Question

A protein has been isolated as a salt with the formula \(\mathrm{Na}_{20} \mathrm{P}\) (this notation means that there are \(20 \mathrm{Na}^{+}\) ions associated with a negatively charged protein \(\mathrm{P}^{20-}\) ). The osmotic pressure of a \(10.0-\mathrm{mL}\) solution containing \(0.225 \mathrm{~g}\) of the protein is 0.257 atm at \(25.0^{\circ} \mathrm{C}\). (a) Calculate the molar mass of the protein from these data. (b) Calculate the actual molar mass of the protein.

Short answer

Molar mass of protein is 415,482 g/mol; actual molar mass 415,022 g/mol.

Step-by-step solution

Understanding Osmotic Pressure

Osmotic pressure (\(\Pi\)) is given by the formula \(\Pi = iMRT\), where \(i\) is the van't Hoff factor, \(M\) is molarity, \(R\) is the ideal gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in Kelvin. Since the protein salt dissociates into \(20\,Na^+\) and \(P^{20-}\), \(i = 21\). The temperature \(T = 25.0\,^{\circ}C = 298\,K\).

Calculate Molarity

We know that \(\Pi = 0.257\,atm\), so \(0.257 = 21 \times M \times 0.0821 \times 298\). Solving for \(M\), we get \(M = 0.257/(21 \times 0.0821 \times 298)\). After performing the calculation, \(M \approx 0.00005413\,mol/L\).

Determine Number of Moles

Since molarity \(M\) is moles per liter and we have a solution volume of \(10.0\,\text{mL} = 0.0100\,L\), the number of moles \(n = M \times \text{volume in L}\). Thus, \(n = 0.00005413 \times 0.0100\). Therefore, \(n \approx 5.413 \times 10^{-7}\,mol\).

Calculate Molar Mass of the Protein

Molar mass (\(Mm\)) is mass per mole. Given that the protein mass is \(0.225\,g\), \( Mm = \frac{0.225\,g}{5.413 \times 10^{-7}\, mol} \). Performing the division, \( Mm \approx 415482\,g/mol \).

Determine Actual Molar Mass

Since the actual protein has 20 charges and there are also \(20\,Na^+\) ions associated, the actual molar mass needs to exclude the 20 sodium ions. Molar mass of sodium is \(22.99\,g/mol\). Total sodium mass is \(20 \times 22.99 = 459.8\,g/mol\). Thus, protein molar mass without sodium is \(415482 - 459.8\).

Final Actual Molar Mass

Finally, subtracting gives the hint towards the actual molar mass of the protein, \(M_m = 415022.2\,g/mol\).

Key concepts

Protein Molar Mass

In chemistry, molar mass is the mass of one mole of a given substance. When it comes to proteins, which are large biomolecules composed of amino acids, understanding their molar mass is critical. Calculating the molar mass of a protein generally requires knowing the mass of the sample and the number of moles present. This is especially useful in biochemistry to determine the sizes of proteins or other macromolecules. In the problem, the molar mass was initially calculated using the concept of osmotic pressure. With a solution containing 0.225 grams of protein, and given the osmotic pressure, we used the formula for osmotic pressure to determine the molarity, and hence the number of moles. By dividing the mass of the protein sample by the number of moles, the molar mass was calculated to be approximately 415,482 g/mol. This value includes contributions from both the protein and sodium ions. However, for proteins like the one discussed, it's essential to correct for the presence of sodium ions when determining the actual molar mass. By subtracting the total mass contribution of sodium ions ( 20 sodium ions times their individual molar mass) from the initial calculation, we get the protein's molar mass alone.

Van't Hoff Factor

The Van't Hoff factor (\(i\)) is used in the discussion of colligative properties to quantify the effect of solute disassociation in a solution. It essentially represents the number of particles a compound dissociates into in solution. For example, when a salt like NaCl dissolves, it dissociates into two ions: Na\(^+\) and Cl\(^-\), giving it an \(i\)-value of 2.In our specific case, a protein associated with sodium ions is dissolved, the Van't Hoff factor is considerably larger due to the formula \(\mathrm{Na}_{20}\mathrm{P}\). This dissociates into one negatively charged protein \(\mathrm{P}^{20-}\), and twenty positively charged sodium ions \(20\mathrm{Na}^+\). Therefore, \\(i\) for this solute is 21, as it accounts for each particle's effect on the solution properties. This factor is fundamental in calculating properties like osmotic pressure since it allows us to understand how much the solute will affect the solvent behavior when dissolved.

Dissociation in Solution

Dissociation in solution refers to the process by which ionic compounds split into their constituent ions when dissolved in a solvent like water. This process is crucial in understanding how solutes affect the properties of solutions, such as conductivity and osmotic pressure.When we talk about dissociation in this problem, the protein complex \(\mathrm{Na}_{20}\mathrm{P}\) dissociates into intricately charged ions within the solution. Each compound of 20 sodium ions and one protein molecule separates into 21 ions: \(20\mathrm{Na}^{+}\) and \(\mathrm{P}^{20-}\). The extent of this dissociation directly influences the concentration of particles in solution, and thus impacts properties that depend on particle number rather than identity, like osmotic pressure.Understanding dissociation also helps explain why some substances have a greater effect on boiling point elevation or freezing point depression in comparison to molecular solutes with the same initial concentration.

Ideal Gas Constant

Central to calculations involving gases and solutions is the Ideal Gas Constant (\(R\)). It appears in equations that describe ideal gas law behavior and osmotic pressure relations, serving to relate multiple physical properties of a system.In the context of our solution, \(R\) is used as part of the formula for calculating osmotic pressure: \\(\Pi = iMRT\). Here, \(R\) has a value of 0.0821 L atm/mol K, providing a bridge between temperature, molarity, and pressure.This constant allows easy transition from theoretical ideas to practical chemical calculations, allowing us to solve problems involving a diverse range of gaseous and dissolved scenarios. It underpins a wide array of chemical comprehension, turning theoretical principles about particle interactions into actionable, calculable results that describe real-world systems accurately.