Application of the general rate law model to the sonolytic degradation of nonvolatile organic pollutants in aqueous media

Abstract

The pseudo-first and pseudo-second order equations have been the most commonly used models to characterize the sonolytic disappearance kinetics of nonvolatile pollutants in aqueous media. In this work, the general rate law model, i.e., pseudo-nth order kinetics equation, was applied for the first time to the sono-decomposition of different nonvolatile organic pollutants, naphthol blue black (NBB), furosemide (FSM), 4-isopropylphenol (4-IPP), and rhodamine B (RhB), in water. It was shown that the general rate law for a chemical reaction would apply to the kinetics of sonochemical decomposition. It is not feasible to set the order of ultrasonic pollutant degradation kinetics to pseudo-first or pseudo-second, as is typically used in numerous studies. The sonochemical oxidation reaction has a fractional order, the order is often non-integer, which frequently indicates a complex sonolytic decomposition reaction mechanism. The degradation mechanism of NBB and RhB does not change with the initial substrate concentration. They are ultrasonically degraded by hydroxyl radicals both in the bulk liquid solution and at the liquid/bubble interfacial layer. The destruction mechanism of FSM and 4-IPP changes as the initial contaminant concentration changes. At low initial substrate concentrations, these pollutants are oxidized mainly by reaction with hydroxyl radicals in the bulk liquid solution and at the interfacial shell of the cavitation bubbles. At high initial substrate concentrations, FSM and 4-IPP are degraded by thermal destruction in the liquid/bubble interfacial layer and by ^•OH radicals both in the bulk liquid solution and at the liquid/bubble interfacial layer. Additionally, the pseudo-nth order model predicts very well the sonolytic degradation at various sonication frequencies and intensities. The general rate law expression should be used to assess the real kinetics order of the sonolytic destruction process without any predetermined assumptions or constraints.

1. Introduction

With the accelerated growth of industrial and agricultural production in recent decades, many organic contaminants have been released into the environment. Organic contaminants have toxic, carcinogenic, and teratogenic effects that have raised concerns about public health and environmental protection. One of the most effective ways to eliminate these contaminants from water is through advanced oxidation processes (AOPs) [1], [2], [3].

AOPs have been extensively utilized in real contaminated waters due to their fast, environmentally friendly and effective removal of the contaminants. The removal of these pollutants under ambient pressure and temperature conditions, without the addition of chemicals, is accomplished by sonolysis [4]. A solution is sonicated to cause acoustic cavitation bubbles to form, expand, and then abruptly collapse [5], [6], [7], [8], [9]. Stable and transient cavitation are the two kinds of cavitation. There are two ways to define stable cavitation: the first is that bubbles persist for a very long time, and the second is that these bubbles are inert during chemical reactions and sonoluminescence (light emission) [5]. As a result, transient cavitation has two definitions. The first is that the acoustic bubble is short-lived, lasting only a single or a few cycles before splitting into daughter bubbles [5], [10]. The second is that the bubbles are involved in chemical processes that produce radicals or sonoluminescence [5]. Consequently, the hot spots of bubble collapse generate enormous pressures and temperatures [5], [11], [12]. A variety of reactive species and radicals are formed by the existence of water vapor and non-condensable gases in hot regions, which are then capable of initiating various secondary chemical processes [6], [13], [14], [15], [16]. Nevertheless, nonvolatile hydrophobic and/or hydrophilic contaminants are mainly removed by interaction with ^•OH radicals in the bulk liquid solution or at the liquid/bubble interfacial layer, while volatile chemicals are likely to be destroyed by pyrolysis within the bubble (gas phase) [5], [17].

