Question

A \(4.7 \times 10^{-2}\) 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 75,440 g/mol.

Step-by-step solution

Convert the given values into appropriate units

First, we need to convert the given values into appropriate units: Mass of protein = \(4.7 \times 10^{-2}\) mg = \(4.7 \times 10^{-5}\) g (1 g/1000 mg) Volume of solution = 0.25 mL = 0.00025 L (1 L/1000 mL) Osmotic pressure = 0.56 torr = 0.000736 atm (1 atm/760 torr) Temperature = 25°C = 298.15 K (273.15 + 25)

Calculate the number of moles of the protein using osmotic pressure formula

Using the osmotic pressure formula, we can calculate the number of moles of the protein: Osmotic pressure = (Molarity) × (R) × (Temperature) Rearrange the formula to find molarity: Molarity = Osmotic pressure / (R × Temperature) Plug in the given values: Molarity = 0.000736 atm / (0.0821 L·atm/mol·K × 298.15 K) Molarity = \(2.49 \times 10^{-6}\) mol/L

Find the number of moles of protein in the given volume of solution

Now, we can find the number of moles of the protein in the 0.00025 L solution using the calculated molarity: Number of moles = Molarity × Volume of solution Number of moles = \(2.49 \times 10^{-6}\) mol/L × 0.00025 L Number of moles = \(6.23 \times 10^{-10}\) mol

Calculate the molar mass of the protein

Finally, use the mass of the protein sample and the number of moles to find the molar mass of the protein: Molar mass = Mass of protein / Number of moles Molar mass = \(4.7 \times 10^{-5}\) g / \(6.23 \times 10^{-10}\) mol Molar mass = 75,440 g/mol The molar mass of the protein is approximately 75,440 g/mol.

Key concepts

Molar Mass Calculation

Calculating the molar mass of a substance involves determining the mass of one mole of that substance. It connects mass with the number of moles to give us understanding about the substance at a molecular level. In our case, we are dealing with a protein whose mass and number of moles are known. To find the molar mass (M) of the protein, we use the formula:\[\text{Molar Mass} = \frac{\text{Mass of Sample}}{\text{Number of Moles}}\]Here:- The mass of the protein sample is given as \(4.7 \times 10^{-5}\) grams.- The number of moles of protein, calculated from osmotic pressure, is \(6.23 \times 10^{-10}\) moles.By dividing the mass by the number of moles, we obtain the molar mass, a quantity crucial for understanding the macroscopic properties of the protein in question.

Molarity

Molarity is a measure of the concentration of a solute within a solution. It provides the number of moles of solute per liter of solution. Understanding molarity is essential when dealing with solutions, especially when calculating properties like osmotic pressure.In this exercise, we determine molarity to help find the osmotic pressure value. To calculate molarity (M), we use the rearranged formula from the osmotic pressure equation:\[\text{Molarity} = \frac{\text{Osmotic Pressure}}{R \times T}\]Where:- Osmotic Pressure is converted into atmospheres (\(0.000736\) atm).- \(R\) is the ideal gas constant (\(0.0821\) L·atm/mol·K).- \(T\) is the temperature in Kelvin (\(298.15\,\text{K}\)).The molarity of the solution comes out to \(2.49 \times 10^{-6}\) mol/L, which informs us about the concentration of our protein in the solution.

Unit Conversion

Unit conversion is a fundamental step in scientific calculations, ensuring consistent units across all measurements. For this problem, several conversions were necessary to align with standard units used in scientific formulas.

Why Convert Units?

To use formulas like the one for osmotic pressure, measurements must be in specific units. Mismatched units can lead to incorrect results.

Conversions Needed

• Mass of Protein: The protein's mass was converted from milligrams to grams to match with the mole unit in molar mass calculations. \(1\,\text{mg} = 10^{-3}\,\text{g}\).
• Volume of Solution: Milliliters were converted to liters because molarity is moles per liter. \(1\,\text{mL} = 10^{-3}\,\text{L}\).
• Osmotic Pressure: Torr was converted to atmospheres, a standard unit for pressure in the ideal gas law. \(1\,\text{atm} = 760\,\text{torr}\).
• Temperature: Celsius was converted to Kelvin to be compatible with the ideal gas constant \((R)\), adding \(273.15\) to the Celsius value.

These conversions ensure that every part of our equation works together seamlessly.

Protein Chemistry

Protein chemistry is a branch of biochemistry that studies the structure, function, and properties of proteins. Proteins are essential molecules, made up of amino acids and involved in nearly all cellular processes. Understanding their molar mass helps in elucidating their function and interaction in biological systems.

Why Molar Mass Matters

Molar mass provides insights into the protein's size and composition. It is crucial for understanding how proteins function in various environments. For instance, molar mass influences: - Solubility: Larger, more complex proteins may have different solubility properties. - Structural Behavior: Knowing the molar mass can hint at the protein's folding and stability. - Interaction with Other Molecules: The size and mass of a protein alter its binding capacities and interaction sites. This exercise, through calculating the molar mass from osmotic pressure, gives a practical example of applying theoretical concepts to understand a protein's characteristics.