A protein molecule, \(\mathrm{P}\), of molar mass \(\mathscr{A}\) dimerizes when it is allowed to stand in solution at room temperature. A plausible mechanism is that the protein molecule is first denatured (that is, loses its activity due to a change in overall structure) before it dimerizes: $$ \begin{array}{rlr} \mathrm{P} & \stackrel{k}{\longrightarrow} \mathrm{P}^{*}(\text { denatured }) & \text { slow } \\ 2 \mathrm{P}^{*} \longrightarrow \mathrm{P}_{2} & \text { fast } \end{array} $$ where the asterisk denotes a denatured protein molecule. Derive an expression for the average molar mass (of \(\mathrm{P}\) and \(\mathrm{P}_{2}\) ), \(, \overline{\mathscr{M}}\), in terms of the initial protein concentration \([\mathrm{P}]_{0}\) and the concentration at time \(t,\) \([\mathrm{P}]_{,}\) and \(\mathscr{A} .\) Describe how you would determine \(k\) from molar mass measurements.

Step by step solution

01

Determine \(k\)

To determine the rate constant \(k\) from molar mass measurements, first measure the concentration of protein P at various times and then use these values to calculate the rate at which the protein is denatured. Knowing the initial concentration of the protein P, the rate law for the first step of the reaction can be written as: \(\text{rate of denaturation} = k [\mathrm{P}]\). As the first step is the slow (rate-determining) step, the rate of the overall reaction should also be equal to the rate of denaturation. By comparing the experimental rate of the reaction with the theoretical rate, the rate constant \(k\) can be determined.

02

Calculate the mole fraction

To find the average molar mass, we need to consider both the individual protein molecules P and the protein dimer \(P_{2}\). Compute the mole fraction of the dimer using the initial and final protein concentrations, \(x_{P} = \frac{[\mathrm{P}]}{[\mathrm{P}]_0}\) and \(x_{P_{2}} = 1 - x_{P}\). This considers that at the beginning all molecules are in the form of protein P and that over time some portion of protein P is converted into dimer \(P_{2}\).

03

Compute the average molar mass

The average molar mass \(\overline{\mathscr{M}}\) is the sum of the product of the mole fraction and molar mass of each species. Knowing that the molar mass of \(P_{2}\) is twice the molar mass of P, compute \(\overline{\mathscr{M}} = x_{P}\mathscr{A} + x_{P_{2}}2\mathscr{A} = \mathscr{A} + (2\mathscr{A}-\mathscr{A})x_{P_{2}} = \mathscr{A} + \mathscr{A}(1 - x_{P})\). This shows that the average molar mass depends on the initial concentration \([\mathrm{P}]_{0}\), the concentration at time \(t,\) \([\mathrm{P}]\), and the molar mass of the protein \(\mathscr{A}\).

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