The pseudo-first order rate law equation is the most widely used model to describe the kinetics of sonolytic destruction of nonvolatile organic contaminants in aqueous media. Most works described the pollutant concentration decay as exponential and automatically used the pseudo-first order rate law. However, there was no linear relation between the initial contaminant concentration and the initial sonolytic degradation rate, in contrast to what a pseudo-first order kinetics law would predict [18], [19], [20], [21], [22], [23], [24], [17], [25], [26], [27]. Additionally, the determined rate constant varies with the initial contaminant concentration, demonstrating that sono-oxidation does not conform a pseudo-first order kinetics law. Thus, it is obvious that the sono-decomposition process does not obey the pseudo-first order rate law and neither can be satisfactorily designated using a rate constant expressed as 1/time [18], [19], [20], [21], [22], [23], [24], [17], [25]. Other studies have fitted the sonochemical degradation kinetics by using the pseudo-second order rate law equation [28], [29], [30], [31], [32]. Contrary to what a pseudo-second order kinetics law would suggest, the initial sonolytic decomposition rate augmented with the initial substrate concentration and then reached a plateau [18], [19], [20], [21], [22], [23], [24], [17], [25], [26], [27], [33].

The sonolytic destruction kinetics of pollutants in the aqueous phase is directly related to the characteristics of the active acoustic cavitation bubbles. These properties remain unknown, despite the active investigation of yields and efficiencies under various operating settings at both the single-bubble and multi-bubble scales, because acoustic cavitation is a complex process that can be affected by a number of variables, even if only one parameter is changed. Furthermore, the sono-degradation of contaminants is a complex process. The transfer of contaminant molecules from the liquid solution bulk to the cavitation bubble plays a significant role in the sono-degradation reaction rate, and the mass transfer resistance with respect to the contaminant properties; i.e., volatility and hydrophobicity, can play a dominant role and control the rate. In addition, the sono-degradation zone and mechanism are significantly influenced by the initial pollutant concentration. Therefore, it was expected that equations like the pseudo-first and pseudo-second order models could not predict the complex phenomena of sonoyltic destruction of contaminants in the aqueous phase.

We are not aware of any previous report describing the sonolytic destruction kinetics of pollutants in the aqueous phase using the general rate law model, i.e., the pseudo-nth order kinetics model. Additionally, the majority of the works have applied the pseudo-first and pseudo-second order models by using the linear least squares method, taking the correlation coefficient values as the paramount parameter to compare the best fit. This technique requires the assumption of the order of the destruction reaction, which is a real constraint, and the results obtained, selected on the basis of the higher value of the correlation coefficient, will be totally biased. Furthermore, the approach that uses nonlinear curve fitting examination to estimate the model parameters is more appropriate because, in addition to potentially violating the normality assumptions of the classical least squares approach and the error variance, the transformation of a nonlinear equation into a linear expression implicitly affects its error structure. Therefore, the general rate law expression should be used to estimate the true kinetics order of the sonolytic destruction process without any predetermined assumptions or constraints.

Two dyes (naphthol blue black (NBB) and rhodamine B (RhB)) and two less hydrophilic compounds (furosemide (FSM) and 4-isopropylphenol (4-IPP)) were used as models of contaminants in this work. NBB is an acid diazo dye with high thermal and photostability. It is widely used in the textile industry for dyeing silk, nylon, wool and textile printing [34]. Other industrial uses include wood stains, anodized aluminum and casein coloration, writing inks, and soap coloration [34]. RhB is a primary xanthene class dye known as a water tracer and is commonly used as a textile colorant. It is a respiratory, eye, and skin irritant and is toxic to humans and animals if ingested [35]. FSM, a sulfamoylanthranilic acid derivative, is a drug commonly used as a diuretic to alter the volume and/or composition of body fluids in a variety of conditions, including heart failure, nephritic syndrome, kidney failure, cirrhosis, and hypertension [35], [36], [37]. 4-IPP is an alkylphenol with acute and chronic toxicity that has been identified as a contaminant in rivers [38]. Therefore, it was considered desirable to undertake systematic efforts to effectively remove these pollutants from waterways, taking into account their harmful effects and hazardous nature.

In the present study, the application of the general rate law model, i.e., the pseudo-nth order kinetics equation, to the description of the sonolytic destruction of nonvolatile organic contaminants in aqueous phase using the nonlinear technique was investigated for the first time. The impacts of initial substrate concentration and sonication frequency and intensity were studied.

2. Experimental

2.1. Chemicals

The chemicals employed in this investigation were of the purest grade. Sigma-Aldrich provided naphthol blue black (NBB), rhodamine B (RhB), furosemide (FSM), and 4-isopropylphenol (4-IPP), and Riedel-de Haën supplied acetonitrile. All tests were performed with ultrapure water.

