Question

Debye lengths. \(\ln\) the Debye-H?ckel theory of monovalent salt solutions, there is a characteristic length quantity \(\kappa\), defined by \(\kappa^{2}=\left(2 e^{2} n_{\infty}\right) /\left(\varepsilon_{0} D k T\right)\), where \(n_{\infty}\) is the salt concentration. (a) Express \(\kappa\) in terms of the Bjerrum length \(\ell_{B}\). (b) For water at room temperature, \(\ell_{B}=7.13 \mathrm{~A}\). Compute the Debye length \(1 / \kappa\) in \(A\) for a solution of \(n_{\text {es }}=1 \mathrm{~mol} \mathrm{~L}^{-1}\).

Short answer

\(\kappa = \sqrt{\dfrac{2 n_{\infty}}{\ell_{B}}} \) and the Debye length is approximately 2.67 Å for the given solution.

Step-by-step solution

Define the Bjerrum length

The Bjerrum length \(\ell_{B}\) is defined as \(\ell_{B} = \dfrac{e^2}{\varepsilon_{0} D k T}\). This represents the distance at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy.

Substitute Bjerrum length into \(\kappa\)

Using the definition of \(\ell_{B}\), express \(\kappa\) in terms of \(\ell_{B}\). Substitute \(\ell_{B} = \dfrac{e^2}{\varepsilon_0 D k T}\) into \(\kappa^{2} = \dfrac{2 e^{2} n_{\text{es}}}{\varepsilon_{0} D k T}\). This gives us: \[\kappa^2 = \dfrac{2 n_{\infty}}{\ell_{B}}\].

Express \(\kappa\) in simplified form

Solve for \(\kappa\): \[\kappa = \sqrt{\dfrac{2 n_{\infty}}{\ell_{B}}}\].

Calculate the Debye length

For water at room temperature, \(\ell_{B} = 7.13 Å = 7.13 \times 10^{-10} m\). Given \(n_{\text{es}} = 1 \mathrm{~mol} \mathrm{~L}^{-1}\), convert to \(m^{-3}\): \(n_{\text{es}} = 1 \times 10^3 \mathrm{~mol} \mathrm{~m}^{-3}\). Substitute the values into the Debye length formula \(\dfrac{1}{\kappa}\): \[\dfrac{1}{\kappa} = \dfrac{1}{\sqrt{\dfrac{2 \times 1 \times 10^3}{7.13 \times 10^{-10}}}} \approx 2.67 Å\].

Key concepts

Debye-Hückel theory

The Debye-Hückel theory is essential for understanding ionic interactions in solutions, especially those involving salts. This theory helps us predict how ions behave in an electrolyte. It is significant because it lays the foundation for calculating important quantities like the Debye length.
The central idea is that ions in a solution are surrounded by a 'cloud' of oppositely charged ions. This cloud influences the electrostatic potential around a given ion. The Debye-Hückel theory quantitatively describes this potential, which decreases exponentially with distance.
The characteristic screening length is termed the Debye length, represented as \(\dfrac{1}{\kappa}\). This length determines how quickly the electrostatic interactions fall off around an ion. A shorter Debye length means stronger screening and thus weaker effective interactions between ions at larger distances.

Bjerrum length

The Bjerrum length is another critical concept in understanding electrostatic interactions in ionic solutions. It is the distance at which the thermal energy of ions is equivalent to their electrostatic interaction energy. Mathematically, it's expressed as:
\(\ell_B = \dfrac{e^2}{\varepsilon_0 D k T}\).
Here,

• \(e\) is the elementary charge,
• \(\varepsilon_0\) is the vacuum permittivity,
• \(D\) is the dielectric constant of the medium,
• \(k\) is the Boltzmann constant,
• and \(T\) is the absolute temperature.

In simple terms, the Bjerrum length defines how far apart two ions can be while still feeling a significant electrostatic force due to each other. For instance, in water at room temperature, the Bjerrum length is approximately 7.13 Å. This is relatively short, meaning that ions can interact strongly at even small distances.

Electrostatic Interaction

Electrostatic interaction is a fundamental force that acts between charged particles. In the context of ionic solutions, it describes the forces that arise due to the presence of charged ions. These interactions can be attractive or repulsive depending on the sign of the charges.
Electrostatic interactions are governed by Coulomb's law, which states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance separating them.
In formulas, this appears as:
\(F = \dfrac{e^2}{4 \pi \varepsilon r^2}\), where

• \(F\) is the force between the ions,
• \(e\) is the elementary charge,
• \(\varepsilon\) is the permittivity of the medium,
• and \(r\) is the distance between the charges.

The interaction can be strong when ions are close, but the presence of other ions and the solvent reduces this interaction. Hence, understanding electrostatic interactions is crucial for analyzing processes like solvation, conductivity, and stability of ionic solutions.