Measurements (calibration functions)

Overview of the supported calibration functions follows. Notice the defined order of parameters (ϑ) and specifiers¶ of the functions (n is the number of reactants, m is the number of species, q ∼ (m, n) is the matrix of stoichiometric coefficients, c_{i} is the free concentration of the i-th component†).

Type 1

y = E_{0} + S log c_{1} + j_{1}c_{1} + j_{2} \frac{β_{1}}{c_{1}}

ϑ ≡ E_{0}, S, j_{1}, j_{2}

Note: Function of type 1 describes the dependence of the difference y of potentials of a “hydrogen” (glass) electrode and a reference electrode in a cell with non-negligible liquid-junction potential on the free concentration c_{1} (β_{1} ≡ K is the ionic product of water). Thus, H+ must be the 1st reactant (1st column of the matrix q), OH− must be the 1st complex (1st row of the matrix q).

Type 2

y = E_{0} + S log c_{1}

(ϑ ≡ E_{0}, S) / (Type 3)

y = E_{0} + S log c_{k} + j_{1}c_{1} + j_{2} \frac{β_{1}}{c_{1}}

ϑ ≡ E_{0}, S, j_{1}, j_{2}

(k ∈ ⟨1, n⟩) / (Type 4)

y = E_{0} + S log c_{k}

ϑ ≡ E_{0}, S

(k ∈ ⟨1, n⟩) / (Type 5)

y = E_{0} + S log \frac{c_{k}}{c_{l}} + j_{1}c_{1} + j_{2} \frac{β_{1}}{c_{1}}

ϑ ≡ E_{0}, S, j_{1}, j_{2} k ≠ l ∈ ⟨1, n⟩

Type 6

y = E_{0} + S log \frac{c_{k}}{c_{l}}

ϑ ≡ E_{0}, S k ≠ l ∈ ⟨1, n⟩

Note: Functions of types 5 and 6 describe the dependence of the difference y of potentials of a redox (platinum) electrode and a reference electrode on the ratio of free concentrations c_{k}, c_{l} of two reactants (i.e. the simplest case of a redox system, e.g. Fe3+/Fe2+, Sn4+/Sn2+, Ce4+/Ce3+ etc.).

Type 7

y = ∑_{i=1}^{n+m} a_{i}c_{i} = a^{T}c

ϑ ≡ a_{i}, i = 1, …, n + m

¶ usually k (and l or d), but the problem is more complicated with calibration functions of types 9, −3, −4, −5 and −6, where k = −i for i ≤ n and k = i − n for i > n (l = −j for j ≤ n and l = j − n for j > n)

† n + m components are ordered such that the first n of them are reactants and the next (last) m are species

Type 8

y = d ∑_{i=1}^{n+m} a_{i}c_{i} = d a^{T}c

ϑ ≡ a_{i}, i = 1, …, n + m d ∈ R

Note: Functions of types 7 and 8 describe the dependence of an \underline{extensive} linearly additive quantity y (absorbance in some “slow exchange” spectrometry (in the case of inseparable peaks), usually UV/VIS spectrophotometry, reaction heat in enthalpimetry, conductivity in conductimetry etc.) on the free concentrations c of all components.

Type 9

y = \frac{c_{i}}{c_{j}}

i ≠ j ∈ ⟨1, n + m⟩

“Subtype” 9ln

y = − ln \frac{c_{i}}{c_{j}}

i ≠ j ∈ ⟨1, n + m⟩

Type −1

y = − log c_{1}

Type −2

y = − log c_{k}

k ∈ ⟨1, n⟩

Type −3

y = αc_{i}

ϑ ≡ α i ∈ ⟨1, n + m⟩

Type −4

y = dαc_{i}

ϑ ≡ α d ∈ R, i ∈ ⟨1, n + m⟩

Type −5

y = ∑_{i=1}^{n+m} a_{i}p_{i}c_{i}/C = a^{T}Pc/C ¶†

P = diag(p_{1}, …, p_{n+m})

ϑ ≡ a_{i}, i = 1, …, n + m

¶

p_{i} ∈ N+ {0} , i = 1, …, n , p_{n+i} = ∑_{j=1}^{n} p_{j}q_{ij} , i = 1, …, m , C = ∑_{i=1}^{n+m} p_{i}c_{i}

† A little more complicated definition (using almost general diagonal matrix P) enables correct description of the dependent variable no matter what linear transformation of the matrix q is actually in use. Moreover, dynamic definition of the analytical concentration C of the “active” component (not exclusively reactant or complex) is \underline{necessary} for the calculation of unknown chemical parameters c, n, V, c’, q (the value of C changes).

Note: Function of type −5 describes the dependence of an \underline{intensive} linearly additive quantity y (chemical shift or relaxation rate, i.e. reciprocal relaxation time, in “fast exchange” NMR spectroscopy, effective mobility in capillary zone electrophoresis of a labile system etc.) on the free concentrations c of all components.

“Subtype” −5ln

y = − ln a_{i} − ln p_{i} − ln c_{i} + ln C ¶

Type −6

y = p_{i}c_{i}/C †

i ∈ ⟨1, n + m⟩

Note: Function of type −6 describes the dependence of an \underline{intensive} quantity y (intensity of an isolated (separated) peak in some ”slow exchange” spectrometry, usually NMR spectroscopy, intensity of an isolated peak in capillary zone electrophoresis of an inert system etc., normalized to the sum of intensities of all the peaks embracing \underline{all} components containing the active component) on the free concentrations c of all components (but mainly on c_{i}).

“Subtype” −6ln

y = − ln p_{i} − ln c_{i} + ln C

Type −7

y = α ∑_{i=1}^{n+m} c_{i}

ϑ ≡ α

Note: Function of type −7 describes the dependence of an \underline{extensive} linearly additive quantity y, usually a change of some colligative property (difference of resistances in VPO, osmotic pressure in membrane osmometry etc.) on the free concentrations c of all components.

¶ p_{i} ≠ 0, ∀ j ≠ i : a_{j} = 0 † p_{i} ≠ 0