Comments on “Solubility and Dissolution Thermodynamic Data of Cefpiramide in Pure Solvents and Binary Solvents”

Abstract

Errors are found in the mathematical correlation based on the combined Jouyban–Acree and Modified Apelblat models for describing the variation in the mole fraction solubility of cefpiramide with temperature and solvent composition for the binary aqueous–ethanol solvent system. The equation coefficients given by Tang and coworkers, when substituted into the model equation, do not yield the authors’ calculated mole fraction solubilities of cefpiramide.

In a recent paper appearing in This Journal Tang and coworkers [1] reported the solubility of cefpiramide in five neat mono-solvents (water, ethanol, 1-propanol, 1-butanol and 2-propanol) and in two binary aqueous–organic solvent mixtures. The two organic solvents are ethanol and 2-propanol. Solubilities were measured at six temperatures from 278.2 to 303.2 K using a spectroscopic method of chemical analysis. The authors used the combined Jouyban–Acree and Modified Apelblat models:

$$ \ln (x_{\text{A}} )_{m,T} = A_{1} + \frac{{A_{2} }}{T} + A_{3} \ln T + A_{4} x_{\text{B}}^{0} + A_{5} \frac{{x_{\text{B}}^{0} }}{T} + A_{6} \frac{{(x_{\text{B}}^{0} )^{2} }}{T} + A_{7} \frac{{(x_{\text{B}}^{0} )^{3} }}{T} + A_{8} \frac{{(x_{\text{B}}^{0} )^{4} }}{T} + A{}_{9}x_{\text{B}}^{0} \ln T $$

(1)

to describe how the measured mole fraction solubility of cefpiramide, (x_A)_m,T, varied with both temperature, T, and initial mole fraction composition of the binary solvent mixture, $x_{\text{B}}^{0}$. The curve-fit equation coefficients, A _i , were determined by regressing the experimental mole fraction solubility data in accordance with Eq. 1. The authors tabulated the calculated curve-fit equation coefficients in Table 8 of their published paper [1]. Only the statistically significant coefficients were tabulated. The authors stated in the manuscript that Eqs. 2 and 3 below (Eqs. 15 and 16 in the published paper):

$$ \ln (x_{\text{A}} )_{m,T} = A_{1} + A_{3} x_{\text{B}}^{0} + A_{5} \frac{{(x_{\text{B}}^{0} )^{2} }}{T} + A_{7} \frac{{(x_{\text{B}}^{0} )^{3} }}{T} + A_{8} \frac{{(x_{\text{B}}^{0} )^{4} }}{T} $$

(2)

$$ \ln (x_{\text{A}} )_{m,T} = A_{1} + A_{3} x_{\text{B}}^{0} + A_{5} \frac{{x_{\text{B}}^{0} }}{T} + A_{6} \frac{{(x_{\text{B}}^{0} )^{2} }}{T} + A_{7} \frac{{(x_{\text{B}}^{0} )^{3} }}{T} + A_{8} \frac{{(x_{\text{B}}^{0} )^{4} }}{T} + A{}_{9}x_{\text{B}}^{0} \ln T $$

(3)

were the final equations for predicting the solubility of cefpiramide in binary aqueous–ethanol and aqueous–2-propanol solvent mixtures in the solvent mole fraction composition range from $x_{\text{B}}^{0} = 0.0$ to $x_{\text{B}}^{0} = 0.9$, respectively.

The purpose of this commentary is to alert journal readers to several errors in the authors’ mathematical correlations. Careful examination of Eqs. 1–3 reveals that the curve-fit equation coefficients are identified differently in Eq. 1 than in Eqs. 2 and 3. The A₃ equation coefficient in Eq. 1 corresponds to the coefficient in the A₃ ln T term, whereas in Eqs. 2 and 3 the A₃ coefficient corresponds to the $A_{3} x_{\text{B}}^{0}$ term. There is a similar problem with the A₅ coefficient in Eqs. 1 and 2. The change in symbolism can lead to confusion when it comes to substituting the numerical values for the equation coefficients. For example, in Table 7 of the published paper [1] the authors gave numerical values of A₁ = −93.252, A₃ = 13.927, A₅ = 3559.445, A₇ = 10448.710 and A₈ = 8314.726 as the coefficients for binary aqueous–ethanol solvent mixture. Does one substitute the numerical values into Eq. 1 to give:

