Consider a protein sphere with a radius of 1.8 nm, and charge Q=-10 e, in an aqueous solution of

step 1 Boltzman's equation Boltzman formula s is equal to KBLN says that the entropy of microscopic sta

Consider a protein sphere with a radius of 1.8 nm, and...

Consider a protein sphere with a radius of $1.8 \mathrm{nm},$ and charge $Q=-10 e,$ in an aqueous solution of $c_{\infty}=0.05 \mathrm{M}$ $\mathrm{NaCl}$ at $25^{\circ} \mathrm{C}$. Consider the small ions as point charges and use the linear approximation to the Poisson-Boltzmann equation. (a) Fill in the steps leading to Equations 9.70 and $9.72,$ and derive the expression for the potential $V(r)$ of a charged sphere in a salty solution. (b) What is the surface potential of the protein in units $k_{\mathrm{B}} T / e ?$ (c) What is the concentration of $\mathrm{Na}^{+}$ ions and of $\mathrm{Cl}^{-}$ ions at the surface of the protein? (d) What is the concentration of $\mathrm{Na}^{+}$ and $\mathrm{Cl}^{-}$ ions at a distance of $0.3 \mathrm{nm}$ from the protein surface? (Adapted from Problem 23.2 of $\mathrm{K}$. Dill and S. Bromberg, Molecular Driving Forces, 2 nd ed. Garland Science, $2011 .$ )

Consider a protein sphere with a radius of $1.8 \mathrm{nm},$ and charge $Q=-10 e,$ in an aqueous solution of $c_{\infty}=0.05 \mathrm{M}$ $\mathrm{NaCl}$ at $25^{\circ} \mathrm{C}$. Consider the small ions as point charges and use the linear approximation to the Poisson-Boltzmann equation.
(a) Fill in the steps leading to Equations 9.70 and $9.72,$ and derive the expression for the potential $V(r)$ of a charged sphere in a salty solution.
(b) What is the surface potential of the protein in units $k_{\mathrm{B}} T / e ?$
(c) What is the concentration of $\mathrm{Na}^{+}$ ions and of $\mathrm{Cl}^{-}$ ions at the surface of the protein?
(d) What is the concentration of $\mathrm{Na}^{+}$ and $\mathrm{Cl}^{-}$ ions at a distance of $0.3 \mathrm{nm}$ from the protein surface? (Adapted from Problem 23.2 of $\mathrm{K}$. Dill and S. Bromberg, Molecular Driving Forces, 2 nd ed. Garland Science, $2011 .$ )

Key Concepts

Boltzmann Distribution for Ion Concentration

The Boltzmann distribution explains how the concentration of ions in a solution is affected by the local electrostatic potential. In an electrolyte, the ions distribute themselves according to the electrostatic energy relative to thermal energy. This results in a modification of the ion concentration near charged surfaces compared to the bulk value. The distribution is key to predicting local ionic concentrations and hence the local chemical environment near a charged protein or surface in the presence of an external potential.

Debye Screening Length

The Debye screening length is a measure of the distance over which electrostatic interactions are screened in an electrolyte solution. It represents the characteristic length scale over which the potential decays exponentially from a charged object. This concept is crucial in understanding how ions in solution rearrange themselves around charged objects, leading to a reduction in the effective range of electrostatic interactions. In the context of the linearized Poisson-Boltzmann equation, the Debye length naturally appears in the mathematical formulation of the potential decay.

Yukawa (Screened Coulomb) Potential

The Yukawa potential, also known as the screened Coulomb potential, arises as a solution to the linearized Poisson-Boltzmann equation for a point or spherical charge in an ionic solution. It describes how the electrostatic potential decays exponentially away from the charged object due to the presence of the salt ions. This form of the potential is particularly important in colloid and biophysics, as it quantifies how long-range Coulomb interactions are modified in the presence of a screening medium.

Poisson-Boltzmann Equation

The Poisson-Boltzmann equation is a fundamental relation that combines Poisson’s equation from electrostatics with the Boltzmann distribution from statistical mechanics. It describes how the electrostatic potential varies in an ionic solution due to a fixed charge distribution. This equation is central to understanding the behavior of charged objects in electrolytes and is used to calculate the spatial variation of the potential resulting from the interaction between the fixed charges on a molecule or surface and the mobile ions in solution.

Linearization of the Poisson-Boltzmann Equation

Under conditions where the electrostatic potential is relatively small compared to thermal energy, the Poisson-Boltzmann equation can be approximated by linearizing the exponential term in the Boltzmann factor. This simplification permits analytical solutions and makes it easier to derive expressions for the electrostatic potential in the presence of ionic screening. The linearized version still captures the essential physics of charge screening but is mathematically more tractable for systems such as charged spheres in salt solutions.

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