Question

Use the concentration of an isotonic saline solution, \(0.92 \% \mathrm{NaCl}(\mathrm{mass} / \text { volume }),\) to determine the osmotic pressure of blood at body temperature, \(37.0^{\circ} \mathrm{C}\). [Hint: Assume that \(\mathrm{NaCl}\) is completely dissociated in aqueous solutions.]

Short answer

The osmotic pressure of blood at body temperature is approximately 7.98 atm.

Step-by-step solution

Understanding the problem and gathering information

Given concentration of NaCl is 0.92%, which means 0.92 g of NaCl are present in 100 g of solution. As this is a mass/volume percent, and density of water is approximately 1 g/cm^3 or 1 g/ml, we can consider this as 0.92 grams in 100 ml solution. Temperature is given as 37.0 °C, which in Kelvin (K) is \(37.0 + 273.15 = 310.15 K\) (since absolute temperature is required for calculations involving gas laws). Following the hint, we know that NaCL dissociates completely in solution into its ions, Na+ and Cl-, making the Van't Hoff factor (i) 2.

Conversion of concentration

We first convert the concentration of NaCl from g/100ml to moles/liter (M), since osmotic pressure is typically expressed in these units. We use the molar mass of NaCl, which is approximately 58.44 g/mol. \( \text {Convert mass of NaCl to moles} = \frac{0.92 \text{ g NaCl}}{58.44 \text{ g/mol NaCl}} = 0.0157 \text{ moles NaCl} \). Next, convert the volume from milliliters to liters: \(0.1 \text { liters} \). Thus, the molar concentration is \( \frac{0.0157 \text{ moles}}{0.1 \text{ liters}} = 0.157 \text{ M NaCl} \)

Calculation of osmotic pressure

We then use the formula for osmotic pressure: \( \text{Osmotic pressure} = i \cdot n/V \cdot R \cdot T\) where \(n/V\) is the molar concentration (in moles/liter), R is the gas constant (0.0821 liter·atm / (mol·K)), and T is the absolute temperature. Substituting the given values: \( \text{Osmotic pressure} = 2 \cdot 0.157 \text { M} \cdot 0.0821 \text { liter·atm / (mol·K)} \cdot 310.15 \text { K} = 7.98 \text { atm} \).

Key concepts

Isotonic Solutions

Isotonic solutions are essential in the medical and biological world due to their equilibrium with bodily fluids. When a solution is isotonic, it means that it has the same osmotic pressure as another solution, typically body fluids like blood or interstitial fluid.
When solutions are isotonic, they ensure a balanced state where there is no net movement of water across cell membranes. This is crucial to prevent cells from swelling or shrinking. Hospitals use isotonic saline to replenish fluid levels in patients without disturbing their body's balance.
Understanding isotonic solutions helps in creating solutions that can safely interact with the body, mimicking the natural environment of cells. This characteristic makes them extremely useful for intravenous infusions. The key takeaway is their balance in osmotic pressure with bodily fluids, which helps maintain cellular health and proper function.

Dissociation of NaCl

The dissociation of NaCl, or table salt, in water is a fundamental process. When NaCl is dissolved, it separates into sodium (Na\(^+\)) and chloride (Cl\(^-\)) ions. This complete dissociation is what makes NaCl a strong electrolyte.
This process occurs because water molecules, being polar, surround and stabilize the Na\(^+\) and Cl\(^-\) ions, pulling them apart and allowing them to act independently in solution.
Such dissociation is critical in various applications, notably in the creation of isotonic solutions. Since NaCl dissociates into two ions, the number of particles in the solution doubles, impacting properties like osmotic pressure. Understanding this concept helps explain why NaCl solutions are effective in medical treatments and experiments, as they mimic the ionic strength and concentration found in biological systems.

Van't Hoff Factor

The Van't Hoff factor, denoted by the symbol \(i\), is a crucial element in calculating colligative properties like osmotic pressure. For NaCl, which dissociates into two ions, Na\(^+\) and Cl\(^-\), the Van't Hoff factor is 2.
This factor reflects the actual number of particles produced in a solution from a solute under ideal conditions. For non-electrolytes, \(i\) is typically 1, as they do not dissociate in solution. However, for strong electrolytes like NaCl, where dissociation occurs completely, \(i\) can be greater than 1.
In calculating osmotic pressure, the Van't Hoff factor plays a pivotal role as it adjusts the concentration of particles to consider all dissociated ions. It's a multiplier in the osmotic pressure formula, emphasizing the increased effect due to dissociation of solutes like NaCl in solutions.

Molar Concentration Calculation

Calculating the molar concentration of a solution involves converting the mass of the solute into moles and considering the volume of the solution it is dissolved in. For NaCl, this is done by using its molar mass, which is about 58.44 g/mol.
To find the molarity (M), you need to first determine how many moles of NaCl are present. This requires dividing the mass of the NaCl by its molar mass. In the case of an isotonic solution containing 0.92 grams of NaCl per 100 ml, the moles of NaCl are calculated as \(\frac{0.92}{58.44}\).
Next, convert the volume from milliliters to liters, then use this to find the molarity by dividing the number of moles by the volume in liters. For given problem specifications, this results in a molarity of 0.157 M.
Understanding molar concentration is key in preparing solutions with precise properties, especially when working with isotonic solutions in medical settings. It's a foundational aspect of solution chemistry that affects how substances interact in a solution.