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Dipyrrin titration mass balance keeps track of the amount of dye while acid addition changes its protonation states and the solution volume. An absorbance decrease at one wavelength can result from redistribution or dilution, rather than destruction of dipyrromethene units. In a primary study of fused derivatives, acid titration produced several distinct spectral responses and, for some derivatives, more than one protonation step.^[ref]

For an original two-species model containing neutral $\mathrm{LH}$ and protonated $\mathrm{L}{\mathrm{H}}_2^{+}$, conservation givesThe relevant acid dissociation constant supplies a second relation. In an ideal dilute aqueous model,This chosen model does not assign an aqueous constant to the cited dyes in an organic solvent. If pH is not independently fixed, acid inventory and electroneutrality must also be solved; adding a known volume of acid does not prescribe the final free hydrogen ion concentration by itself.

The Beer-Lambert law predictsFor a stipulated 0.10 micromole of dye initially in 10 mL, adding 2.5 mL gives $C_L=8$ micromolar. At a measured condition with $f_H=1/2$, $l=1$ cm, ${ϵ}_H=20000$ and ${ϵ}_N=60000\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$, $A=0.32$. Omitting dilution predicts 0.40 despite exactly the same dye amount and species fraction.

Aliquot-aware dye inventory in titration adds another correction. If aliquots of volume $v_j$ are removed from well-mixed solutions having total dye concentrations $C_j$, the remaining amount is $n_0-{\sum }_j{v}_j{C}_j$, unless other gains or losses occur. Adsorption, precipitation or chemical transformation require additional inventories. For species containing two dye units, their concentration contributes twice to the conserved dye-unit sum.

A useful titration records additions, withdrawals and full spectra, then checks whether returning the acid-base conditions recovers the predicted inventory-corrected response. Conservation of dye units, charge balance and the spectral measurement equation impose different constraints; agreement with only one does not establish the others.

Electroneutrality means that positive and negative charges balance on the length scale under consideration. For a bulk electrolyte with ionic concentrations $c_i$ and charge numbers $z_i$, the approximation is ${\sum }_i{z}_i{c}_i=0$.

A solution containing sodium and sulfate ions therefore needs twice as many singly charged sodium ions as doubly charged sulfate ions if these are its only ions. Local charge separation still occurs near an electrode in an electric double layer. Thus bulk electroneutrality does not mean that every microscopic region is neutral, and it should not be imposed pointwise when modeling a charged interface.

When a single ion species supplies the net charging of a cell membrane, its bulk concentration increment is $\Delta c=C_{m}A\Delta V/(zF\mathcal{V})$. Opposing ionic Electric currents can make gross entry exceed this minimum net electric charge budget. A large membrane potential change can coexist with approximate bulk electroneutrality.

A membrane potential can coexist with near electroneutrality in the bulk cytosol and external chemical solution. The small imbalance producing the voltage lies near the cell membrane; most positive and negative electric charges balance in the bulk.^[ref]

For a separate original capacitor model, choose a total membrane capacitance $C=5\mathrm{pF}$ and $V_m=-70\mathrm{mV}$. These are stipulated parameters, not measured properties of a particular biological cell, and this voltage is not asserted to equal the preceding $E_{\mathrm{K}}$. Model the separated electric charge relative to an uncharged reference byUsing the elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$, this representsexcess monovalent negative-charge equivalents inside. These can represent a deficit of positive ions, rather than the arrival of that many negative ions.

Independently choose an internal reservoir volume $\mathcal{V}=1\mathrm{pL}=10^{-12}\mathrm{L}$ containing $140\mathrm{mM}=0.140\mathrm{mol}{\mathrm{L}}^{-1}$ potassium, with balancing negative electric charge. With the Avogadro constant $N_A=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$, its potassium inventory isThus, if charging were entirely due to outward monovalent potassium transport, about one in $38600$ of these potassium ions would suffice. This is a comparison with the chosen potassium inventory, not a fraction of all charged particles or a measurement of spatial charge density. No shape, surface area or thickness has been inferred from the chosen capacitance and volume. The calculation illustrates why substantial voltage does not require a strongly charged bulk cytosol.

Suppose initially $c_1>c_2$. Positive ions initially move by diffusion from side 1 to side 2. Since negative counterions cannot cross, side 1 acquires a negative interfacial charge and side 2 a positive one. The resulting electrical force opposes further positive-ion transfer. The two contributions balance when the electrochemical potentials are equal; individual ions still cross in both directions, but there is no net electric current of the permeant chemical species.

A simple capacitor model makes the feedback quantitative. If $n$ moles of monovalent positive ions move from initially neutral side 1 to side 2, the charge on side 1 is $-Fn$, hence $V=-Fn/C$. The free-energy stationarity condition isAs $n$ increases, both the concentration imbalance and the voltage drive toward reversal diminish the original diffusion tendency. The equilibrium lies before the bulk concentrations become equal. This demonstrates why a nonzero Nernst potential can coexist with very little change in bulk composition and approximate bulk electroneutrality. A concentration difference alone does not maintain an electrical voltage if both counterions can freely follow each other; selective permeability is essential to this model.

