Problem 46 - Chemistry: Principles and Reactions

Question

Lysozyme, extracted from egg whites, is an enzyme that cleaves bacterial cell walls. A 20.0-mg sample of this enzyme is dissolved in enough water to make \(225 \mathrm{~mL}\) of solution. At \(23^{\circ} \mathrm{C}\) the solution has an osmotic pressure of \(0.118 \mathrm{~mm} \mathrm{Hg}\). Estimate the molar mass of lysozyme.

Step-by-Step Solution

Answer

The estimated molar mass of lysozyme is approximately 24,310 g/mol.

Step 1: Write down the given data

Write down the information given in the problem: Mass of lysozyme: \(20.0\mathrm{~mg}\) Volume of solution: \(225\mathrm{~mL}\) Temperature: \(T = 23^{\circ}\mathrm{C} = 23 + 273.15 = 296.15\mathrm{~K}\) Osmotic pressure: \(\Pi = 0.118\mathrm{~mm~Hg}\)

Step 2: Convert osmotic pressure to atm

To use the osmotic pressure in calculations, it needs to be in atm. Convert osmotic pressure from mm Hg to atm using the conversion factor 1 atm = 760 mm Hg: \(\Pi = 0.118\mathrm{~mm~Hg} \times \frac{1\mathrm{~atm}}{760\mathrm{~mm~Hg}} = 1.553\times10^{-4}\mathrm{~atm}\)

Step 3: Convert mass of lysozyme to grams

The mass of lysozyme needs to be in grams for our calculations. Mass of lysozyme: \(20.0\mathrm{~mg} \times \frac{1\mathrm{~g}}{1000\mathrm{~mg}} = 0.020\mathrm{~g}\)

Step 4: Use the osmotic pressure formula to solve for molar mass

The formula to use for osmotic pressure, \(\Pi\), is given by: \(\Pi = CRT\) Where: \(C\) is the molar concentration of the solute, \(R\) is the ideal gas constant (\(0.08206 \mathrm{~L~atm~mol^{-1}~K^{-1}}\)), \(T\) is the temperature in Kelvin. First, we need to find the concentration \(C\) with the formula: \(C = \frac{n}{V}\) Where \(n\) is the number of moles of solute and \(V\) is the volume of the solution in liters. The number of moles, \(n\), can be expressed as the mass of the solute, \(m\), divided by its molar mass, \(M\). Therefore, we have the concentration as: \(C = \frac{m}{MV}\) Now, we can substitute this expression for \(C\) into the osmotic pressure formula: \(\Pi = \frac{mRT}{MV}\) We want to find the molar mass \(M\), so we can rearrange this equation for \(M\): \(M = \frac{mRT}{\Pi V}\) Now plug in the values we found earlier: \(M = \frac{0.020\mathrm{~g} \times 0.08206\mathrm{~L~atm~mol^{-1}~K^{-1}} \times 296.15\mathrm{~K}}{1.553\times10^{-4}\mathrm{~atm} \times 0.225\mathrm{~L}}\)

Step 5: Calculate the molar mass

Calculate the molar mass of lysozyme: \(M = \frac{0.020\mathrm{~g} \times 0.08206\mathrm{~L~atm~mol^{-1}~K^{-1}} \times 296.15\mathrm{~K}}{1.553\times10^{-4}\mathrm{~atm} \times 0.225\mathrm{~L}} \approx 24310\mathrm{~g~mol^{-1}}\) The estimated molar mass of lysozyme is around \(24,310 \mathrm{~g~mol^{-1}}\).

Key Concepts

Molar Mass Calculation

Molar mass calculation is an essential part of understanding chemical compositions. It is the process used to determine the mass of one mole of a substance. In the context of the provided exercise with lysozyme, the molar mass is calculated using the osmotic pressure formula. Osmotic pressure, \( \Pi \), is the pressure required to prevent the flow of solvent into a solution through a semipermeable membrane.

To calculate the molar mass \( M \) of lysozyme, we use the formula for osmotic pressure: \[ \\Pi = \frac{mRT}{MV} \\]Where:

• \( m \) is the mass of the solute.
• \( R \) is the ideal gas constant (usually given as \( 0.08206 \ \mathrm{L \, atm \, mol^{-1} \, K^{-1}} \)).
• \( T \) is the temperature in Kelvin.
• \( V \) is the volume of the solution in liters.

By rearranging the osmotic pressure formula, we get the equation for calculating the molar mass:\[ \M = \frac{mRT}{\Pi V} \\]

This calculated value helps understand the molecular weight of the lysozyme enzyme and aids in its characterization.

Lysozyme

Lysozyme is a fascinating enzyme, mostly recognized for its ability to break down the cell walls of bacteria. It serves as a natural defense mechanism in many organisms. Extracted from egg whites, lysozyme acts by cleaving peptidoglycan, a component of bacterial cell walls. This ability makes it an essential agent in the study of bacteria and infection control.

In the context of the exercise, lysozyme's properties allow us to explore its molar mass through osmotic pressure, which is a colligative property. Molar mass helps understand the size and structure of the molecules in this enzyme and gives insight into its action mechanism. Lysozyme's relatively large molecular weight relates to its complex structure, which is designed for specific biological functions.

Lysozyme's role in biochemistry extends beyond simple structural analysis, influencing how we develop antibiotics and understand metabolic pathways.

Ideal Gas Constant

The ideal gas constant, symbolized as \( R \,\), is an integral part of many equations in chemistry, particularly those concerning gases and solutions. It provides a link between the macroscopic and microscopic properties of gas molecules. Its value, commonly used as \( 0.08206 \ \mathrm{L \, atm \, mol^{-1} \, K^{-1}} \), is essential for calculations involving moles, temperature, volume, and pressure.

In the calculation of the molar mass of lysozyme, the ideal gas constant is crucial. It allows us to relate the osmotic pressure to these variables in the formula \( \Pi = CRT \). This constant ensures that the derived molar mass is accurate by maintaining consistency in units across physical measurements.

Understanding the ideal gas constant reinforces the connection between gas laws and solution chemistry, enabling calculations in various biochemical and physical contexts.