$$ \ln (x_{A} )_{m,T} = - 93.252 + 13.927\ln T + 3559.445\frac{{x_{\text{B}}^{0} }}{T} + 10448.710\frac{{(x_{\text{B}}^{0} )^{3} }}{T} + 8314.726\frac{{(x_{\text{B}}^{0} )^{4} }}{T} $$

(4)

or does one substitute the numerical values into Eq. 2 to yield:

$$ \ln (x_{A} )_{m,T} = - 93.252 + 13.927x_{\text{B}}^{0} + 3559.445\frac{{(x_{\text{B}}^{0} )^{2} }}{T} + 10448.710\frac{{(x_{\text{B}}^{0} )^{3} }}{T} + 8314.726\frac{{(x_{\text{B}}^{0} )^{4} }}{T} $$

(5)

What I have done is to calculate the solubility of cefpiramide for the binary aqueous–ethanol solvent system at T = 298.2 K using both Eqs. 4 and 5. The results of my calculations are summarized in the third and fourth columns of Table 1 of this commentary, along with the calculated values that the authors gave in Table 1 of their published paper for water and for the five binary solvent compositions studied. According to the headings in Table 3 of the published paper [1], the authors’ calculated values are presumably based on Eq. 5 (which would be Eq. 15 in the published paper with the coefficients inserted). Careful examination of the numerical entries in the last three columns of Table 1 reveals that neither Eq. 4 nor Eq. 5 reproduce the authors’ calculated values. In the case of Eq. 5 the calculated mole fraction solubility of cefpiramide would be the same at all six temperatures for $x_{\text{B}}^{0} = 0.0$ as only the first term would contribute to the calculation. The remaining four terms would equal zero at $x_{\text{B}}^{0} = 0.0$. Equation 4 on the other hand gives a calculated value of $10^{6} \times \,(x_{\text{A}} )_{m,T}^{\text{calc}}$ close to the value reported by the authors for $x_{\text{B}}^{0} = 0.0$; however, calculated values at the larger mole fractions of solvent component B exceed unity. Mole fraction compositions cannot exceed unity. There are clearly problems with the A _i equation coefficients given in the paper by Tang and coworkers [1] for the aqueous–ethanol solvent system. I suspect that one of the authors’ tabulated equation coefficients (perhaps the A₇ coefficient) is missing a negative sign.

Table 1 Comparison between the experimental mole fraction solubilities of cifpiramide, $\,(x_{\text{A}} )_{m,T}^{{}}$, calculated values reported by Tang and coworkers [1], and calculated values based on Eqs. 4 and 5

There is also an error in the symbolism associated with the equation coefficients for the binary aqueous–2-propanol solvent mixture reported in reference 1. The numerical value for the A₃ coefficient should pertain to the A₃ ln T term, and not the $A_{3} x_{\text{B}}^{0}$ term as implied by Eq. 16 in the authors’ published paper [1]. If the A₃ coefficient were to apply to the $A_{3} x_{\text{B}}^{0}$ term then the calculation would yield ln (x_A)_m,T = − 89.733 at $x_{\text{B}}^{0} = 0.0$, which would correspond to an aqueous mole fraction solubility of (x_A)_m,T = 1.07 × 10⁻³⁹ for all six temperatures studied. The authors’ calculated value for T = 298.2 K is much larger, e.g., $10^{6} \times \,(x_{\text{A}} )_{m,T}^{\text{calc,authors}} = 0.9568$. As an informational note, the authors’ tabulated coefficients (using A₃ for the A₃ ln T term) for the binary aqueous–2-propanol system are much better at reproducing the calculated mole fraction solubilities at T = 298.2 K reported in Table 4 of the published paper. I only checked only the calculations for T = 298.2 K.