume that the erythrocytes are in osmotic equilibrium with the surrounding NaCl solution and that the NaCl solution is dilute. Using the given freezing point depression, we can determine the molality of the NaCl solution. Then, using the van't Hoff equation and the assumptions, we can calculate the osmotic pressure of the solution inside the erythrocytes, which is approximately 21.36 atm. The assumptions are reasonable for a dilute NaCl solution, but their accuracy might vary for more concentrated solutions or different solutes.

Step 1: Determine the molality of the NaCl solution

Using the given freezing point depression for the NaCl solution, we can find the molality using the formula: ΔT_f = K_f * m, where ΔT_f is the freezing point depression, K_f is the freezing point depression constant of water, and m is the molality of the NaCl solution. The given freezing point depression is -0.406 °C and the value of K_f for water is 1.86 °C kg/mol. Rearrange the equation to find the molality: \(m = \frac{ΔT_f}{K_f}\)

Step 2: Calculate the molality of the NaCl solution

Plug the values into the equation to find the molality of the NaCl solution: \(m = \frac{-0.406 °C}{1.86 °C kg/mol} = -0.218 \mathrm{mol/kg}\) Since we need a positive value for molality, we can assume that the negative sign is an error and that molality should be 0.218 mol/kg.

Step 3: Assume osmotic equilibrium and use the van't Hoff equation

We assume that the erythrocytes are in osmotic equilibrium with the surrounding NaCl solution, meaning that the osmotic pressure inside the erythrocytes is equal to the osmotic pressure of the NaCl solution. We also assume that the NaCl solution is dilute, meaning that we can directly use the molality to find the osmotic pressure using the van't Hoff equation: Π = n * i * R * T, where Π is the osmotic pressure, n is the number of moles of solute, i is the van't Hoff factor (which is 2 for NaCl, as it dissociates into two ions), R is the ideal gas constant (8.314 J/(K mol)), and T is the temperature in Kelvin (298.15 K, since T = 25 + 273.15).

Step 4: Calculate the osmotic pressure of the solution inside the erythrocytes

We know the molality, so we can rearrange the van't Hoff equation and plug in the values: Π = n * i * R * T = (0.218 mol/kg) * 2 * (8.314 J/(K mol)) * (298.15 K) = 2163.64 J/kg. Now, to convert the osmotic pressure from J/kg to atm, we can use the conversion factor (1 atm = 101325 J/m^3) and the density of the solution (assuming it's close to the density of water, 1000 kg/m^3): Π = (2163.64 J/kg) * (1 atm / 101325 J/m^3) * (1000 kg/m^3) = 21.36 atm. Under the assumptions made, the osmotic pressure of the solution inside the erythrocytes is approximately 21.36 atm. The assumption is reasonable for a dilute NaCl solution, but for more concentrated solutions or for different solutes, the assumption might not be as accurate.

Key Concepts

van't Hoff equation

Understanding the van't Hoff equation is crucial when exploring the behavior of solutions under various conditions. This equation demonstrates the relationship between the number of moles of solute, temperature, and the osmotic pressure of a solution. It is presented as: \[ \Pi = n \times i \times R \times T \] where \(\Pi\) is the osmotic pressure, \(n\) represents the number of moles of solute, \(i\) is the van't Hoff factor (indicating the number of particles the solute dissociates into), \(R\) is the ideal gas constant, and \(T\) is the absolute temperature in Kelvin.

When applied, this equation allows us to calculate the osmotic pressure, which is essentially the pressure needed to prevent the flow of a solvent into the solution via osmosis. Osmotic pressure is particularly important in biological processes, like when determining the proper saline concentration for an intravenous solution, which must match the osmotic pressure of the patient's blood to avoid cell damage.

For a solution like sodium chloride (NaCl), it's important to remember that the van't Hoff factor \(i\) is typically 2, because NaCl dissociates into two ions: Na+ and Cl-. However, the actual value may deviate due to ion pairing and other interactions at higher concentrations.

colligative properties

Colligative properties are a fascinating group of properties in solutions that depend purely on the ratio of the number of solute particles to the number of solvent molecules, not on the identity of the solute itself. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.

The significance of colligative properties lies in their practical applications. For instance, salting roads in winter relies on freezing point depression to prevent ice formation. In the medical field, the isotonicity of IV solutions with blood is crucial to ensuring that cells do not shrink or swell, which is a direct application of osmotic pressure – another colligative property.

The importance lies in the fact that these properties can give valuable insights into the molecular weight of solutes, the solute-solvent interactions, and the number of particles resulting from solute dissociation in solution. Moreover, understanding colligative properties aids in determining solution concentrations that are imperative in various aspects of chemistry and biology.

molality

Molality is a measure of the concentration of a solution that is defined as the number of moles of solute per kilogram of solvent. Not to be confused with molarity, which is moles per liter of solution, molality is especially useful in situations involving temperature changes because it does not change with temperature, unlike volume-based measures which can expand or contract with temperature fluctuations.

Calculated using the formula \( m = \frac{n}{w} \) where \(m\) is molality, \(n\) is the number of moles of solute, and \(w\) is the mass of the solvent in kilograms, molality is critical in calculating changes in boiling points and freezing points of solutions (colligative properties).

It is important to be precise when calculating molality to ensure accuracy in predicting colligative properties, such as the freezing point depression in the example of the saline solution affecting erythrocytes. An accurate calculation of molality is also required for correct determination of the osmotic pressure within cells using the van't Hoff equation.

freezing point depression

Freezing point depression is another member of the colligative properties family; it describes the phenomenon where the addition of a solute to a solvent results in a lower freezing point for the solution compared to the pure solvent. The extent to which the freezing point is lowered is not only proportional to the number of solute particles but also depends on a property known as the freezing point depression constant, represented as \(K_f\), which varies with the solvent.

This principle is illustrated by applying the formula \( \Delta T_f = K_f \times m \) where \(\Delta T_f\) represents the change in freezing point, \(K_f\) is the constant for the solvent, and \(m\) is the molality of the solution. In practical scenarios, this is used when de-icing airplanes and roads, or in creating 'freeze mixtures' for cooling purposes.

Understanding how to use this formula is vital for accurately determining concentrations and properties of solutions, and ultimately, for ensuring biological systems, like erythrocytes in saline solution, maintain proper function without cellular damage due to incorrect osmotic conditions.