# Penicillins’ Solubility in Supercritical Carbon Dioxide: Modeling by Cubic Equations of States Revisited

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{2}) rather than organic ones [1,2,3,4,5,6,7,8,9]. Drugs such as antibiotics are chemical substances for which SCF technology involves appropriate formulation to increase the drug’s efficiency based specially on particle size, morphology and surface structure [2,10,11]. At present, the oldest and most frequently used antibiotics are Penicillins [12] which belong to the α-lactam group [13] because they have in their chemical structure an α-lactam ring fused to a thiazolidine ring [14]. Many years after their discovery by Fleming, penicillins are still the most effective treatment of diseases due to bacterial infections as Staphylococus and Syphilis [15,16,17]. Among them, the most important on the commercial plan are Penicillin G and Penicillin V [18].

_{2}is long and time-consuming, which explains the researchers’ interest in mathematical modeling. In their reviews, Brenneck and Eckert [29] and Johnston et al. [30] have discussed globally the analysis and modeling of this special phase equilibrium. To represent the thermodynamic behavior of these mixtures considered as largely asymmetric, models frequently applied are cEoSs, because they represent a fundamental tool [31,32]. However, disadvantages associated to this type of equation are many, even when solubility data are available because it is important(in the majority of cases) to insert other adjustable parameters [10,33,34]. The results of predicting solubility data in SC CO

_{2}using cEoSs related to different mixing rules are affected by the estimation of some physical properties as acentric factors, critical constants and sublimation pressures of the solid drugs [35]. Here, the focus is on the sublimation pressure of Penicillin G and Penicillin V, which is the principal influencer on solubility and considered as an ad hoc adjustable parameter. To present this review, the steps aforementioned in the abstract are followed and detailed gradually.

## 2. Modeling and Thermodynamic Basis

## 3. cEoSs Needed

- Redlich-Kwong (RK cEoS):

- Soave-Redlich-Kwong (SRK cEoS):$$P=\frac{RT}{\left(v-b\right)}-\frac{a\alpha \left({T}_{r},\omega \right)}{v\left(v+b\right)};\alpha \left({T}_{r},\omega \right)={\left[1+s\left(1-\sqrt{{T}_{r}}\right)\right]}^{2};{T}_{r}=\frac{T}{{T}_{c}};\phantom{\rule{0ex}{0ex}}s=0.48+1.574\omega -0.176{\omega}^{2}$$
- Peng-Robinson (PR cEoS):$$P=\frac{RT}{v-b}-\frac{a}{v\left(v+b\right)+b\left(v-b\right)}\left(ationsneeded\right)$$$$a={\displaystyle \sum}_{i}{\displaystyle \sum}_{j}{y}_{i}{y}_{j}{a}_{ij};b={\displaystyle \sum}_{i}{y}_{i}{b}_{i}$$$$\{\begin{array}{c}{a}_{ij}=0.42748\frac{{R}^{2}{T}_{{C}_{ij}}^{2,5}}{{P}_{{C}_{ij}}};{T}_{{C}_{ij}}=\left(1-{k}_{ij}\right)\sqrt{{T}_{{C}_{i}}{T}_{{C}_{j}}}\\ {P}_{{C}_{ij}}=\frac{{Z}_{{C}_{ij}}R{T}_{{C}_{ij}}}{{V}_{{C}_{ij}}};{Z}_{{C}_{ij}}=\frac{{Z}_{{C}_{i}}+{Z}_{{C}_{j}}}{2};{V}_{{C}_{ij}}={\left(\frac{{V}_{{C}_{i}}^{1/3}+{V}_{{C}_{j}}^{1/3}}{2}\right)}^{2}\end{array}$$

## 4. Review and Discussion of Literature Results

#### 4.1. Penicillin G

_{2}, to obtain an estimated ${P}_{2}^{sub}$ by regression with RK cEoS and SRK cEoS. Table 1 presents their results.

#### 4.2. Penicillin V

_{2}) and (Penicillin V–CO

_{2}), deviations of AARD are relatively large and very pointed at high temperatures (333.15 K and 334.85 K). However, the question is what is the reliable value of sublimation pressure as long as the deviations AARD are close?

