Calculation of Diffusion Coefficients in Ternary Systems from Diaphragm Cell Experiments

Abstract

Using a generalized least square procedure, a method is developed to estimate diffusion coefficients, D_ij, from concentration measurements made in a set of experiments of varying duration but with the same initial conditions. The method requires neither approximations nor restrictions in the flow equations. The D_ij and their errors are calculated simultaneously with the adjustment of the weighted observables – concentrations of the two solute components and time. A procedure is described to insure convergence in a given experimental set. The effect of the duration of the longest experiment, and errors in the observables on the estimated values of D_ij and their errors, are investigated. Application of the method to available data yielded values for D_ij in agreement with those previously calculated by more involved procedures.

1. Introduction

The methods reported for the calculation of diffusion coefficients in ternary systems from diaphragm cell data require either restrictions in the experimental conditions [1, 2]¹ or approximations in the flux equations [3]. A new method is reported here that obviates the limitations of previous procedures when concentrations of the diffusing solutes are directly measured in the cell compartments. Although the present method was derived independently of a procedure recently reported by Cussler and Dunlop [4], the two approaches to the problem are similar. The present derivations, however, allow a more complete and direct treatment of the subject, particularly for the calculation of errors.

2. Theory

The general equations that define diffusion flows, (J_i)_V, in ternary systems relative to a volume-fixed frame of reference are [5, 6].

$${\left(J_i\right)}_V=-\sum _{j=1}^2{\left(D_{ij}\right)}_V\frac{\partial C_j}{\partial x}(i=1,2)$$ (1)

in which (D_ij)_V are practical diffusion coefficients and C_j is the concentration of component j. Integration of eq (1), considering the conditions for the diaphragm cell, assuming linear concentration gradients and no changes in volume during the diffusion process, gives the general solution [7]

$$\Delta C_i=A_i{e}^{-\beta m_{1}t}+B_i{e}^{-\beta m_{2}t}(i=1,2)$$ (2)

in which ΔC_i is the concentration difference (molar scale is used throughout these derivations) of component i between the two compartments at any time t, β is the cell constant and the other quantities are defined by

$$\begin{array}{l}{A}_i=\frac{\Delta C_i^{°}\left(D_{ii}-m_2\right)+D_{ij}\Delta C_j^{°}}{m_1-m_2}\hfill \\ B_i=\frac{\Delta C_i^{°}\left(D_{ii}-m_1\right)+D_{ij}\Delta C_j^{°}}{m_2-m_1}\hfill \end{array}\}\begin{array}{ll}i=1,2\hfill & (3)\hfill \\ j=1,2\hfill & \hfill \\ i\ne j\hfill & (4)\hfill \end{array}$$

$$\begin{gathered} m_1=\frac{1}{2}\left[D_{11}+D_{22}+\sqrt{{\left(D_{11}+D_{22}\right)}^2-4\left(D_{11}{D}_{22}-D_{12}{D}_{21}\right)}\right] \\ {m}_2=\frac{1}{2}\left[D_{11}+D_{22}-\sqrt{{\left(D_{11}+D_{22}\right)}^2-4\left(D_{11}{D}_{22}-D_{12}{D}_{21}\right)}\right] \end{gathered}$$ (5)

In eqs (3) and (4), ΔC° denotes concentration differences at t = 0, and D_ij are the integral cell diffusion coefficients.²

The calculation of D_ij is accomplished by a generalized least squares procedure [8] using eq (2) as the mathematical model. Thus, the condition equations are of the form

$$F=\left(X_1-X_2\right)-A_1{e}^{-\beta m_{1}T}-B_1{e}^{-\beta m_{2}T}$$ (6)

$$L=\left(Y_1-Y_2\right)-A_2{e}^{-\beta m_{1}T}-B_2{e}^{-\beta m_{2}T}$$ (7)

in which X and Y represent the concentrations of components 1 and 2, respectively, in the compartments indicated by their subscripts at the observation time T.³

If observations (determinations of X and Y) are made at the end of N diffusion experiments (with the same initial conditions) of duration T₁, T₂, …, T_N, then there will be 2N condition equations. The observations corresponding to different values of T are statistically independent, since for each duration, a new experiment is conducted. The number of condition equations may not be a multiple of 2, however, if either X or Y is not determined at the end of some of the experiments.