2.2. Sono-reactors

In a 500 mL capacity cylindrical jacketed glass reactor, a high-frequency transducer was positioned at the bottom. The transducer functioning at 585 (5.3 cm diameter of active area) is capable of producing sonication irradiation with adjustable intensity (model E/805/T/M from Meinhardt Ultrasonics). The 300 kHz transducer consisted of a piezo-electric ceramic disc (4 cm diameter) attached to a Pyrex dish surface. These frequencies were selected because they are the most efficient for sono-degradation of contaminants in water as reported in the literature [39]. In all studies, a chiller was used to circulate cooling solution throughout the sonication reactor envelope to maintain the reacting medium temperature constant (25 ± 1 °C) during ultrasonication.

2.3. Sonolytic oxidation of contaminants

Sonolytic destruction of the contaminants of interest was studied in different sono-reactors using 300 mL of substrate solutions. Samples of the reacting media were collected at various times to determine the concentrations of the substrates. To maintain a constant sonication power density delivered to the dye solutions (naphthol blue black and rhodamine B), the dye solution samples were returned to the sono-reactors after analysis. The volume of samples taken for HPLC analyses (furosemide and 4-isopropylphenol) was small and did not affect the sonication power density.

Experiments were conducted at various initial contaminant concentrations ranging from 3 to 80 mg/L for NBB, 0.5–20 mg/L for FSM, 0.5–60 mg/L for 4-IPP, and 1–40 mg/L for RhB.

Tests were carried out at pH 6.0, 5.2, 5.3, and 6.5 for NBB, FSM, RhB, and 4-IPP, respectively. These values were the natural pHs obtained by simply dissolving the solutes in ultrapure water.

2.4. Analytical procedures

Quantitative analyses of 4-isopropylphenol and furosemide were performed on Waters Associates 590 and YL9100 HPLC instruments, respectively, using an LC-18 Supelcosil column. UV detectors (YL9120 for furosemide and model 440 for 4-isopropylphenol) were set at 254 and 230 nm for 4-isopropylphenol and furosemide, respectively. Mobile phases used in isocratic mode were acetonitrile/water at 50/50 (v/v) for 4-isopropylphenol and acetonitrile/water at 30/70 (v/v) and formic acid (0.1%) for furosemide.

For dye analysis, a Biochrom WPA (Lightwave II) spectrophotometer was utilized to determine the optical density at the wavelengths of maximum absorption (620 nm for naphthol blue black and 551 nm for rhodamine B).

The mean value is reported for each experiment conducted in triplicate. Typically, the standard deviation does not exceed 3%.

3. General rate law model

The kinetics of the sono-oxidation reaction (the rate of disappearance of the contaminant per unit volume) can be expressed by the pseudo-first order rate law model and is written as follows:

$$\mathrm{r}=-\frac{\mathrm{d}\mathrm{C}}{\mathrm{d}\mathrm{t}}={\mathrm{k}}_1\mathrm{C}$$ (1)

where r (mg/L·min) is the sono-oxidation rate (the rate of disappearance of the contaminant), C (mg/L) is the contaminant concentration, t (min) is the time, and k₁ (1/min) is the pseudo-first order rate constant.

Eq. (2) is obtained by separating the variables and integrating Eq. (1).

$$\mathrm{l}\mathrm{n}\frac{\mathrm{C}}{{\mathrm{C}}_0}=-{\mathrm{k}}_1\mathrm{t}$$ (2)

Eq. (2) is the most commonly used form for representing the sono-decomposition of nonvolatile organic contaminants in aqueous media.

The second form of the pseudo-first order rate law expression (Eq. (3)) is obtained by exponentiating each side of Eq. (2).

$$\mathrm{C}={\mathrm{C}}_0{\mathrm{e}}^{-{\mathrm{k}}_1\mathrm{t}}$$ (3)

Eq. (4) denotes the time required for the initial substrate concentration to be reduced by half, i.e., C₀/2, commonly referred to as the half-life time.

$${\mathrm{t}}_{1/2}=\frac{\mathrm{l}\mathrm{n}2}{{\mathrm{k}}_1}$$ (4)

The half-life time, t_1/2, for a pseudo-first order kinetics does not depend on the initial substrate concentration.

