Problem 43 - Chemistry: An Atoms-Focused Approach

Question

Osmotic Pressure of Red Blood Cells One of the functions of the alkali metal cations \(\mathrm{Na}^{+}\) and \(\mathrm{K}^{+}\) in cells is to maintain the cells' osmotic pressure. The concentration of NaCl in red blood cells is approximately \(11 \mathrm{m} M .\) Calculate the osmotic pressure of this solution at body temperature \(\left.\left(37^{\circ} \mathrm{C}\right) . \text { (Hint: See Equation } 11.2 .\right)\)

Step-by-Step Solution

Answer

Answer: The osmotic pressure of the NaCl solution inside red blood cells at 37°C is approximately 28.48 J/lt.

Step 1: Convert temperature to Kelvin

To work with the van't Hoff equation, we need the temperature in Kelvin. We convert the given temperature: $$ T = 37 + 273.15 = 310.15 \; \mathrm{K} $$

Step 2: Convert concentration to mol/L

The given concentration is in mM, so we need to convert it to mol/L. $$ C = 11 \cdot 10^{-3} \; \mathrm{M} $$

Step 3: Use the van't Hoff equation to calculate osmotic pressure

Now, we can plug the values of C, R, and T into the van't Hoff equation to find the osmotic pressure, \(\Pi\). $$ \Pi = CRT = (11 \cdot 10^{-3} \; \mathrm{M}) \cdot (8.31 \; \frac{\mathrm{J}}{\mathrm{mol} \; \mathrm{K}}) \cdot (310.15 \; \mathrm{K})$$

Step 4: Calculate the osmotic pressure

Now, calculate the osmotic pressure by multiplying the given values: $$ \Pi = (11 \cdot 10^{-3}) \cdot 8.31 \cdot 310.15 \approx 28.48 \; \frac{\mathrm{J}}{\mathrm{litre}} $$ The osmotic pressure of the NaCl solution inside red blood cells at body temperature is approximately 28.48 J/lt.

Key Concepts

van't Hoff equation

The van't Hoff equation is a crucial tool for calculating the osmotic pressure of solutions. It relates osmotic pressure (\( \Pi \)) to solute concentration, temperature, and the gas constant. The formula is given by:
\[\Pi = CRT\]
where:

• \( \Pi \): Osmotic pressure, expressed in units such as J/L or atm
• \( C \): Molar concentration of the solute (mol/L)
• \( R \): Universal gas constant \( (8.31 \, \text{J} / \text{mol} \, \text{K}) \)
• \( T \): Temperature in Kelvin

To successfully calculate osmotic pressure using this equation, it is necessary to ensure all units are consistent. The solute concentration should be in mol/L, and temperature must be converted to Kelvin. This formula explains how solutions exert pressure through osmosis, moving water across a semipermeable membrane from areas of lower solute concentration to higher concentration.

red blood cells

Red blood cells are crucial to our bodies because they carry oxygen from the lungs to tissues and return carbon dioxide back to the lungs. An important aspect of their function is maintaining osmotic balance.
This is controlled partially by ions like \(\text{Na}^{+}\) and \(\text{K}^{+}\). These ions help regulate the water movement in and out of cells through osmosis. Red blood cells have a characteristic concentration of solutes such as NaCl, usually around 11 mM. When looking at osmotic pressure,

• If red blood cells were placed in a hypertonic solution (higher external concentration), they would shrivel as water exits the cell.
• If cells were placed in a hypotonic solution (lower external concentration), they would swell and possibly burst as water enters the cell.

Thus, the osmotic pressure is vital to keeping red blood cells in a stable environment, ensuring they function properly in transporting gases.

NaCl concentration

The concentration of NaCl is critical for maintaining osmotic pressure within red blood cells. In the given exercise, the NaCl concentration in red blood cells is 11 mM, which is quite precise. To work with osmotic pressure calculations, this concentration is converted to mol/L, resulting in 0.011 M.
Concentration impacts the number of particles in a solution and, therefore, its osmotic pressure. NaCl, when dissolved, disassociates into two ions, \( \text{Na}^{+} \) and \( \text{Cl}^{-} \), which effectively doubles the particle concentration affecting osmosis. Understanding the NaCl concentration in solutions helps explain how cells manage water movement across their membranes:

• This balance is essential for cells like red blood cells to prevent drastic changes in shape or size.
• Such changes could disrupt their ability to transport oxygen efficiently.

In physiological conditions, maintaining this concentration facilitates correct cellular function and overall homeostasis.