When observations are made in only one of the compartments, for example compartment 2, the condition equations take the form

$$F^{\prime }=\left(\Delta X^{°}+2X_2^{°}-2X_2\right)-A_1{e}^{-\beta m_{1}T}-B_1{e}^{-\beta m_{2}T}$$ (8)

$$L^{\prime }=\left(\Delta Y^{°}+2Y_2^{°}-2Y_2\right)-A_2{e}^{-\beta m_{1}T}-B_2{e}^{-\beta m_{2}T}$$ (9)

in which ΔX° and ΔY° are concentration differences for the two components at time T = 0.

The aim of the procedure is to compute the adjustments of the observables and the corrections for the initial estimates of the diffusion coefficients in such a way that

$$\begin{gathered} F^{(n)}\left(x_1^{(n)},x_2^{(n)},t^{(n)},\Delta x^{°},\Delta y^{°},D_{11},D_{12},D_{21},D_{22}\right)\approx 0 \\ (n=1,2,\dots ,N) \end{gathered}$$ (10)

and

$$L^{(n)}\left(y_1^{(n)},y_2^{(n)},t^{(n)},\Delta x^{°},\Delta y^{°},D_{11},D_{12},D_{21},D_{22}\right)\approx 0$$ (11)

in which x, y, t, Δx°, and Δy° are adjusted observables, and D_ij are the final estimates of coefficients. This adjustment is done subject to the least squares criterion that the weighted sum of squares of the residuals (observed value – adjusted value) of the observables X, Y, T, ΔX°, and ΔY° be a minimum. In these derivations, the cell constant, β, is considered as an “external parameter,” that is, a quantity obtained from independent experiments, and therefore, not susceptible of adjustment. However, the error in β is taken into account in the estimation of the error bounds for D_ij, as explained below.

Linearization of the condition equations (6) and (7) is done about the observed points (X₁, X₂, T, ΔX°, ΔY°, $D_{11}^{°},D_{12}^{°},D_{21}^{°},D_{22}^{°}$) and (Y₁, Y₂, T, ΔX°, ΔY°, $D_{11}^{°},D_{12}^{°},D_{21}^{°},D_{22}^{°}$) using the best available initial estimates, $D_{ij}^{°},$ for the diffusion coefficients. A procedure to obtain good initial estimates for D_ij is described later.

The 2N reduced condition equations, resulting from the linearization mentioned above, may be written, in matrix notation,

$$BV+P\alpha =M$$ (12)

in which B is the matrix (2N × [5N + 2]) of the derivatives of the condition functions [6] and [7] with respect to the observables, V is the column vector ([5N + 2] × 1) of the residuals, P is the matrix (2N × 4) of the derivatives of the condition functions with respect to D_ij, α is the column vector (4 × 1) of the corrections for the $D_{ij}^{°}$ (i.e., ${\alpha }_{ij}=D_{ij}^{°}-D_{ij}$) and M is the column vector (2N × 1) of the values of the condition functions. The condition functions and their derivatives are evaluated using the experimental values of the observables and the initial estimates for the diffusion coefficients.

The sum of squares of the residuals, S², is given by

$$S^2=\tilde{V}WV$$ (13)

in which W is the diagonal matrix of the weights, W_obs, for the 5N + 2 observables and the tilde represents transposition. These weights are calculated from

$$W_{\text{obs}}=\frac{{\sigma }_{\text{int}}^2}{{\sigma }_{\text{obs}}^2}$$ (14)

in which σ_obs is the standard deviation for the observable in question and σ_int is an arbitrary constant conveniently selected for calculation purposes.

Minimization of S² (eq (13)) subject to the conditions (12), effected by the use of a Lagrange multiplier, Λ, and the subsequent elimination of V, results in 2N + 4 normal equations with 2N unknown λ’s (the elements of column vector (Λ), and four unknown α_ij. As shown by Guest [9] this leads to the following equations:

$$\left[\begin{array}{cc}R& P\\ \tilde{P}& 0\end{array}\right]\times \left[\begin{array}{c}\Lambda \\ \alpha \end{array}\right]=\left[\begin{array}{c}M\\ 0\end{array}\right]$$ (15)

or

$$G\times \left[\begin{array}{l}\Lambda \hfill \\ \alpha \hfill \end{array}\right]=\left[\begin{array}{l}M\hfill \\ 0\hfill \end{array}\right]$$

in which $R=B{W}^{-1}\tilde{B},$ of dimension 2N × 2N.

