Problem 44 - Chemistry The Science in Context

Question

Calculate the osmotic pressure exerted by a \(92 \mathrm{mM}\) solution of KCl in a red blood cell at body temperature \(\left.\left(37^{\circ} \mathrm{C}\right) . \text { (Hint: See Equation } 11.13 .\right)\)

Step-by-Step Solution

Answer

Answer: The osmotic pressure exerted by a 92 mM solution of KCl in a red blood cell at body temperature (37°C) is approximately 2.34 atm.

Step 1: Convert temperature to Kelvin

To use the van't Hoff equation, we need to convert the given temperature from Celsius to Kelvin. The conversion formula is: \(T_{K} = T_{C} + 273.15\) So, we have: \(T_{K} = 37 + 273.15 = 310.15\, \mathrm{K}\)

Step 2: Convert solution concentration from mM to M

The given concentration of KCl is provided in mM (millimolar). In order to use it in the van't Hoff equation, it needs to be in moles per liter (M, or molar). To convert from mM to M, simply divide the value by 1000: \(c_{M} = \frac{c_{mM}}{1000} = \frac{92\, \mathrm{mM}}{1000} = 0.092\, \mathrm{M}\)

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

Now that we have the temperature in Kelvin and the concentration in moles per liter, we can apply the van't Hoff equation to calculate the osmotic pressure: \(\Pi = cRT\) Plug in the values that we obtained in the previous steps: \(\Pi = (0.092\, \mathrm{M})(0.0821\, L\, atm\, (K\, mol)^{-1})(310.15\, \mathrm{K})\)

Step 4: Calculate the osmotic pressure

Performing the calculation for the osmotic pressure: \(\Pi \approx 2.34\, \mathrm{atm}\) The osmotic pressure exerted by a 92 mM solution of KCl in a red blood cell at body temperature (37°C) is approximately 2.34 atm.

Key Concepts

van't Hoff equation

The van't Hoff equation is essential when discussing osmotic pressure calculations in solutions. It relates the osmotic pressure (\( \Pi \)) of a solution to its molar concentration (\( c \)), the temperature in Kelvin (\( T \)), and the ideal gas constant (\( R \)). The equation is expressed as:\[\Pi = cRT\]Here's how each part of this equation contributes:

• \( \Pi \) is the osmotic pressure that you want to find. It represents the force per unit area that prevents the movement of solvent across a semipermeable membrane.
• \( c \) is the molar concentration of the solute in the solution, reflecting how many moles of solute are present per liter.
• \( R \) is the ideal gas constant, with a common value of 0.0821 L atm K⁻¹ mol⁻¹, which connects the units of pressure (atm), temperature (Kelvin), and molarity.
• \( T \) is the absolute temperature measured in Kelvin.

To effectively use the van't Hoff equation, one must ensure to have concentrations in molarity and temperatures in Kelvin.

temperature conversion

Converting temperature from Celsius to Kelvin is a crucial step when using the van't Hoff equation. The formula to convert Celsius to Kelvin is simple:\[ T_{K} = T_{C} + 273.15 \]Here's why this conversion is necessary:

• The Kelvin scale is an absolute temperature scale, which means that zero Kelvin is the absolute minimum temperature, also known as absolute zero.
• This scale is essential in scientific calculations because physical equations, like the van't Hoff equation, demand absolute temperature for accuracy.
• The conversion simply involves adding 273.15 to the Celsius temperature, thus aligning it with the Kelvin scale.

In the case of our exercise, converting body temperature (37°C) to Kelvin results in 310.15 K. Remembering these conversions ensures precise and reliable calculations.

molar concentration

Molar concentration, often referred to as molarity, is the measurement of the concentration of a solute in a solution in terms of amount of substance in a given volume. It is expressed in moles per liter (M).When given a concentration in millimoles (mM), converting it to molarity involves a straightforward calculation:\[ c_{M} = \frac{c_{mM}}{1000} \]Here's a breakdown of molar concentration:

• Millimolar (mM) is a convenient unit for lab-scale experiments, but when performing calculations like the ones in the van't Hoff equation, molarity (M) is preferred.
• The conversion from mM to M involves dividing by 1000, translating millimoles into standard moles per liter.
• Accurate molarity is crucial for predicting properties such as osmotic pressure accurately.

In the provided exercise, a 92 mM solution of KCl is converted to 0.092 M for use in further calculations, demonstrating how crucial this conversion is.