The general rate law model (pseudo-nth order) describes the kinetics of the sono-oxidation reaction (the rate of disappearance of the contaminant per unit volume) and is written as follows:

$$\mathrm{r}=-\frac{\mathrm{d}\mathrm{C}}{\mathrm{d}\mathrm{t}}={\mathrm{k}}_{\mathrm{n}}{\mathrm{C}}^{\mathrm{n}}$$ (5)

where k_n (mg¹⁻ⁿ/L¹⁻ⁿ·min) is the pseudo-nth order rate constant.

After separating, integrating, and rearranging Eq. (5), the following expression can be obtained:

$${\mathrm{C}}^{1-\mathrm{n}}={\mathrm{C}}_0^{1-\mathrm{n}}-(1-\mathrm{n}){\mathrm{k}}_{\mathrm{n}}\mathrm{t}$$ (6)

By rearranging Eq. (6), the following expressions can be obtained:

$$\frac{1}{{\mathrm{C}}^{\mathrm{n}-1}}-\frac{1}{{\mathrm{C}}_0^{\mathrm{n}-1}}=\left(\mathrm{n}-1\right){\mathrm{k}}_{\mathrm{n}}\mathrm{t}$$ (7)

$$\mathrm{C}={\left[\left(\mathrm{n}-1\right){\mathrm{k}}_{\mathrm{n}}\mathrm{t}+{\mathrm{C}}_0^{1-\mathrm{n}}\right]}^{1/(1-\mathrm{n})}$$ (8)

Eqs. (6), (7), (8) are not valid for n = 1.

The time taken to achieve a half decomposition of the initial substrate concentration is obtained by replacing C by C₀/2 in Eq. (7) and rearranging.

$${\mathrm{t}}_{1/2}=\frac{{\mathrm{C}}_0^{1-\mathrm{n}}(2^{\mathrm{n}-1}-1)}{(\mathrm{n}-1){\mathrm{k}}_{\mathrm{n}}}$$ (9)

Eq. (9) is undefinable for n = 1.

By taking the natural logarithm of Eq. (9):

$$\ln \left({\mathrm{t}}_{1/2}\right)=\left(1-\mathrm{n}\right)\mathrm{l}\mathrm{n}{\mathrm{C}}_0+\mathrm{l}\mathrm{n}\frac{(2^{\mathrm{n}-1}-1)}{{\left(\mathrm{n}-1\right)\mathrm{k}}_{\mathrm{n}}}$$ (10)

Eq. (10) is indefinable for n = 1.

The ln(t_1/2) vs. lnC₀ plot must be linear. A change in the slope (discontinuity) of the ln(t_1/2) vs. lnC₀ plot indicates a change in the degradation mechanism as the initial contaminant concentration changes.

For n = 2, Eq. (7) becomes:

$$\frac{1}{\mathrm{C}}=\frac{1}{{\mathrm{C}}_0}+{\mathrm{k}}_2\mathrm{t}$$ (11)

where k₂ (L/mg·min) is the pseudo-second order rate constant.

Eq. (11) represents the pseudo-second order rate law model.

For a pseudo-second order kinetics, t_1/2 is given by:

$${\mathrm{t}}_{1/2}=\frac{1}{{{\mathrm{k}}_2\mathrm{C}}_0}$$ (12)

The half-life time (t_1/2) is inversely proportional to the initial substrate concentration for a pseudo-second order rate law.

The sono-degradation results for the non-volatile contaminants studied (NBB, RhB, FSM, and 4-IPP) were examined for consistency with the general rate law model (Eq. (8)) using OriginPro® software and the nonlinear curve fitting analysis technique, which is the best method for determining the equation parameters. Once the model parameters are determined, the degradation kinetics courses are reconstructed using the determined values. The generated curves are a superimposition of the experimental results (dots) and the theoretically expected data (lines). The functions utilized to assess the goodness of fit of the pseudo-nth order kinetics model were the nonlinear chi-square test and the correlation coefficient.