The solution of the above system of equations yields

$$\left[\begin{array}{l}\Lambda \hfill \\ \alpha \hfill \end{array}\right]=G^{-1}\left[\begin{array}{l}M\hfill \\ 0\hfill \end{array}\right].$$ (16)

Since G⁻¹ is symmetric, like G, it can be partitioned similarly as follows:

$$G^{-1}=\left[\begin{array}{rr}\hfill Q& \hfill H\\ \hfill \tilde{H}& \hfill C\end{array}\right].$$ (17)

Then, the solutions for Λ and α can be written as,

$$\Lambda =QM$$

$$\alpha =\tilde{H}M$$

The values thus obtained for α_ij can be now used to calculate new estimates for D_ij, since

$$D_{ij}=D_{ij}^{°}-{\alpha }_{ij}.$$ (18)

These new estimates of D_ij are used iteratively as initial estimates (substituting for $D_{ij}^{°}$) until convergence in S², calculated by eq (13), is obtained. At this point the final residuals are calculated from $V=W^{-1}\tilde{B}\Lambda .$ The error vector Δ, (ΔD₁₁, ΔD₁₂, ΔD₂₁, ΔD₂₂), can be found from the 4×4 error submatrix, E=−C, of G⁻¹:

$$\Delta ={\sigma }_{\text{ext}}\sqrt{E_{jj}}$$ (19)

in which the standard error, σ_ext, is given by $\sqrt{\left\{S^2/\left(2N-4\right)\right\}},$ 2N − 4 being the degrees of freedom.

A discrepancy between the ratio ${\sigma }_{\text{ext}}^2/{\sigma }_{\text{int}}^2$ and unity is an indication of disagreement between the variances assumed in the calculations (based on previous experience) and the error variances actually existing in the experiment [10].

The errors as obtained from eq (19) do not reflect the influence of the error, σ_β, in the independently determined cell constant β. Therefore, calculations are repeated using successively the values β + σ_β and β − σ_β for the cell constant. A t-test is then performed between the D_ij obtained in this fashion and those obtained when β was considered free of error. If the t-test indicates no significant difference, at a selected level of significance, the errors obtained for the D_ij with the assumption that σ_β = 0 may be adopted. Otherwise, a confidence band for the D_ij should be estimated.

A requirement for the convergence of the iterative procedure in the calculation of D_ij is that the initial estimates of these parameters be close to the final estimates. Various procedures have been suggested [11, 12, 13] to insure convergence in problems of this nature. In this investigation it was found that, for ternary systems, good initial estimates for only two of the four coefficients are required to insure convergence, their identity being determined by the initial experimental conditions. Three cases of initial conditions are possible in a ternary system: (1) Both diffusing components have nonzero concentration gradients; (2) Component 1 has a nonzero gradient and component 2 has a zero gradient; (3) Component 2 has a nonzero gradient and component 1 has a zero gradient. The two diffusion coefficients for which good initial estimates are required in each one of the three cases are: for case (1) D₁₁ and D₂₂; for case (2) D₁₁ and D₂₁; and for case (3) D₂₂ and D₁₂. The initial estimates for the remaining 2 coefficients in each case can be taken as zero. In the flux equations (1) the main contribution to the flux of one component is given by those terms involving the component for which there is an initial concentration gradient. Then, ignoring, as a first approximation, those terms that contribute the least to the diffusion fluxes, performing the corresponding integrations, the resulting expressions may be written in the logarithmic form,

$$\text{ln}\frac{\Delta C_i}{\Delta C_j^{°}}=-D_{ij}\beta t\begin{array}{l}i=1,2\hfill \\ j=1,2\hfill \end{array}$$ (20)

which covers the three cases given above. The required initial estimates of the diffusion coefficients are then obtained by a simple linear regression using the appropriate experimental data. In this way, convergence was obtained after 4 to 6 cycles in all the cases tested.

3. Test of the Method

The procedure was thoroughly tested by constructing artificial “experimental” data. The condition functions (8) and (9) were used assuming a value of zero for $X_2^{°}$ and $Y_2^{°}.$ Exact solutions of eqs (2) were obtained for different arbitrary sets of D_ij, β, and initial concentration differences. The “true” points thus obtained were transformed into “experimental” observations by the use of a table of normal deviates, and the foregoing method was applied to these data.⁴ Thus the following factors affecting the estimated values for D_ij and their errors were investigated: (a) duration of the longest experiment in a given set of experiments, (b) number of experiments (total number of observations) in sets having the same duration for the longest experiment, and (c) errors in the observables.

