Question

Calculate the molecular weight of a pure isoelectric protein if a \(1 \%\) solution gives an osmotic pressure of \(46 \mathrm{~mm}\) of \(\mathrm{H}_{2} \mathrm{O}\) at \(0^{\circ} \mathrm{C}\). Assume that it yields an ideal solution.

Short answer

The molecular weight of the pure isoelectric protein can be calculated using the van't Hoff equation. After converting given units to SI units and plugging into the equation, the molecular weight is found to be approximately \(49.36 \mathrm{~g/mol}\).

Step-by-step solution

Understand the van't Hoff equation

The van't Hoff equation is given by \[π = cRT\], where \(π\) is the osmotic pressure, \(c\) is the concentration of the solution, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Convert given units

Convert the given units to SI units to make it compatible with the ideal gas constant value, \(R = 8.314\) J/(mol K): - Osmotic pressure: \(46\) mm of \(\mathrm{H}_2\mathrm{O}\) is equivalent to \(\approx 4.5586 \times 10^3\) Pa (1 mmHg = 133.322 Pa) - Temperature: \(0^{\circ} \mathrm{C}\) is equivalent to \(273.15\) K - Concentration: \(1 \%\) solution means there are 1g of protein in 100g of solution, which can be expressed as g/mL or g/L.

Calculate the concentration in mol/L

In order to use the van't Hoff equation, we need to express the concentration in mol/L. We know the concentration is 1 g/mL. Let \(M\) be the molecular weight of the protein. Therefore, the concentration in moles per liter is \[\frac{1 \frac{\text{g}}{\text{mL}}}{M \frac{\text{g}}{\text{mol}}}\times10^3 \frac{\text{mL}}{\text{L}}= \frac{10^3}{M} \frac{\text{mol}}{\text{L}}.\]

Plug the values into the van't Hoff equation and solve for the molecular weight

Substitute the given values and the derived concentration in terms of molecular weight into the van't Hoff equation: \[\pi = c\cdot R \cdot T \Rightarrow 4.5586 \times 10^3 \text{ Pa} = \frac{10^3}{M}\frac{\text{mol}}{\text{L}} \cdot 8.314 \frac{\text{J}}{\text{mol}\cdot\text{K}} \cdot 273.15 \text{ K}.\] Now, solve for the molecular weight, \(M\): \[M = \frac{10^3}{\pi} \cdot R \cdot T = \frac{10^3}{4.5586 \times 10^3 \text{ Pa}} \cdot 8.314 \frac{\text{J}}{\text{mol}\cdot\text{K}} \cdot 273.15 \text{ K} \approx 49.36 \frac{\text{g}}{\text{mol}}.\] So, the molecular weight of the protein is approximately \(49.36 \mathrm{~g/mol}\).

Key concepts

van't Hoff equation

The van't Hoff equation is a crucial tool in physical chemistry that relates osmotic pressure to solute concentration. It is expressed as \[ \pi = cRT \]where

• \( \pi \) is the osmotic pressure, usually measured in Pascals (Pa).
• \( c \) is the concentration of the solution in moles per liter (mol/L).
• \( R \) is the ideal gas constant, valued at 8.314 J/(mol K).
• \( T \) is the temperature in Kelvin (K).

This equation is similar to the ideal gas law, because solutions behave like gases when it comes to pressure and concentration. It helps us calculate how pressure changes as we dissolve solutes in a solvent. For students learning about this equation, understanding each component individually helps grasping how they collectively determine osmotic pressure. Always ensure the units are consistent to achieve accurate results.

osmotic pressure

Osmotic pressure is the force that a solute exerts on a solvent to balance concentration differences. This occurs when solutions of different concentrations are separated by a semipermeable membrane. Osmotic pressure drives the solvent from a lower-concentration area to a higher-concentration area. This process is important in biological systems. For example, it regulates fluid balance in cells, ensuring they neither shrivel nor burst. In practical terms, when you calculate osmotic pressure using the van’t Hoff equation, the formula provides an estimate of how strong the solvent needs to "push" to reach equilibrium. Understanding osmotic pressure is essential for figuring out molecular weights and concentrations. When working with proteins or large molecules, knowing their osmotic properties helps in fields like pharmacology and biochemistry.

ideal solution

An ideal solution is a concept where the interactions between solute molecules and solvent molecules are similar to those within pure components. In such solutions, properties like vapor pressure, boiling point, and osmotic pressure are predictable with simple equations. For calculating osmotic pressure, assuming the solution to be ideal makes calculations straightforward. This is because deviations in ideality are small and often negligible when the concentration is low or the solute does not have strong intermolecular forces. When given an exercise or problem, if an ideal solution is assumed, it simplifies the calculation with the van’t Hoff equation. However, in real-world scenarios, understanding when and why solutions deviate from ideality is vital for more precise results. Recognizing ideal vs. real considerations can help chemists design better experiments.

unit conversion

Unit conversion is a necessary step to ensure all measurements in a chemical formula are uniform and correct. In the context of the van't Hoff equation, unit conversion allows us to calculate correctly using known constants like the ideal gas constant.Here are common conversions seen in these problems:

• For pressure: Convert from mmHg to Pascals (Pa) using the relation: 1 mmHg = 133.322 Pa.
• For temperature: Convert degrees Celsius (°C) to Kelvin (K) by adding 273.15.
• For concentration: Express in mass per volume, such as grams per liter (g/L), then convert to moles per liter (mol/L) based on the substance’s molecular weight.

Consistent units prevent errors in calculations, ensuring results are accurate. It's critical to convert units where necessary, especially when constants like \(R\) are involved, which require specific units to work correctly.