Question

A \(4.7 \times 10^{-2} \mathrm{mg}\) sample of a protein is dissolved in water to make \(0.25 \mathrm{~mL}\) of solution. The osmotic pressure of the solution is \(0.56\) torr at \(25^{\circ} \mathrm{C}\). What is the molar mass of the protein?

Short answer

The molar mass of the protein is approximately 6,241 g/mol.

Step-by-step solution

Convert the given information into appropriate units

First, we'll convert the given information into appropriate units: - Protein mass: \(4.7 \times 10^{-2} \mathrm{mg}\) Convert to grams: \(\frac{4.7 \times 10^{-2} \mathrm{mg}}{1000} = 4.7 \times 10^{-5} \mathrm{g}\) - Volume of the solution: \(0.25 \mathrm{mL}\) Convert to liters: \(\frac{0.25 \mathrm{mL}}{1000}= 2.5 \times 10^{-4} \mathrm{L}\) - Temperature: \(25^{\circ} \mathrm{C}\) Convert to Kelvin: \(25 + 273.15 = 298.15 \mathrm{K}\) - Osmotic pressure: \(0.56 \mathrm{torr}\) Convert to atm: \(\frac{0.56 \mathrm{torr}}{760} = 7.368 \times 10^{-4} \mathrm{atm}\) Now we have all the given quantities in suitable units.

Rearrange the osmotic pressure equation to solve for the number of moles

Rearrange the van 't Hoff equation: \[n = \frac{ΠV}{RT}\] Plug in the values: \[n = \frac{7.368 \times 10^{-4} \mathrm{atm} \times 2.5 \times 10^{-4} \mathrm{L}}{0.0821 \mathrm{L \cdot atm/mol \cdot K} \times 298.15 \mathrm{K}}\] Compute the number of moles: \[n = 7.53 \times 10^{-6} \mathrm{mol}\]

Calculate the molar mass of the protein

Now we can calculate the molar mass (M) of the protein using the number of moles (n) and the mass of the protein (m): \[M = \frac{m}{n}\] Plug in the values: \[M = \frac{4.7 \times 10^{-5} \mathrm{g}}{7.53 \times 10^{-6} \mathrm{mol}}\] Compute the molar mass: \[M \approx 6,241 \mathrm{g/mol}\] So the molar mass of the protein is approximately 6,241 g/mol.

Key concepts

Osmotic Pressure

Osmotic pressure is a fundamental concept in chemistry and biology, essential for understanding how fluids and their dissolved substances behave. Imagine two solutions separated by a semi-permeable membrane that allows only the solvent (like water) but not the solute (like our protein) to pass through. Osmotic pressure is the pressure that must be applied to the solution to prevent the inward flow of water across the membrane.

When a solute is dissolved in a solvent, the solution's concentration increases, which naturally leads to water moving into it to dilute it -- this is osmosis. The pressure exerted by the solution to draw water in is its 'osmotic pressure'. It is determined by factors including the concentration of the solute in the solution, the temperature of the solution, and the nature of the solute and solvent involved. In the case of our protein, a higher concentration would mean a higher osmotic pressure, and this property is exploited when determining the molar mass of dissolved substances.

Van 't Hoff Equation

The van 't Hoff equation is the perfect bridge between thermodynamics and the practical world of solutions. It connects osmotic pressure (Π) with the number of moles (n) of solute present, the temperature (T) in Kelvin, and the volume (V) of the solution, allowing us to express the behavior of dilute solutions through the ideal gas law.

The equation is written as: \[ΠV = nRT\] where 'R' is the gas constant. To solve for the number of moles 'n', we can rearrange the equation to \[n = \frac{ΠV}{RT}\], as seen in the textbook solution. By determining 'n', and knowing the mass of the solute, we can calculate its molar mass. This equation underscores the proportional relationship between osmotic pressure and solute concentration. Understanding this equation is key for anyone studying physical chemistry, especially when exploring properties of solutions.

Conversion of Units

Accuracy in science is closely tied to the precision of our measurements, and that precision is reflected in the units we use. Conversion of units is an essential skill, as calculations often require measurements to be in specific, consistent units to make sense. For instance, in our example, converting mass from milligrams to grams, volume from milliliters to liters, temperature from Celsius to Kelvin, and pressure from torr to atmospheres ensures we're using the same system (SI units) throughout the calculation.

These conversions are straightforward if one remembers the correct factors: there are 1,000 milligrams in a gram, 1,000 milliliters in a liter, and for temperature, 0 degrees Celsius equates to 273.15 Kelvin. Pressure requires knowing that 1 atm is equivalent to 760 torr. By accurately converting your units prior to calculation, you ensure that the numbers inputted into formulas like the van 't Hoff equation are compatible and ready for a smooth calculation process.