For the case of binary systems, expressions have been reported [14, 15] to calculate an optimum experimental time at which the fractional error in the diffusion coefficient becomes minimum. Derivation of similar expressions for ternary systems involves such complexity that it is practically unmanageable. In the present investigation this subject was studied by calculations of D_ij for different sets of experiments, varying the duration of the longest experiment in each set. A sample of the results obtained is shown in table 1. The time given at the top of each column corresponds to the longest experiment in a set consisting of eight experiments, the time increment for each successive experiment being one-eighth of the longest experiment. Values for the percent decrease of the initial concentration differences of components 1 and 2, R₁ and R₂, at the end of the longest experiment for a particular set are given in the second and third rows. In order to facilitate the interpretation of these calculations, the same random numbers were used to estimate the “experimental” observations for each set. For this reason, the ratio ${\sigma }_{\text{ext}}^2/{\sigma }_{\text{int}}^2$ is fairly constant in this table and, as might have been anticipated, its value is close to unity.

Table 1.

Effect of duration of the longest experiment on the estimated values and errors of the diffusion coefficients

Duration of longest exp., hr*	24	120	216	312	408	504	600	800	True diff. coeffs.
R1 = 100 ΔX/ΔX°	83.2	39.5	18.3	8.1	3.4	1.2	0.2	−0.3
R2 =100 ΔY/ΔY°	90.5	61.2	42.0	29.1	20.4	14.3	10.1	4.9
D11 × 105	1.576 ±0.155	1.935 ±0.033	1.959 ±0.021	1.967 ±0.018	1.971 ±0.016	1.973 ±0.016	1.974 ±0.017	1.975 ±0.019	2.000
D12 × 105	0.888 ±0.252	0.296 ±0.049	0.255 ±0.030	0.240 ±0.023	0.233 ±0.021	0.229 ±0.020	0.226 ±0.020	0.221 ±0.022	0.200
D2l × 105	0.062 ±0.052	0.103 ±0.011	0.103 ±0.008	0.103 ±0.006	0.103 ±0.006	0.103 ±0.006	0.103 ±0.006	0.103 ±0.006	0.100
D22 × 105	1.062 ±0.085	0.997 ±0.017	0.997 ±0.011	0.998 ±0.008	0.998 ± 0.007	0.998 ±0.007	0.998 ±0.007	0.999 ±0.007	1.000
$\frac{{\sigma }_{\text{ext}}^2}{{\sigma }_{\text{int}}^2}$	0.72	1.07	1.08	1.08	1.08	1.07	1.08	1.09
$\left(\Sigma {\Delta }_{ij}^2\right)\times {10}^{13}$	97.614	3.909	1.489	0.946	0.770	0.724	0.744	0.916

Initial conditions: ΔX° = 0.025; ΔY° = 0.015. Cell constant: β = 0.1.

Total number of experiments in a set: 8.

Adopted errors: E_{X, Y} = ±0.2%; E_T = ±60 s; E_{ΔX°, ΔY°} = ±0.1%; E_β = 0.1%.

^*Same set of random numbers used for each column.

Table 1 shows that as the duration of the longest experiment increases, the estimated D_ij approach the values for the “true” diffusion coefficients. Furthermore, the last row in the table shows that there is a minimum in the sum of squares of the errors in D_ij, $\Sigma {\Delta }_{ij}^2,$ corresponding to a set with an experiment of maximum duration between 200 and 600 hr. This type of calculation was also performed with a number of systems varying the initial conditions and the (true) values for D_ij. In all cases a minimum was found but its position varied from system to system and there was no apparent trend correlating the minimum with the parameters used or calculated. In some of the systems studied, the approach to and departure from the minimum value of $\Sigma {\Delta }_{ij}^2$ occurred very gradually as the duration of the longest experiment increased. In other systems, however, the valley region was very narrow. Typical examples of these features are shown in figure 1 in which the quantity $\Sigma {\Delta }_{ij}^2$ is plotted against the duration of the longest experiment in various experimental sets. Initial conditions and related parameters are given in the figure caption. The lower curve corresponds to the example given in table 1; however, both plots are restricted to the regions adjacent to the minimum. Values for D_ij estimated from the experimental sets represented by the points between arrows in figure 1 do not differ appreciably.

