page_title: Debye–Hückel theory
section_title: Mathematical development
hierarchical_section_title: Debye–Hückel theory / Mathematical development
caption_attribution_description: English: Debye-Hueckel limiting law
context_page_description: The Debye–Hückel theory was proposed by Peter Debye and Erich Hückel as a theoretical explanation for departures from ideality in solutions of electrolytes and plasmas. It is a linearized Poisson–Boltzmann model, which assumes an extremely simplified model of electrolyte solution but nevertheless gave accurate predictions of mean activity coefficients for ions in dilute solution. The Debye–Hückel equation provides a starting point for modern treatments of non-ideality of electrolyte solutions.
context_section_description: The deviation from ideality is taken to be a function of the potential energy resulting from the electrostatic interactions between ions and their surrounding clouds. To calculate this energy two steps are needed.
  The first step is to specify the electrostatic potential for ion j by means of Poisson's equation
  ψ(r) is the total potential at a distance, r, from the central ion and ρ(r) is the averaged charge density of the surrounding cloud at that distance. To apply this formula it is essential that the cloud has spherical symmetry, that is, the charge density is a function only of distance from the central ion as this allows the Poisson equation to be cast in terms of spherical coordinates with no angular dependence.
  The second step is to calculate the charge density by means of a Boltzmann distribution.
  where kB is Boltzmann constant and T is the temperature. This distribution also depends on the potential ψ(r) and this introduces a serious difficulty in terms of the superposition principle. Nevertheless, the two equations can be combined to produce the Poisson–Boltzmann equation.
  Solution of this equation is far from straightforward. Debye and Hückel expanded the exponential as a truncated Taylor series to first order. The zeroth order term vanishes because the solution is on average electrically neutral (so that ∑ nᵢ zᵢ = 0), which leaves us with only the first order term.  The result has the form of the Helmholtz equation
  ,
  which has an analytical solution. This equation applies to electrolytes with equal numbers of ions of each charge. Nonsymmetrical electrolytes require another term with ψ². For symmetrical electrolytes, this reduces to the modified spherical Bessel equation
  The coefficients  and  are fixed by the boundary conditions.  As ,  must not diverge, so .  At , which is the distance of the closest approach of ions, the force exerted by the charge should be balanced by the force of other ions, imposing , from which  is found, yielding
  The electrostatic potential energy, , of the ion at  is
  This is the potential energy of a single ion in a solution. The multiple-charge generalization from electrostatics gives an expression for the potential energy of the entire solution (see also: Debye–Hückel equation). The mean activity coefficient is given by the logarithm of this quantity as follows (see also: Extensions of the theory)
  where I is the ionic strength and a₀ is a parameter that represents the distance of closest approach of ions. For aqueous solutions at 25 °C A = 0.51 mol-1/2dm3/2 and B = 3.29 nm⁻¹mol-1/2dm3/2
  The most significant aspect of this result is the prediction that the mean activity coefficient is a function of ionic strength rather than the electrolyte concentration. For very low values of the ionic strength the value of the denominator in the expression above becomes nearly equal to one. In this situation the mean activity coefficient is proportional to the square root of the ionic strength. This is known as the Debye–Hückel limiting law.
caption: Experimental 
    
      
        
          log
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            γ
            
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      {\displaystyle \log \gamma _{\pm }}
    
   values for KBr at 25°C (points)[14] and Debye–Hückel limiting law (coloured line)
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