## 5. Schmitt and Reid Modified PR cEoS

- Regression of the experimental data (N = 18) for Penicillin G with modified PR cEoS is done by the implementation of ${P}_{2}^{sub}$, one of whichis obtained by SRK cEoS (${P}_{2}^{sub}$-SRK) and the other is obtained by RK cEoS(${P}_{2}^{sub}$-RK). The results are presented in Table 3.

## 6. Use of New Sublimation Pressure

## 7. Conclusions

_{2}using the cubic equations of states approach. This work focused on this predominant thermophysical property.

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Experimental and calculated solubility of Penicillin G versus density. (

**a**) (${P}_{2}^{sub}$-RK). (

**b**) (${P}_{2}^{sub}$-SRK).

**Figure 2.**Experimental and calculated solubility of Penicillin G at T = 313K: (

**a**) (${P}_{2}^{sub}$-RK). (

**b**) (${P}_{2}^{sub}$-SRK).

**Figure 3.**Experimental and calculated solubility of Penicillin V (global representation): (

**a**) (${P}_{2}^{sub}$-PR), (

**b**) (${P}_{2}^{sub}$-Clapeyron).

**Figure 4.**Experimental and calculated solubility of Penicillin V (temperature-by-temperature representation): (

**a**) (${P}_{2}^{sub}$-PR), (

**b**) (${P}_{2}^{sub}$-Clapeyron).

**Figure 5.**Interpolation of the sublimation pressures obtained for Penicillin G by the (Clausius-Clapeyron) equation.

T (K) | ${\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}\text{}\left(\mathbf{Bar}\right)$ RK [40] | AARD (%) [40] | ${\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}\text{}\left(\mathbf{Bar}\right)$ SRK [40] | AARD (%) [40] | $\mathbf{Error}\left[\frac{\left|\left({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathit{S}\mathit{R}\mathit{K}\right)-\left({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathit{R}\mathit{K}\right)\right|}{\left({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathit{S}\mathit{R}\mathit{K}\right)}\right]$ |
---|---|---|---|---|---|

313.15 | $3.55\times {10}^{-12}$ | 23 | $2.82\times {10}^{-12}$ | 21 | 26% |

323.15 | $2.24\times {10}^{-11}$ | 23 | $4.57\times {10}^{-11}$ | 21 | 51% |

333.15 | $1.44\times {10}^{-10}$ | 23 | $3.09\times {10}^{-9}$ | 21 | 95% |

T(K) | ${\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}\left(\mathbf{Bar}\right)$ Clausius [46] | AARD (%) | ${\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}\left(\mathbf{Bar}\right)$ PR [46] | AARD (%) | $\mathbf{Error}\left[\frac{\left|\left({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathit{P}\mathit{R}\right)-\left({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathit{C}\mathit{l}\mathit{a}\mathit{u}\mathit{s}\mathit{i}\mathit{u}\mathit{s}\right)\right|}{\left({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathit{P}\mathit{R}\right)}\right]$ |
---|---|---|---|---|---|

314.85 | $5.53\times {10}^{-10}$ | 37.85 | $1.15\times {10}^{-10}$ | 36.23 | 381% |

324.85 | $1.54\times {10}^{-9}$ | 42.46 | $9.10\times {10}^{-9}$ | 40.25 | 83% |

334.85 | $3.83\times {10}^{-9}$ | 54.38 | $3.93\times {10}^{-7}$ | 41.30 | 99% |

$\mathbf{Using}\text{}({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathbf{SRK})$ | $\mathbf{Using}\text{}({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathbf{RK})$ | |
---|---|---|

${a}_{2}\left(\mathrm{P}\mathrm{a}{\left({\mathrm{m}}^{3}/\mathrm{m}\mathrm{o}\mathrm{l}\right)}^{2}\right)$ | 1.63 × 10^{−4} | 2.19 × 10^{−4} |