4. Results and discussion

The characteristics of the examined contaminants, tabulated in Table 1, demonstrate that the nonvolatile nature of the studied dyes, i.e., NBB and RhB, along with FSM and 4-IPP, prevents them from entering the acoustic cavitation bubbles and forces them to degrade on the outside of the collapsing bubble [18], [19], [17], [40], [25], [41].

Table 1.

Main properties of the studied contaminants.

Properties	Naphthol blue black	Rhodamine B	Furosemide	4-isopropylphenol
CAS number	1064–48-8	81–88-9	54–31-9	99–89-8
Molecular formula	C22H14N6Na2O9S2	C28H31N2O3Cl	C12H11ClN2O5S	C9H12O
Abbreviation	NBB	RhB	FSM	4-IPP
Solubility	10–50 mg/mL (17.7 °C)	34 mg/mL (20 °C)	>49.6 / 73.1 µg/mL (30 °C)	1.102 mg/mL (25 °C)
Vapor pressure (mm Hg) at 25 °C	4.85 × 10−29	1.89 × 10−19	3.11 × 10−11	0.05
Henry’s law constant (atm·m3/mol)	1.25 × 10−31	2.2 × 10−21	3.90 × 10−21	1.09 × 10−6
Octanol-water partition coefficient, log Kow	0.82	1.95	2.03	2.90

4.1. Impact of initial contaminant concentration

Sonolysis of NBB in water at 585 kHz and 3.58 W/cm² was conducted at diverse initial concentrations ranging from 3 to 80 mg/L. The results obtained as well as those predicted by the general rate law model, i.e., the pseudo-nth order equation, using the nonlinear regression method were shown in Fig. 1. The determined parameters of the pseudo-nth order equation are consigned in Table 2. According to Fig. 1, the sono-oxidation kinetics of NBB slowed down as the initial contaminant concentration augmented. A variety of non-volatile organic contaminants have been shown to exhibit the same behavior when subjected to sonolysis in the aqueous phase [22], [39], [42], [43], [44], [45]. Additionally, Fig. 1 and Table 2 demonstrate that the ultrasonic decomposition of NBB in water can be excellently predicted by the general rate law model. This is reflected in lower chi-square test values and higher correlation coefficients. After successfully applying the general rate law model to the experimental data, it was seen that the pseudo-nth order (n) of the oxidation reaction was found to be between 1.15937 and 0.66038 (Table 2). These results confirm that the pseudo-order of the sono-decomposition reaction varies with the initial pollutant concentration and, in most cases, is different from a pseudo-first order kinetics. Additionally, the pseudo-nth order decreased with increasing initial NBB concentration. As the initial contaminant concentration augmented in the interval of 3–30 mg/L, the pseudo-nth order rate constant (k_n) diminished. This tendency is not true for an initial NBB concentration of 80 mg/L.

Fig. 1.

Experimental and predicted results for the sonochemical degradation kinetics of NBB in water at different initial substrate concentrations ((a): 3–15 mg/L, (b): 20–80 mg/L) by the general rate law model (conditions: volume: 300 mL, initial concentrations: 3–80 mg/L, frequency: 585 kHz, acoustic intensity: 3.85 W/cm², temperature: 25 °C, pH: 6.0 (natural)).

Table 2.

General rate law model parameters obtained by nonlinear regression technique for different initial substrate concentrations (conditions: volume: 300 mL, initial concentrations: 3–80 mg/L (NBB), 0.5–20 mg/L (FSM), 0.5–60 mg/L (4-IPP) and 1–40 mg/L (RhB), frequency: 585 kHz (NBB and FSM) and 300 kHz (RhB and 4-IPP), acoustic intensity: 3.85 W/cm² (NBB), 4.3 W/cm² (FSM), 2.04 W/cm² (RhB) and 2.86 W/cm² (4-IPP), temperature: 25 °C, pH: natural: 6.0 (NBB), 5.2 (FSM), 5.3 (RhB) and 6.5 (4-IPP)).

