Question

An aqueous solution containing \(1.00 \mathrm{g}\) of bovine insulin (a protein, not ionized) per liter has an osmotic pressure of \(3.1 \mathrm{mm}\) Hg at \(25^{\circ} \mathrm{C} .\) Calculate the molar mass of bovine insulin.

Short answer

The molar mass of bovine insulin is approximately 6008 g/mol.

Step-by-step solution

Convert Osmotic Pressure

First, we need to convert the osmotic pressure from mm Hg to atmospheres because the ideal gas constant is typically used in these units. Use the conversion factor: \(1 \, ext{atm} = 760 \, ext{mm Hg}\).The osmotic pressure in atm is given by:\[ \Pi = \frac{3.1 \, ext{mm Hg}}{760 \, ext{mm Hg/atm}} \approx 0.00408 \, ext{atm} \]

Use the Van't Hoff Equation

The formula for osmotic pressure is given by the Van't Hoff equation:\[ \Pi = MRT \]where:- \(\Pi\) is the osmotic pressure in atm.- \(M\) is the molarity in mol/L.- \(R\) is the ideal gas constant (approximately \(0.0821 \, ext{L atm/mol K}\)).- \(T\) is the temperature in Kelvin.Convert the temperature to Kelvin:\[ T = 25^{\circ}C + 273.15 = 298.15 \, K \]

Calculate Molarity

Rearrange the Van't Hoff equation to solve for molarity:\[ M = \frac{\Pi}{RT} \]Substitute the values:\[ M = \frac{0.00408 \, ext{atm}}{0.0821 \, ext{L atm/mol K} \times 298.15 \, K} \approx 1.664 \times 10^{-4} \, ext{mol/L} \]

Determine Molar Mass

Molarity is defined as moles of solute per liter of solution. Given that the solution contains \(1.00 \, ext{g}\) of bovine insulin per liter, we can find the molar mass \(M_m\) using the molarity:\[ M_m = \frac{ ext{mass of solute}}{ ext{moles of solute}} = \frac{1.00 \, ext{g}}{1.664 \times 10^{-4} \, ext{mol}} \approx 6008 \, ext{g/mol} \]

Verify Calculation

Double-check all calculations and conversion factors used to ensure accuracy. Make sure the units are consistent throughout the calculations.

Key concepts

Van't Hoff equation

The Van't Hoff equation provides an easy way to relate osmotic pressure to the concentration of solute in a solution. This equation is similar to the ideal gas law and is given by: \[ \Pi = MRT \] where

• \( \Pi \) is the osmotic pressure in atm,
• \( M \) is the molarity of the solution in mol/L,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature in Kelvin.

Using this equation, we can determine different properties of a solution like molarity, which is essential in calculating the molar mass. To use the Van’t Hoff equation effectively, ensure conversion of all units to keep calculations consistent. For example, osmotic pressure should be in atmospheres, and temperature must be in Kelvin. Simple consistency in units helps avoid calculation errors.

Molar Mass Calculation

Molar mass refers to the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It's a critical part of understanding chemical quantities. Using the Van't Hoff equation, we can calculate the molar mass by first finding the molarity of the solution. Here's a quick overview of how to calculate the molar mass:

• First, rearrange the Van’t Hoff equation to solve for molarity \( M \) using \( M = \frac{\Pi}{RT} \).
• With molarity known, and if you know the total mass of solute in the solution, use this mass divided by the molarity to find the molar mass: \[ M_m = \frac{\text{mass of solute}}{\text{moles of solute}} \].

For instance, if you have a solution with known osmotic pressure and temperature, and known grams of solute, you can determine the molar mass step by step using the correct conversions, as shown in the solution provided. Ensuring accuracy in this process is crucial for precise results.

Aqueous Solutions

An aqueous solution is simply a solution where water is the solvent. Water, being a universal solvent, readily dissolves many substances forming aqueous solutions, which are extremely common in chemistry. When studying properties such as osmotic pressure, knowing that a solution is aqueous helps in predicting behavior since water's properties aid in dissolution and provide predictable interaction with solutes. When addressing problems involving aqueous solutions, it's necessary to consider:

• The concentration of solute in water,
• The presence of any ions or molecules that can affect calculations heavily depending on their behavior in a water-based medium.

This understanding lets us use equations like Van't Hoff, assuming that water is the primary medium through which osmotic forces act. This straightforward assumption simplifies calculations and ensures predictable interactions.

Gas Constant

The gas constant, represented by \( R \), has an approximate value of 0.0821 L atm/mol K and is an integral part of the Van't Hoff equation, mirroring its role in the ideal gas law. The importance of \( R \) is its universality, as it applies to a variety of gas-related calculations, bridging concepts between gases and solutions. It helps translate osmotic pressure - a concept typically associated with liquids - into a framework that incorporates volume and temperature, much like it does for gases.To effectively use \( R \) in equations, keep these points in mind:

• Ensure unit consistency: Osmotic pressure should be measured in atmospheres, temperature in Kelvin, and volume in liters to align with the units \( R \) uses.
• Remember that \( R \) is a constant value but represents the complex relationships in thermodynamics and chemistry between energy, temperature, and moles.

Employing \( R \) with precision can lead to accurate calculations in chemistry, cementing its role as a fundamental constant in understanding both gaseous and dissolved states.