Problem 38 - Chemistry and Chemical Reactivity

Question

Estimate the osmotic pressure of human blood at \(37^{\circ} \mathrm{C} .\) Assume blood is isotonic with a \(0.154 \mathrm{M}\) NaCl solution, and assume the van't Hoff factor, \(i\), is 1.90 for \(\mathrm{NaCl}\).

Step-by-Step Solution

Answer

The osmotic pressure of human blood at 37°C is approximately 7.77 atm.

Step 1: Understand the formula for osmotic pressure

The osmotic pressure (\( \Pi \)) can be calculated using the formula: \( \Pi = iMRT \) where \(i\) is the van't Hoff factor, \(M\) is the molarity of the solution, \(R\) is the universal gas constant (0.08206 L·atm/mol·K), and \(T\) is the temperature in Kelvin.

Step 2: Convert temperature to Kelvin

To convert the given temperature from Celsius to Kelvin, use the formula: \( T(K) = T(°C) + 273.15 \). Here, \( 37^{\circ}C \) converts to \( 37 + 273.15 = 310.15 \text{ K} \).

Step 3: Substitute values into the osmotic pressure formula

Now, substitute the known values into the formula \( \Pi = iMRT \). This gives: \( \Pi = (1.90)(0.154 \text{ M})(0.08206 \text{ L·atm/mol·K})(310.15 \text{ K}) \).

Step 4: Calculate osmotic pressure

Perform the calculation: \( \Pi = 1.90 \times 0.154 \times 0.08206 \times 310.15 \). This results in an osmotic pressure \( \Pi \approx 7.77 \text{ atm} \).

Key Concepts

Van't Hoff Factor

The van't Hoff factor, often represented as the symbol \( i \), is an essential component in calculating colligative properties such as osmotic pressure. This factor denotes the number of particles a solute forms in a solution. When a solute dissolves, it can separate into more than one particle; \( i \) quantifies this separation. For example, when sodium chloride (NaCl) dissolves in water, it dissociates into its constituent ions, sodium (Na\(^+\)) and chloride (Cl\(^-\)). This means that \( i \) for NaCl in an ideal situation would be 2. However, due to real-world interactions in solutions such as ion pairs formation, \( i \) can differ from this ideal value. In the case from the exercise, the given van't Hoff factor is 1.90, reflecting these real-world interactions. Understanding \( i \) helps predict and calculate the effects of solutes in solutions, particularly in biological and chemical processes.

Isotonic Solutions

Isotonic solutions are crucial in medical and scientific settings, particularly concerning their osmotic pressure. Two solutions are considered isotonic when they have the same osmotic pressure. This balance is vital in medical treatments, ensuring that the concentration of solutes in intravenous fluids matches that of the body to avoid cell damage. Isotonic conditions mean cells neither gain nor lose water, maintaining their volume and function. In the exercise's context, human blood is said to be isotonic with a 0.154 M NaCl solution. This means their osmotic pressures are the same. The concept of isotonicity prevents harmful outcomes such as cell lysis or crenation, ensuring homeostasis within biological systems. It illustrates how delicate the balance in the human body can be, where even small changes can lead to significant physiological impacts.

Universal Gas Constant

The universal gas constant, denoted \( R \), is a critical element in various equations related to the behavior of gases and solutions. Its value is 0.08206 L·atm/mol·K, a constant that links various properties like pressure, volume, temperature, and number of moles in an equation. In osmotic pressure calculations, \( R \) plays a central role, tying together the impact of temperature and solute concentration in a formula. It's essentially a bridge that helps relate microscopic chemical properties to macroscopic observables such as pressure and volume. Understanding \( R \) allows for accurate calculations in many chemical processes, fundamental in scenarios ranging from laboratory research to industrial applications. In our specific exercise, \( R \) integrates with the van't Hoff factor, molarity, and temperature to calculate the osmotic pressure of blood, highlighting its widespread application and importance in chemical thermodynamics.