Naphthol blue black
Initial concentration (mg/L)
	3	5	15	20	30	80
n	1.15937	1.03942	0.89751	0.92844	0.88058	0.66038
kn*	0.13383	0.09281	0.0862	0.06523	0.06283	0.10082
χ2	0.00359	0.00250	0.00348	0.03346	0.09891	0.52238
R	0.99866	0.99957	0.99992	0.99957	0.99942	0.99956
Furosemide
Initial concentration (mg/L)
	0.5	1	2	5	10	20
n	1.48598	1.1856	0.97313	1.15601	0.96298	0.65585
kn*	0.9599	0.34535	0.20407	0.16181	0.12541	0.12779
χ2	5.57112 × 10−5	8.97815 × 10−4	0.00160	0.00760	0.03188	1.14280
R	0.99845	0.99504	0.99804	0.99874	0.99880	0.99123
4-isoppropylphenol
Initial concentration (mg/L)
	0.5	1	5	10	20	40	60
n	1.34724	1.27867	0.83507	0.84275	0.85409	0.54426	0.49219
kn*	0.30615	0.13609	0.08472	0.07214	0.06732	0.09564	0.08559
χ2	2.19751 × 10−4	4.14525 × 10−4	0.002290	0.023890	0.05759	0.16827	0.05389
R	0.99614	0.99781	0.99954	0.99929	0.99929	0.99950	0.99950
Rhodamine B
Initial concentration (mg/L)
	1	2	3	4	5	6
n	1.02449	0.99692	1.11850	1.15187	0.93255	0.89189
kn*	0.04572	0.04281	0.03633	0.03245	0.03618	0.03727
χ2	8.8015 × 10−5	0.00107	0.00245	0.00263	7.8438 × 10−4	0.00537
R	0.99956	0.99865	0.99844	0.99906	0.99984	0.99927
Initial concentration (mg/L)
	7	8	10	15	20	40
n	0.88496	0.88935	0.92607	0.94521	1.04242	1.04167
kn*	0.03868	0.03500	0.03264	0.02864	0.02159	0.01530
χ2	0.00875	0.00574	0.01203	0.02979	0.03533	0.18269
R	0.99914	0.99954	0.99938	0.99931	0.99951	0.99937

^*(mg¹⁻ⁿ/L¹⁻ⁿ·min).

The pseudo-nth order model, i.e., the general rate law equation, was also applied to the decomposition of FSM in water at various initial contaminant concentrations ranging from 0.5 to 20 mg/L by ultrasonication at the same sonication frequency as NBB and 4.3 W/cm² using the nonlinear regression technique. The simulated curves as well as the experimental results were shown in Fig. 2, and the parameters of the general rate law model, the nonlinear chi-square test values, and the correlation coefficients are regrouped in Table 2. The lower chi-square test values and higher correlation coefficients indicate that the general rate law model fits the experimental data of FSM sonolytic destruction in aqueous phase very well. The pseudo-nth order varies in the initial concentration range examined and is different from a pseudo-first order. The pseudo-nth order was 1.48598, 1.18560, 0.97313, 1.15601, 0.96298, and 0.65585, respectively, for initial FSM concentrations of 0.5, 1, 2, 5, 10, and 20 mg/L. The pseudo-nth order decreased with increasing initial contaminant concentration, except at 5 mg/L FSM. The pseudo-nth order rate constant values were 0.9599, 0.34535, 0.20407, 0.16181, 0.12541, and 0.12779 mg¹⁻ⁿ/L¹⁻ⁿ·min at initial FSM concentrations of 0.5, 1, 2, 5, 10, and 20 mg/L, respectively. The pseudo-nth order rate constant decreased as the initial substrate concentration increased, and almost the same value was obtained at initial FSM concentrations of 10 and 20 mg/L.

Fig. 2.

Experimental and predicted results for the sonochemical degradation kinetics of FSM in water at different initial substrate concentrations ((a): 0.5–1 mg/L, (b): 5–20 mg/L) by the general rate law model (conditions: volume: 300 mL, initial concentrations: 0.5–20 mg/L, frequency: 585 kHz, acoustic intensity: 4.3 W/cm², temperature: 25 °C, pH: natural: 5.2 (natural)).

