Problem 101 - Chemistry The Science in Context

Question

Physiological Saline One hundred milliliters of a solution of physiological saline \((0.92 \%\) NaCl by mass) is diluted by the addition of \(250.0 \mathrm{mL}\) of water. What is the osmotic pressure of the final solution at \(37^{\circ} \mathrm{C} ?\) Assume that \(\mathrm{NaCl}\) dissociates completely into \(\mathrm{Na}^{+}(a q)\) and \(\mathrm{Cl}^{-}(a q)\)

Step-by-Step Solution

Answer

Answer: The osmotic pressure of the final solution at 37°C is approximately 2.29 atm.

Step 1: Calculate the mass of NaCl in 100 ml solution

Firstly, we need to determine the mass of NaCl in the 100 mL of the solution prior to dilution. It's given that the solution contains 0.92% NaCl by mass, so we have: mass of NaCl = 0.0092 * mass of solution To calculate the mass of the solution, we need to find the total volume and use the density of water (≈1 g/mL) to convert volume to mass. Since the volume before dilution was 100 mL, the mass of the solution is approximately 100 g. mass of NaCl = 0.0092 * 100 g = 0.92 g

Step 2: Find the number of moles of NaCl

To convert the mass of NaCl to moles, we use its molar mass, which is 58.44 g/mol, as follows: moles of NaCl = mass of NaCl / molar mass of NaCl moles of NaCl = 0.92 g / 58.44 g/mol = 0.01574 mol

Step 3: Calculate the concentration of NaCl after dilution

The total volume after the dilution is 100 mL (original) + 250 mL (added) = 350 mL. To obtain the concentration (in mol/L) after the dilution, we use the formula: conc. of NaCl (post-dilution) = moles of NaCl / total volume in L conc. of NaCl (post-dilution) = 0.01574 mol / 0.350 L ≈ 0.04497 mol/L

Step 4: Calculate the concentration of dissolved particles

Since NaCl dissociates completely into Na+ and Cl-, the concentration of dissolved particles (i.e., of ions) will be double the concentration of NaCl. So, we have: conc. of dissolved particles = 2 * conc. of NaCl (post-dilution) = 2 * 0.04497 mol/L = 0.08994 mol/L

Step 5: Find the osmotic pressure

Now, we can use the osmotic pressure formula: osmotic pressure = conc. of dissolved particles * R * T Where R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature in Kelvin (37°C = 310.15 K). osmotic pressure = 0.08994 mol/L * 0.0821 L atm/mol K * 310.15 K ≈ 2.29 atm Thus, the osmotic pressure of the final solution at 37°C is approximately 2.29 atm.

Key Concepts

Physiological Saline Solution

A physiological saline solution is a mixture commonly used in medical treatments that simulates the salt concentration found in the body. This type of solution typically contains 0.9% sodium chloride (NaCl) by mass, which is isotonic to human blood, preventing cell dehydration or swelling when it's used as an intravenous infusion. The salt concentration in physiological saline is crucial as it ensures the osmolarity stays very close to that of the body's own fluids, providing a safe and stable medium when introduced to the body's circulatory system.

When assessing the impact on cells, you should be aware that any deviation from the isotonic condition can lead to cells either shrinking due to water exit (in hypertonic solutions) or swelling and possibly bursting due to water entry (in hypotonic solutions). Therefore, the accurate preparation and dilution of physiological saline are essential for its safe application in medical settings.

Molar Concentration after Dilution

Dilution refers to the process of reducing the concentration of a solute in a solution, typically by adding more solvent. In the context of molar concentration, this process affects the number of moles of solute per liter of solution. After dilution, the total volume of the solution increases, while the number of moles of solute remains the same. Consequently, the molar concentration decreases.

To calculate the new concentration after dilution, you use the formula:
\[ \text{Concentration (M)} = \frac{\text{moles of solute}}{\text{total volume of solution (L)}} \]
It's important to convert the volume into liters when dealing with molar concentration, and understanding that while the number of moles doesn't change during the dilution process, the concentration does because of the increased volume.

Dissociation of NaCl

When sodium chloride (NaCl) dissolves in water, it dissociates into its constituent ions, sodium (Na+) and chloride (Cl-). This dissociation is complete in a physiological saline solution, meaning that each NaCl unit separates to form an Na+ ion and a Cl- ion.

This behavior is crucial for the calculation of osmotic pressure since the number of particles in solution affects the osmosis process. When NaCl dissociates, it doubles the number of particles in solution compared to the undissociated salt, thus impacting the osmotic pressure. This concept should always be kept in mind when preparing such solutions and calculating osmotic pressures.

Osmotic Pressure Formula

Osmotic pressure is a fundamental concept in chemistry and biology that describes the pressure required to prevent the inward flow of water across a semipermeable membrane. It is a colligative property, meaning it depends on the number of particles in solution rather than their identity.

The formula to calculate osmotic pressure (\(\Pi\)) is given by:
\[ \Pi = i \cdot c \cdot R \cdot T \]
Where:

• \(i\) represents the van't Hoff factor, which is the number of particles the solute forms in solution. For NaCl, this is 2 because NaCl dissociates into two ions.
• \(c\) is the molar concentration of the dissolved particles in mol/L.
• \(R\) is the gas constant, which in units of L atm/(mol K) is 0.0821.
• \(T\) is the absolute temperature in Kelvin.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This equation emphasizes the direct relationship between the number of dissolved particles and the osmotic pressure: if the number of particles increases (due to dissociation, for example), the osmotic pressure goes up, assuming temperature remains constant.

In practical terms, understanding and applying the osmotic pressure equation helps with creating solutions that match the body’s osmolarity or explain behaviors such as why plant cells stiffen when water flows in due to osmosis.