Figure 1. Effect of duration of longest experiment on errors in D_ij.

For both plots: Number of experiments per set, initial conditions, errors in observables, and cell constant, as given in table 1. For 1a, “true” D_ij as in table 1. For 1b, “true” D_ij × 10⁵: D₁₁ = 8.0, D₁₂ = 1.0, D₂₁ = −2.0, D₂₂ = 4.0.

The actual position of the minimum in $\Sigma {\Delta }_{ij}^2$ for a given system shifts for different values of the cell constant, β, but it always occurs at approximately the same values of R₁ and R₂. In the absence of analytical expressions to calculate optimum diffusion times in ternary systems, the procedure reported here can be used to determine the duration for which a minimum in the errors for D_ij is obtained. This approach, of course, requires estimates for the values of D_ij, errors in the observables, initial conditions, and the cell constant. With these parameters, curves like those in figure 1 can be computed and a decision about the diffusion time can be made.

The effect of the number of experiments in a set on the estimated values of D_ij is shown in table 2. Calculations for this table were made using the same initial conditions and set of parameters used in table 1. The longest time used in any particular set was 516 hr, and the time increment for the N singular experiments was 516/N. The calculations were carried out several times for each set using different sets of random numbers for the estimation of the “experimental” observations. Results for the two sets of calculations displaying the maximum differences in D_ij are given in the table. It is apparent that as the number of experiments in each set is increased, the variations in D_ij and Δ_ij due to experimental errors decrease although the duration of the longest experiment in all the sets was the same, 516 hr.

Table 2.

Effect of number of experiments in a set on the estimated values and errors of the diffusion coefficients^*

Number of experiments in a set	3		5		8		True diff. coeffs.
D11 × 105	2.005 ±0.027	1.938 ±0.026	2.051 ±0.015	2.010 ±0.022	2.002 ±0.010	1.973 ±0.016	2.000
D12 ×105	0.197 ±0.029	0.261 ±0.028	0.151 ±0.018	0.202 ±0.025	0.189 ±0.012	0.228 ±0.020	0.200
D21 × 105	0.114 ±0.009	0.088 ±0.009	0.097 ±0.005	0.096 ±0.008	0.103 ±0.003	0.103 ±0.006	0.100
D22 × 105	0.986 ±0.010	1.012 ±0.010	1.003 ±0.006	1.004 ±0.009	0.994 ±0.004	0.998 ±0.007	1.000
$\frac{{\sigma }_{\text{ext}}^2}{{\sigma }_{\text{int}}^2}$	0.58	0.58	0.47	0.99	0.40	1.08

Initial conditions: ΔX° = 0.025; ΔY° = 0.015. Cell constant: β = 0.1.

Longest experiment in each set: 516 hr.

Adopted errors: E_{X, Y} = ±0.2%; E_T = ±60 s; E_{ΔX°, ΔY°} = ±0.1%; E_β = ±0.1%.

^*Entries in subcolumns are extreme values of D_ij obtained with 6 to 10 sets of randomized data.

The errors in the observables not only affect the estimated values for D_ij and Δ_ij, but also the time at which a minimum in $\Sigma {\Delta }_{ij}^2$ is obtained for a given system. Such effects are shown in table 3; the values for D_ij were calculated from sets having an experiment with a maximum duration equal to the time for which a minimum in $\Sigma {\Delta }_{ij}^2$ was observed. With large errors in the observables, the values for D_ij obtained by the adjustment procedure may be very inaccurate; this situation is particularly true for the cross-term diffusion coefficients. Although definite shifts in the location of the minima in $\Sigma {\Delta }_{ij}^2$ are apparent, they are hardly significant for experimental planning.

Table 3.

Effect of errors in observables on the estimated values and errors of the diffusion coefficients

Ex,Y	2.0%*	0.2%*	0.02%*	True diff. coeffs.
D11 × 105	1.742 ±0.147	1.973 ±0.016	1.997 ±0.002	2.000
D12 × 105	0.465 ±0.181	0.228 ±0.020	0.203 ±0.002	0.200
D21 × 105	0.130 ±0.055	0.103 ±0.006	0.100 ±0.001	0.100
D22 × 105	0.978 ±0.068	0.998 ±0.007	1.000 ±0.001	1.000
$\left(\Sigma {\Delta }_{ij}^2\right)\times {10}^{13}$	62.025	0.723	0.008
Duration of longest experiment, hr	532	516	536

Initial conditions: ΔX° = 0.025; ΔY° = 0.015. Cell constant: β = 0.1.