Since the experimental sonolysis results of the oxidation of NBB and FSM were excellently described by the general rate law, the model was also applied to the sono-decomposition of RhB and 4-IPP because they show different properties. From Table 1, Table 4-IPP has higher fugacity, octanol/water partition coefficient, and Henry's law constant than FSM, and RhB has the same tendency compared to NBB. Consequently, 4-IPP may be more present in the interfacial layer of the acoustic cavitation bubbles compared to FSM, and an identical predisposition can be observed for RhB compared to NBB. Sonolytic decomposition of 4-IPP and RhB was conducted at 300 kHz and 2.86 W/cm² for 4-IPP and 2.04 W/cm² for RhB. The initial contaminant concentration investigated varies in the intervals of 0.5–60 mg/L for 4-IPP and 1–40 mg/L for RhB. The predicted sono-decomposition kinetics of 4-IPP and RhB determined using the general rate law model compared to the experimental data were shown in Fig. 3, Fig. 4, respectively. The parameters of the general rate law model, as well as the correlation coefficients and the chi-square values, were presented in Table 2. From these figures as well as the results consigned in Table 2, an excellent prediction of the experimental data of both contaminants was obtained by the general rate law model, highlighted by lower chi-square values and higher correlation coefficient values. After successfully applying the general rate law model to the kinetics data, it was determined that the pseudo-nth order of the sono-decomposition reaction was in the range of 0.49219–1.34724 in the case of 4-IPP and in the interval of 0.88496–1.15187 in the case of RhB.

Table 4.

General rate law model parameters obtained for sonolytic degradation of furosemide by nonlinear regression technique for different ultrasonic intensities (conditions: volume: 300 mL, initial concentration: 10 mg/L, frequency: 585 kHz, temperature: 25 °C, pH: 5.2 (natural)).

	Ultrasonic intensity (W/cm2)
	0.83	2.64	4.3
n	1.08738	0.91310	0.93448
kn (mg1−n/L1−n·min)	0.02340	0.06221	0.13978
χ2	0.00723	0.00309	0.00819
R	0.99959	0.99988	0.99966

Fig. 3.

Experimental and predicted results for the sonochemical degradation kinetics of 4-IPP in water at different initial substrate concentrations ((a): 0.5–10 mg/L, (b): 20–60 mg/L) by the general rate law model (conditions: volume: 300 mL, initial concentrations: 0.5–60 mg/L, frequency: 300 kHz, acoustic intensity: 2.86 W/cm², temperature: 25 °C, pH: 6.5 (natural)).

Fig. 4.

Experimental and predicted results for the sonochemical degradation kinetics of RhB in water at different initial substrate concentrations ((a): 1–8 mg/L, (b): 20–40 mg/L) by the general rate law model (conditions: volume: 300 mL, initial concentrations: 1–40 mg/L, frequency: 300 kHz, acoustic intensity: 2.04 W/cm², temperature: 25 °C, pH: 5.3 (natural)).

For all the investigated contaminants and in the examined initial substrate concentration ranges, the sonochemical oxidation reaction has a fractional order, the order is often not an integer, which often indicates a complex reaction mechanism. The complexity of the sonolytic oxidation mechanism is expected because, despite active investigation of degradation yields and efficiencies under various operating conditions, the properties of acoustically active bubbles remain elusive due to the complexity of acoustic cavitation, which can be affected by multiple variables even when only one parameter is changed.