The equilibrium molecular-acid concentration is $\boxed{[{\mathrm{H}}_2{\mathrm{N}}_2{\mathrm{O}}_2]=0.162\mathrm{mol}{\mathrm{L}}^{-1}}$. From the supplied pH convention,With only these monovalent ions appreciably present, electroneutrality requires $[\mathrm{H}{\mathrm{N}}_2{\mathrm{O}}_2^{-}]=[{\mathrm{H}}^{+}]$. Substitution in the acid dissociation constant givesThis calculation assumes ideal thermodynamic activities for the acid and its conjugate base as well as for ${\mathrm{H}}^{+}$, negligible water autoionization, negligible second dissociation, and no additional electrolyte. The acid's analytical concentration would include its dissociated fraction and be approximately $0.16220\mathrm{M}$, rather than exactly the molecular-acid concentration requested.

Bulk electroneutrality gives $\boxed{[\mathrm{N}{\mathrm{O}}_3^{-}]=[{\mathrm{H}}^{+}]+2[\mathrm{Z}{\mathrm{n}}^{2+}]}$. Each dissolved zinc ion contributes two positive elementary charges, whereas each nitrate contributes one negative charge.

This assumes complete nitric acid dissociation, only ${\mathrm{H}}^{+}$ and $\mathrm{Z}{\mathrm{n}}^{2+}$ as significant cations, and only nitrate as a significant anion. It neglects $\mathrm{O}{\mathrm{H}}^{-}$ at acidic pH, other dissolved salts, zinc complexes and additional ionic nitrogen products. It is a bulk charge balance; a thin charged electrode interface does not invalidate bulk electroneutrality.

The particle carries positive surface charge density $\sigma =Q/(4\pi a^2)$. Its surrounding electric double layer has an excess of negative chloride counterions and a deficit of positive magnesium coions. Thus $\varphi (r)>0$ decreases to zero, $n_{\mathrm{Cl}}(r)$ decreases towards its bulk value, and $n_{\mathrm{Mg}}(r)$ increases towards its bulk value. The diffuse charge density is negative and tends to zero.

If $c$ denotes the bulk number of dissolved salt formula units per unit volume, the far-field number densities are $n_{\mathrm{Mg},\infty }=c$ and $n_{\mathrm{Cl},\infty }=2c$. The bulk is neutral even though these number densities differ. In the dilute mean-field picture the total diffuse electric charge is $-Q$, ensuring overall electroneutrality. A molecularly bound compact layer would require additional surface-chemistry information and is not assumed in the supplied potential.

Let $\epsilon ={\epsilon }_0{\epsilon }_r$ be the permittivity, $\beta =(k_{B}T{)}^{-1}$, $e>0$ the elementary charge, and $c$ the bulk salt formula-unit number density. The Boltzmann distribution givesConsequently the exterior Poisson-Boltzmann equation isFor small $Q$, meaning $2\beta e|\varphi (a)|\ll 1$, linearize the exponentials. The Debye–Hückel approximation givesThis is also ${\kappa }^2=\beta e^2{\sum }_i{z}_i^2{n}_{i,\infty }/\epsilon$: the divalent cation contributes $4c$ and the two monovalent anions contribute $2c$. In terms of molar salt concentration $C$ in $\mathrm{mol}{\mathrm{m}}^{-3}$, replace $c$ by $N_{A}C$; if $C$ is in $\mathrm{mol}{\mathrm{L}}^{-1}$, use $c=10^3{N}_{A}C$. The molar ionic strength is $3C$ in the same concentration units.

The boundary conditions are $\varphi (\infty )=0$ and, by Gauss's law,Here the surface charge is uniform and the spherically symmetric interior field is zero. Set $u=r\varphi$; the radial equation reduces to $u^{\prime \prime }={\kappa }^{2}u$. Decay excludes the growing exponential, so write $\varphi =B{e}^{-\kappa (r-a)}/r$. Then ${\varphi }^{\prime }(a)=-B(1+\kappa a)/a^2$, givingIt solves the linearized Poisson-Boltzmann equation, not the full nonlinear equation at finite $Q$. The radial ion profiles, consistently to first order, areThe exponential Boltzmann distribution formulas above express the full mean-field relation, but substituting a linearized potential into them is accurate only to the retained order. A useful electroneutrality check isThe result is the Debye-Hückel potential of a charged sphere.

Potassium sulfate is the ionic chemical compound ${\mathrm{K}}_2\mathrm{S}{\mathrm{O}}_4$. Two singly charged potassium ions balance one doubly charged sulfate ion.

The formula illustrates electroneutrality in an ionic material. On dissolution, the formula-unit ratio must be distinguished from the total concentration of separate dissolved ions.