Problem 52 - Chemistry and Chemical Reactivity

Question

Estimate the osmotic pressure of human blood at \(37^{\circ} \mathrm{C}\) Assume blood is isotonic with a \(0.154 \mathrm{M} \mathrm{NaCl}\) solution, and assume the van't Hoff factor, \(i,\) is 1.9 for \(\mathrm{NaCl}\).

Step-by-Step Solution

Answer

The osmotic pressure of human blood at 37°C is approximately 7.8 atm.

Step 1: Understand the Problem

We need to estimate the osmotic pressure of human blood. Blood is assumed to be isotonic with a 0.154 M NaCl solution, and the temperature is given as 37°C. We will use the van't Hoff equation for osmotic pressure, which is \[ \Pi = iMRT \]where \(i\) is the van't Hoff factor, \(M\) is the molarity, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Convert Temperature to Kelvin

First, convert the temperature from Celsius to Kelvin. The conversion formula is \[ T(K) = T(^{\circ}C) + 273.15 \]Given that the temperature is 37°C, \[ T = 37 + 273.15 = 310.15 \, \text{K} \]

Step 3: Determine Constants

We will use the following constants:1. Molarity \(M\) of the NaCl solution: 0.154 M2. van't Hoff factor \(i\) for NaCl: 1.93. Ideal gas constant \(R\): 0.0821 L·atm/(K·mol) (since we will solve for osmotic pressure in atm)4. Temperature \(T\) in Kelvin: 310.15 K

Step 4: Apply the van't Hoff Equation

Substitute the constants into the van't Hoff equation\[ \Pi = iMRT = (1.9)(0.154 \, \text{mol/L})(0.0821 \, \text{L·atm/(K·mol)})(310.15 \, \text{K}) \]Simplifying gives \[ \Pi = 1.9 \times 0.154 \times 0.0821 \times 310.15 \approx 7.8 \, \text{atm} \]

Step 5: Conclusion

The estimated osmotic pressure of human blood at 37°C is approximately 7.8 atm.

Key Concepts

van't Hoff equation

The van't Hoff equation is a fundamental equation that helps calculate the osmotic pressure of a solution. Osmotic pressure is the pressure required to stop the movement of solvent molecules through a semi-permeable membrane. This equation is particularly useful because it relates osmotic pressure to the concentration of solute in a solution. The equation is expressed as:\[ \Pi = iMRT \]where:

• \(\Pi\) is the osmotic pressure in atm
• \(i\) is the van't Hoff factor, which accounts for the degree of dissociation of the solute
• \(M\) is the molarity of the solution in mol/L
• \(R\) is the ideal gas constant
• \(T\) is the temperature in Kelvin

To put it simply, the van't Hoff equation allows us to predict how much pressure would be needed to prevent the natural flow of solvent and keep two solutions separate. Knowing this equation is essential in fields such as chemistry and biology.

isotonic solutions

Isotonic solutions have identical osmotic pressures, which means there is no net movement of solvent molecules across a semi-permeable membrane. In practical terms, when two solutions are isotonic, they have similar concentrations of solutes. In the context of human blood, it is isotonic to a 0.154 M NaCl (salt) solution. In healthcare, isotonic solutions are crucial because they ensure that cells neither swell nor shrink when they are in contact with a solution. Key characteristics of isotonic solutions include:

• Equilibrium: Solvent molecules move in and out of cells at equal rates.
• No volume change: Cells maintain their size and shape.
• Safe for the body: Commonly used in IV drips to hydrate patients without disrupting cellular balance.

Understanding isotonicity is critical for medical treatments and laboratory research where maintaining cellular integrity is paramount.

molarity

Molarity, denoted as \(M\), is a way to express the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters. The formula for molarity is:\[ M = \frac{n}{V} \]where:

• \(n\) is the number of moles of solute
• \(V\) is the volume of the solution in liters

Molarity is important in various chemical calculations because it directly correlates with the chemical reactivity of the solution. For instance, a higher molarity means a higher concentration of solute and usually greater chemical reactivity. In our original exercise, the molarity of the NaCl solution was given as 0.154 M, indicating its concentration. Knowing how to calculate and interpret molarity is a fundamental skill in chemistry.

ideal gas constant

The ideal gas constant, denoted as \(R\), is a crucial component in various equations in chemistry and physics, including the van't Hoff equation for calculating osmotic pressure. Its value can vary based on units used, but in the context of osmotic pressure calculations, it is often given as:\[ R = 0.0821 \text{ L}\cdot\text{atm/(K}\cdot\text{mol)} \]This constant allows us to relate pressure, volume, and temperature of gases, and by extension, dilute solutions. It essentially standardizes the relationship between these variables, making it easier to predict one if the others are known. Here are some critical aspects of the ideal gas constant:

• Universal: It applies to ideal gases and solutions in similar conditions.
• Bridge for calculations: Integrates into formulas that describe the physical behavior of substances.
• Adaptable: The constant has different values for various unit systems, maintaining accuracy across different mathematical contexts.

Understanding \(R\) and its role is essential for solving a variety of thermodynamic and solution-based problems.