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 \(4.66 \times 10^3 \mathrm{g/mol}\).

Step-by-step solution

Write down the osmotic pressure formula

The osmotic pressure formula is given by: \[\Pi = \cfrac{nRT}{V}\] Where: \(\Pi\) = osmotic pressure, \(n\) = number of moles of solute, \(R\) = gas constant, \(T\) = temperature in Kelvin, and \(V\) = volume of the solution.

Convert temperature to Kelvin

We know that the temperature is given in Celsius, and we need to convert it to Kelvin. To do this, add 273.15 to the given temperature: \[T = 25 + 273.15 = 298.15 \mathrm{K}\]

Convert torr to atmosphere

The osmotic pressure is given in torr, we need to convert it to atmosphere (atm), because the gas constant R is in atm when used for osmotic pressure calculations. Use the conversion factor 1 atm = 760 torr: \[\Pi = \cfrac{0.56 \mathrm{torr}}{760 \mathrm{torr/atm}} = 7.37 \times 10^{-4} \mathrm{atm}\]

Convert volume to liters

The volume is given in mL, we need to convert it to liters to work with the osmotic pressure formula. Use the conversion factor 1 L = 1000 mL: \[V = \cfrac{0.25 \mathrm{mL}}{1000 \mathrm{mL/L}} = 2.50 \times 10^{-4} \mathrm{L}\]

Find the number of moles

Use the osmotic pressure formula to find the number of moles of the protein: \[n = \cfrac{\Pi V}{RT}\] Where: \(R\) = 0.0821 \(\mathrm{L\cdot atm / (mol\cdot K)}\), Plug in the values of \(\Pi\), \(T\), and \(V\) we found earlier: \[n = \cfrac{7.37 \times 10^{-4} \mathrm{atm} \times 2.50 \times 10^{-4} \mathrm{L}}{0.0821 \mathrm{L\cdot atm / (mol\cdot K)} \times 298.15 \mathrm{K}} = 1.01 \times 10^{-5} \mathrm{mol}\]

Calculate molar mass

We are given the mass of the protein sample in mg, so we will first convert it to grams using the conversion factor 1 g = 1000 mg: \[m = 4.7 \times 10^{-2} \mathrm{mg} \times \cfrac{1\mathrm{g}}{1000\mathrm{mg}} = 4.7 \times 10^{-5} \mathrm{g}\] Then, we can use the number of moles we found in step 5 to calculate the molar mass of the protein: \[M = \cfrac{m}{n} = \cfrac{4.7 \times 10^{-5} \mathrm{g}}{1.01 \times 10^{-5} \mathrm{mol}} = 4.66 \times 10^3 \mathrm{g/mol}\] The molar mass of the protein is approximately \(4.66 \times 10^3 \mathrm{g/mol}\).

Key concepts

Molar Mass Determination

Molar mass is a crucial concept in chemistry used to identify the molecular weight of substances. It's the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). To determine the molar mass of a protein or any solute when its osmotic pressure is known, follow this systematic approach:

• Firstly, utilize the osmotic pressure formula: \[ \Pi = \frac{nRT}{V} \]. Here \( \Pi \) stands for osmotic pressure, \( n \) is the number of moles, \( R \) is the gas constant, \( T \) is temperature in Kelvin, and \( V \) is the volume of the solution in liters.
• Calculate the number of moles \( n \) from the osmotic pressure to find the amount of substance in the solution.
• Next, convert the given mass from milligrams to grams before dividing it by the number of moles, as molar mass \( M \) is determined by: \( M = \frac{m}{n} \).
• Through this process, you can accurately ascertain the molar mass of your compound, which, in a sample exercise, was found to be approximately \(4.66 \times 10^3\) g/mol.

Understanding these steps and calculations can greatly aid in solving chemistry problems involving molarity.

Gas Constant

In the realm of chemistry and physics, the gas constant (\( R \)) is a fundamental factor in the study of gases and solutions. It relates energy scales in the context of gases. In osmotic pressure calculations, \( R \) serves as a bridge between pressure, volume, and temperature.
For osmotic pressure calculations, \( R \) is commonly used as 0.0821 L·atm / (mol·K), which suits the pressures expressed in atmospheres. It's important to ensure that other units in your calculation align with \( R \) for coherence.

• \( R \) provides consistency across various scientific calculations involving gases and solutions.
• Its standardized value enables simplified and uniform calculations across experiments and theoretical work.

Remember, sticking to correct unit conversions in line with \( R \) is vital to ensure accurate results.

Solution Concentration

Solution concentration is a measure of the amount of solute present in a given quantity of solvent or solution. In this context, having a keen understanding of concentration is essential to calculate properties such as osmotic pressure markedly. In chemistry:

• Concentration is often measured in terms of molarity, which involves moles per liter of solution.
• The process of calculating osmotic pressure involves knowing the moles of solute within a specific volume.
• As seen in exercises utilizing osmotic pressure calculations, knowledge of volume conversion and mole determination are essentials.

Thus, mastering concentration nuances assists in detecting solute interactions and predicting solution behaviors.

Unit Conversion

Converting units is a fundamental skill in scientific calculations, ensuring consistency and accuracy. This principle holds particularly true in chemistry, where quantities like mass, volume, temperature, and pressure require specific unit formats.
The following conversions are crucial in solving osmotic pressure problems:

• For temperature, convert Celsius to Kelvin by adding 273.15. This adjustment aligns with the gas constant.
• Osmotic pressure often needs conversion from other units, like torr to atmospheres, as 1 atm equals 760 torr.
• Volume measurements, essential in solution-based exercises, typically need conversion from milliliters to liters, where 1 L equals 1000 mL.
• Mass conversions from milligrams to grams, using 1 g = 1000 mg, ensure uniformity in molar mass calculations.

Each of these conversions is pivotal for maintaining accuracy across calculations, thereby fostering precise and credible results.