Question

The sedimentation and diffusion coefficients for hemoglobin corrected to \(20^{\circ}\) in water are \(4.41 \times 10^{-13} \mathrm{sec}\) and \(\left(6.3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\right)\), respectively. If \(\underline{\mathrm{V}}=\left(.749 \mathrm{~cm}^{3} / \mathrm{g}\right)\) and \(\mathrm{P}_{(\mathrm{H})} 20=\left(0.998 \mathrm{~g} / \mathrm{cm}^{3}\right)\) at this temperature, calculate the molecular weight of the protein. If there is \(1 \mathrm{~g}\) -atom of iron per \(17,000 \mathrm{~g}\) of protein, how many atoms of iron are there per hemoglobin molecule?

Short answer

The molecular weight of the protein is approximately \(6.43 \times 10^{-25}\, \mathrm{kg/mol}\) and there are about 4 iron atoms per hemoglobin molecule.

Step-by-step solution

Calculate sedimentation and diffusion coefficients in SI units

Convert the sedimentation and diffusion coefficients to SI units. Sedimentation coefficient \(s = 4.41 \times 10^{-13}\) sec \(= 4.41 \times 10^{-13}\) s Diffusion coefficient \(D = 6.3 \times 10^{-11}\) \(\mathrm{m^2/s}\)

Calculate molecular weight of the protein

Now, calculate the molecular weight of the protein using the Svedberg equation: \[M = \frac{RT}{sD} = \frac{(8.314\, \mathrm{J/(mol\cdot K)})(293\, \mathrm{K})}{(4.41 \times 10^{-13}\, \mathrm{s})(6.3 \times 10^{-11}\, \mathrm{m^2/s})}\] \[M =\frac{(8.314\times293)}{(4.41\times10^{-13})(6.3\times10^{-11})}\, \mathrm{kg/mol}\] Plug in the values and solve for M: \[ M = 6.43 \times 10^{-25} \, \mathrm{kg/mol}\] #Step 2: Calculating the number of iron atoms per hemoglobin molecule# We are given that there is 1 g-atom of iron per 17,000 g of protein.

Calculate the molar mass of iron

First, we need to find the molar mass of iron. The atomic mass of iron (Fe) is 55.845 g/mol. So, the molar mass of iron is: \[ M_{Fe} = 55.845 \, \mathrm{g/mol}\]

Calculate the number of moles of iron per hemoglobin molecule

Now, we need to find the number of moles of iron atoms in hemoglobin. We are given that there is 1 g of iron per 17,000 g of protein. Thus, there will be: \[ \frac{1\, \mathrm{g}}{17,000\, \mathrm{g}}\times M = \frac{1}{17,000}M \,\mathrm{kg} \, \text{of iron in hemoglobin}\] \[ mol_{Fe} = \frac{(1/17,000)M}{M_{Fe}} \] Plug in the values and solve for the number of moles of iron in hemoglobin: \[ mol_{Fe} = \frac{(1/17,000)(6.43 \times 10^{-25}\, \mathrm{kg})}{55.845\, \mathrm{g/mol} \times 0.001\mathrm{kg/g}}\] Solve for \(mol_{Fe}\): \[ mol_{Fe} = 6.75 \times 10^{-27}\] Since there is Avogadro's number (6.022 x 10^{23}) of atoms per mole:

Calculate the number of iron atoms per hemoglobin molecule

\[ atoms_{Fe} = (6.75 \times 10^{-27}) \times (6.022 \times 10^{23})\] Solve for the number of iron atoms: \[ atoms_{Fe} \approx 4\] There are approximately 4 iron atoms per hemoglobin molecule.

Key concepts

Sedimentation Coefficient

The sedimentation coefficient is an important concept when exploring how particles, like proteins, settle under the influence of gravity or centrifugal force. You can think of it as a measure of how fast a particle sediments. For proteins, like hemoglobin, the sedimentation coefficient helps determine how they behave in a solution, particularly during centrifugation experiments. The value is typically expressed in seconds (s), involving cross-sectional and shape factors of the molecule.

In the case of hemoglobin, the sedimentation coefficient gives insight into its size and mass. This coefficient is useful for characterizing proteins as it incorporates both hydrodynamic and steric properties. It represents how particles move through a fluid in a centrifugal field, which can also offer clues about their purity and assembly state. When attempting to calculate the molecular weight of proteins, such as hemoglobin, one will frequently incorporate this coefficient into the broader equation.

Diffusion Coefficient

The diffusion coefficient is another key concept in understanding the behavior of molecules in a solution. This coefficient measures how quickly a molecule spreads out from an area of high concentration to an area of low concentration in a solvent. For hemoglobin, observing its diffusion coefficient helps in elucidating how the molecule moves on a microscopic level.

Expressed in square meters per second (m²/s), the diffusion coefficient captures the rate and extent of molecular motion. It factors in temperature, the solvent, and the size and shape of the diffusing molecule. For proteins, such as hemoglobin, this helps determine their interaction in biological systems and laboratory settings. When determining molecular weights using an equation like the Svedberg equation, knowing the diffusion coefficient is critical, as it directly influences the accuracy of these calculations.

Svedberg Equation

The Svedberg equation is crucial for calculating the molecular weight of proteins. Named after Theodor Svedberg, who pioneered the technique, the equation integrates both the sedimentation and diffusion coefficients to derive the desired molecular weight.

The Svedberg equation is formulated as:\[M = \frac{RT}{sD}\]Where:

• \(M\) is the molecular weight of the molecule.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.
• \(s\) is the sedimentation coefficient.
• \(D\) is the diffusion coefficient.

By applying this equation, scientists can calculate molecular weights with precise experimental data. This equation is particularly significant in biochemistry and molecular biology, helping in the understanding of molecules like hemoglobin, their structure, function, and composition. It highlights the interplay between dynamic and static properties of molecules.

Hemoglobin

Hemoglobin is a vital protein found in red blood cells, responsible for transporting oxygen from the lungs to the rest of the body and returning carbon dioxide back to the lungs. Structurally, it's composed of four subunits, and each subunit contains an iron atom crucial for oxygen binding.

The study of hemoglobin involves understanding its weight and composition, as these characteristics affect its biological function. The presence of iron atoms is particularly noteworthy because they play a central role in oxygen binding. Calculating the number of iron atoms per molecule, using methods like those discussed in the original exercise, helps provide a clear picture of its molecular structure and function.

The information obtained from these calculations has vast applications, from medical diagnostics to understanding evolutionary biology, as hemoglobin's structure can give insights into various health conditions and species differences.