Problem 25 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A protein has a molar mass of \(69000 \mathrm{gmol}^{-1}\). Assuming ideal behaviour, calculate the osmotic pressure of a solution containing \(20 \mathrm{g} \mathrm{dm}^{-3}\) of the protein at \(298 \mathrm{K}\). (Section 17.4 )

Step-by-Step Solution

Answer

The osmotic pressure is approximately 0.00708 atm or 717.5 Pa.

Step 1: Understand the Osmotic Pressure Formula

The formula for osmotic pressure \( \pi \) of a solution is given by the van't Hoff equation: \[ \pi = \frac{n}{V}RT \]where \( n \) is the number of moles of solute, \( V \) is the volume of the solution in liters, \( R \) is the ideal gas constant (0.0821 L atm mol⁻¹ K⁻¹), and \( T \) is the temperature in Kelvin. Since the concentration \( C \) is given in \( \mathrm{g} \mathrm{dm}^{-3} \), we need to relate this to \( n \) and \( V \).

Step 2: Convert Mass Concentration to Moles

The mass concentration is \( 20 \mathrm{g} \mathrm{dm}^{-3} \). Convert this to molarity (\( n/V \)) using the molar mass of the protein: \[ \text{Molarity} = \frac{20 \mathrm{g} \mathrm{dm}^{-3}}{69000 \mathrm{gmol}^{-1}} \approx 2.8985 \times 10^{-4} \mathrm{mol} \mathrm{dm}^{-3} \] This is the concentration \( C \) that we will use in our calculations.

Step 3: Apply the Van't Hoff Equation

Using the Van't Hoff equation \( \pi = CRT \), substitute the known values: \[ C = 2.8985 \times 10^{-4} \mathrm{mol} \mathrm{dm}^{-3}, \ R = 0.0821 \mathrm{L} \mathrm{atm} \mathrm{mol}^{-1} \mathrm{K}^{-1}, \ T = 298 \mathrm{K} \]Thus, \[ \pi = (2.8985 \times 10^{-4}) \times (0.0821) \times (298) \approx 0.00708 \mathrm{atm} \]

Step 4: Convert Units if Necessary

The osmotic pressure is calculated in atm. If you need it in another unit, like Pascal (1 atm = 101325 Pascal), you can convert it:\[ \pi = 0.00708 \mathrm{atm} \times 101325 \mathrm{Pa/atm} \approx 717.5 \mathrm{Pa} \]

Key Concepts

van't Hoff equation

Osmotic pressure is an important concept in chemistry and biology and can be calculated using the van't Hoff equation. This equation relates the osmotic pressure of a solution to its solute concentration, temperature, and a constant. The formula is expressed as:

• \( \pi = CRT \)

where:

• \( \pi \) is the osmotic pressure.
• \( C \) is the molarity of the solution. This is the concentration expressed in moles of solute per liter of solution.
• \( R \) is the ideal gas constant (explained in detail below).
• \( T \) is the temperature, measured in Kelvin.

With the van't Hoff equation, we can calculate the pressure exerted by particles in solution. This pressure is analogous to a gas pressure in a closed container. Osmotic pressure can cause significant biological phenomena, such as the swelling of cells or the movement of water across cell membranes.

ideal gas constant

The ideal gas constant, represented as \( R \), is a crucial factor in various chemical equations, including the van't Hoff equation. It acts as a proportionality factor that relates the other variables in these equations. The value of \( R \) can vary depending on the units used for pressure, volume, and temperature, but in the context of osmotic pressure calculations:

• \( R = 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1} \)

This specific value is useful when pressure is measured in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K). The idea behind the ideal gas constant is derived from combining historical laws about gases, including Boyle's law, Charles's law, and Avogadro's law. These foundational theories enable us to extend gas laws into solutions, helping us understand the behavior of diluted solutes in solvents, just like gases.

molarity calculation

Calculating molarity involves determining how many moles of a solute are present in a specific volume of solution. Here's how you can calculate molarity with given mass:

• Start with the mass of the solute you have, which in this exercise was 20 g of protein per liter of solution.
• Use the molar mass of the protein (given as 69000 g/mol) to convert this mass into moles. The formula is:
  \[ \text{Molarity} = \frac{\text{mass of solute (g)}}{\text{molar mass (g/mol)}} \]

For example, if you have a 20 g/dm³ protein solution:

• Convert 20 g of protein to moles using \( \text{Molarity} \approx \frac{20}{69000} \approx 2.8985 \times 10^{-4} \text{ mol/dm}^3 \)

This calculation helps you find the concentration of the protein inside the solution, a key step in applying the van't Hoff equation.

molar mass of protein

The molar mass of a compound, especially large macromolecules like proteins, is an essential value that indicates how much one mole of that compound weighs. For proteins, this can be used to convert their concentration from grams per liter to moles per liter, which is crucial for calculations involving molarity.

• In this example, the molar mass of the protein was given as 69000 g/mol.
• This implies that one mole of this protein would weigh 69000 grams.

Knowing the molar mass helps in understanding how substantial or light a molecule is relative to other substances. Proteins usually have high molar masses, which is typical for substances with complex structures, as they are made up of long chains of amino acids. Understanding molar mass enables you to compare different proteins and other molecules, which aids in more complex calculations involving chemical reactions and solution properties.