${b}_{2}\left({\mathrm{m}}^{3}/\mathrm{m}\mathrm{o}\mathrm{l}\right)$ | 1.89 × 10^{−4} | 1.98 × 10^{−4} |

(AARD %) global | 70.86 | 27.98 |

(AARD%) for 313 K | 98.58 | 28.61 |

(AARD%) for 323 K | 95.04 | 23.10 |

(AARD%) for 333 K | 18.95 | 41.51 |

$\mathbf{Using}\text{}({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathbf{PR})$ | $\mathbf{Using}\text{}({\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathbf{Clapeyron})$ | |
---|---|---|

${a}_{2}\left(\mathrm{P}\mathrm{a}{\left({\mathrm{m}}^{3}/\mathrm{m}\mathrm{o}\mathrm{l}\right)}^{2}\right)$ | 1.87 × 10^{−4} | 2.80 × 10^{−4} |

${b}_{2}\left({\mathrm{m}}^{3}/\mathrm{m}\mathrm{o}\mathrm{l}\right)$ | 2.72 × 10^{−4} | 2.93 × 10^{−4} |

(AARD %) global | 79.5 | 48.5 |

(AARD%) for 314.85 K | 99.5 | 41.7 |

(AARD%) for 324.85 K | 100.1 | 43.4 |

(AARD%) for 334.85 K | 48.1 | 60.3 |

**Table 5.**Results of the correlating data with the two empirical models and the newsublimation pressures obtained.

Mendez-Santiago-Teja Model | Bartle’s Model | ||||
---|---|---|---|---|---|

Penicillin G | Penicillin V | Penicillin G | Penicillin V | ||

A’= | −11,475.4 | −5495.9 | a = | 25.1 | 12.6 |

B’= | 165,852.2 | 73,223.7 | b = | −10,260.3 | −5049.7 |

C’= | 26.2 | 12.8 | c = | 1.2 × 10^{−2} | 5.05 × 10^{−3} |

AARD% = | 24.6 | 17.01 | AARD% = | 16.9 | 16.1 |

$\Delta {H}^{s}$_{estimated} (kJ/mol) | 85.3 | 41.9 | |||

$\mathrm{ln}{P}_{2}^{sub}\left(Pa\right)={C}^{\prime}-\frac{\Delta {H}^{S}}{RT}$ | |||||

Penicillin G: ${P}_{2}^{sub}={e}^{\left(26.2-\frac{85.3}{RT}\right)}$ | Penicillin V: ${P}_{2}^{sub}={e}^{\left(12.8-\frac{41.9}{RT}\right)}$ |

AARD% (Penicillin G) | AARD% (Penicillin V) | ||||
---|---|---|---|---|---|

$\mathbf{New}\text{}{\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}$ | ${\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathbf{RK}$ | $\mathbf{New}\text{}{\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}$ | ${\mathit{P}}_{2}^{\mathit{s}\mathit{u}\mathit{b}}-\mathbf{Clapeyron}$ | ||

T = 313.15 K | 27.4 | 28.6 | T = 314.85 K | 26.7 | 41.7 |

T = 323.15 K | 11.1 | 23.1 | T = 324.85 K | 26.4 | 43.4 |

T = 333.15 K | 4.0 | 41.5 | T = 334.85 K | 40.6 | 60.3 |

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**MDPI and ACS Style**

Nasri, L. Penicillins’ Solubility in Supercritical Carbon Dioxide: Modeling by Cubic Equations of States Revisited. *Antibiotics* **2021**, *10*, 1448.
https://doi.org/10.3390/antibiotics10121448

**AMA Style**

Nasri L. Penicillins’ Solubility in Supercritical Carbon Dioxide: Modeling by Cubic Equations of States Revisited. *Antibiotics*. 2021; 10(12):1448.
https://doi.org/10.3390/antibiotics10121448

**Chicago/Turabian Style**

Nasri, Loubna. 2021. "Penicillins’ Solubility in Supercritical Carbon Dioxide: Modeling by Cubic Equations of States Revisited" *Antibiotics* 10, no. 12: 1448.
https://doi.org/10.3390/antibiotics10121448