Total number of experiments in a set: 8.

Adopted errors: E_{ΔX°, ΔY°} = ±0.1%; E_β = ±0.1%; E_T = ±60 s.

^*Same set of random numbers used for each column.

The adjustment procedure described in this paper was applied to the diffusion data reported by Kelly [1] for the ternary system mannitol (1) – NaCl (2) – H₂O, using the condition eqs (6) and (7). Kelly had obtained the values for D_ij from the results of two experimental arrangements with initial conditions given at the head of table 4. His results and the results obtained by the present procedure are shown, side by side, in the same table.

Table 4.

Comparison of Kelly’s estimated values of the diffusion coefficients with those estimated by the present procedure using Kelly’s experimental data for the system mannitol (X) – NaCl (Y) – H₂O

Average conc.	$\overline{X}=0.2M;\overline{Y}=1.5M$				$\overline{X}=0.2M;\overline{Y}=3.0M$
Initial conditions	ΔX° = 0.4 ΔY° = 0.0		ΔX° = 0.4 ΔY° = 0.5		ΔX° = 0.4 ΔY° = 0.0		ΔX° = 0.0 ΔY° = 0.0
System	1 – A 5 experiments		1 – B 4 experiments		2 – A 6 experiments		2 – B 3 experiments
	This paper	Ref. (1)	This paper	Ref. (1)	This paper	Ref. (1)	This paper	Ref. (1)
D11×105	0.557 ±0.003	0.559 ±0.004	1.148 ±0.246		1.006 ±0.016	0.448 ± 0.006	0.439 ±0.625
D12×105	0.025 ±0.039		0.018 ±0.002	0.016 ±0.010	0.512 ± 0.066		0.013 ±0.003	0.013 ± 0.008
D21×105	0.232 ±0.017	0.232 ±0.001	0.138 ±1.918		0.494 ±0.072	0.456 ±0.003	−0.040 ±9.969
D22×105	1.556 ±0.230		1.388 ±0.013	1.390 ±0.002	1.122 ±0.291		1.469 ±0.048	1.469 ±0.003
$\frac{{\sigma }_{\text{ext}}^2}{{\sigma }_{\text{int}}^2}$	5.20		4.32		167.18		38.98

E_{X°, Y°} = ±0.02%; E_{X, Y} = ±0.02%; E_T = ±60 s; E_β = 0.1%.

There is satisfactory agreement between the results obtained by the present procedure and those obtained by Kelly for systems where he could calculate diffusion coefficients, with the exception of the value for D₁₁ in the system 2–A. The values for D_ij estimated from Kelly’s experimental data for system 1–A should have agreed, in theory, with those obtained from the experimental data for the system 1–B. The cause of these discrepancies is that, usually, when one of the components has zero initial concentration gradient, the amount diffused is so small that a reliable estimation of the D_ij is impossible.

The discrepancies in systems 2–A and 2–B are explained on the same basis; however, the large value for the ratio ${\sigma }_{\text{ext}}^2/{\sigma }_{\text{int}}^2$ for system 2–A suggests that the error variances used for the observables are very different from those actually existing in this system; for this reason, the values for D_ij obtained from this experiment are questionable.

Cussler and Dunlop [4] also estimated D_ij from Kelly’s data. It is not clear in their publication whether they combined the data for systems 1–A and 1–B or estimated D_ij from each arrangement separately. The values they obtained are in good agreement with those obtained by the present method and reported in table 4, with the exception of the value for D₂₁ which is about 5 percent lower than the figure reported here. Significant differences, however, do exist between the errors reported by Cussler and Dunlop and those obtained in the present investigation. This is not surprising because in their unweighted least squares procedure, those investigators arrived at estimates of errors for the D_ij by a bound method. Their reported error figures correspond to the maximum difference between the D_ij calculated considering the nominal value of the observables and the D_ij obtained through similar calculation but using the values of the observables as the nominal values minus their errors and plus their errors, respectively. The procedure given in this paper to calculate Δ_ij is not only time saving but also it has a sound statistical basis.