The time required to achieve a half decomposition of the initial substrate concentration, i.e., t_1/2, for the contaminants investigated in the present work is obtained by using Eq. (9). The plots of the natural logarithm of the half degradation time (ln t_1/2) as a function of the natural logarithm of the initial contaminant concentration (ln C₀) for the studied substrates were shown in Fig. 5. A change in the slope (discontinuity) of the ln(t_1/2) vs. lnC₀ plot is an indication of a change in the degradation mechanism as the initial contaminant concentration changes. In the case of the dyes examined, i.e., NBB and RhB, the slopes of the ln(t_1/2) vs. lnC₀ plot were 0.5159 for NBB and 0.2599 for RhB, thus no change in the slopes of the ln(t_1/2) vs. lnC₀ plot was observed over the whole interval of initial dye concentrations examined. These results demonstrate that the sonolytic decomposition mechanism of both dyes does not change when the initial substrate concentration was varied over the interval studied for both dyes. Since hydroxyl radicals are primarily responsible for the sonolytic oxidation of the tested dyes, and the investigated substrates exhibit a nonvolatile behavior (relatively low fugacity, octanol/water partition coefficient, and vapor pressure), it is reasonable to assume that the pollutants will undergo a number of free radical decomposition reactions, but primarily at the interfacial layer of the acoustic cavitation bubbles and to some extent in the bulk liquid solution. In contrast, the ln(t_1/2) vs. lnC₀ plots for FSM and 4-IPP show a change in the slopes for the initial concentration range investigated, indicating a modification in the sono-decomposition mechanism with changing initial substrate concentration. For FSM, the slopes of the ln(t_1/2) vs. lnC₀ plot were 0.7560 in the initial substrate concentration interval of 0.5–2 mg/L and 0.9714 in the initial substrate concentration interval of 5–20 mg/L. For 4-IPP, the slopes of the ln(t_1/2) vs. lnC₀ plot were 0.3940 and 1.1633 in the initial concentration range of 0.5–10 and 20–60 mg/L, respectively. These changes in the slopes of the ln(t_1/2) vs. lnC₀ plot indicate a change in the mechanism of sono-decomposition with a change in the initial contaminant concentration. In the low initial concentration range, i.e., 0.5–2 mg/L for FSM and 0.5–10 mg/L for 4-IPP, the tested contaminants are expected be oxidized by hydroxyl radical attack in the bulk liquid solution and at the interfacial layer of the acoustic cavitation bubbles. On the other hand, the sonolytic decomposition process will occur primarily at the shell of the acoustic cavitation bubbles instead of in the bulk liquid solution at the high initial concentration interval, i.e., 5–20 mg/L for FSM and 20–60 mg/L for 4-IPP. Based on the information collected from the literature, the concentration of ^•OH radicals in the interfacial layer between the liquid and the cavitation bubbles was estimated to be 2 × 10⁻² M [14], [46], [47]. In the high initial concentration range, i.e., 5–20 mg/L for FSM and 20–60 mg/L for 4-IPP, the pollutants gather at the interfacial layer of the cavitation bubbles and interact with the hydroxyl radicals existing at high concentration in this zone. Additionally, it can be predicted that the pollutants, i.e., FSM and 4-IPP, at the high initial concentration interval will also be degraded in the liquid/bubble interfacial layer by thermal decomposition process.

Fig. 5.

Evolution of lnt_1/2 as a function of lnC₀ for the sono-degradation of the studied contaminants.

By analyzing the results of the general rate law model in combination with the ln(t_1/2) vs. lnC₀ plot and the properties of the examined contaminants, the findings of this work are in excellent agreement. The investigated dyes, i.e., NBB and RhB, have very low Henry’s law constant and vapor pressure and high solubility. These dyes are ultrasonically degraded by hydroxyl radicals generated from acoustic cavitation bubbles both in the bulk liquid solution and at the liquid/bubble interfacial layer. The sono-oxidation mechanism of FSM and 4-IPP changes as the initial contaminant concentration changes. These contaminants are hydrophilic and poorly volatile. At low initial substrate concentrations, these pollutants are oxidized principally at the interfacial shell of the acoustic cavitation bubbles by reacting with hydroxyl radicals released in the interfacial zone ([^•OH] = 2 × 10⁻² M [14], [46], [47]) and in the bulk liquid solution (∼10% of the radicals produced in the bubble interior are able to diffuse into the bulk liquid solution [40]). At high substrate initial concentrations, FSM and 4-IPP are destroyed by thermal degradation in the liquid/bubble interfacial layer and by hydroxyl radicals formed by acoustic cavitation bubbles both in the bulk liquid solution and at the liquid/bubble interfacial layer.

Table 2 shows that the correlation coefficients are very good for all tested contaminants and vary in the range of 0.99866–0.99992, 0.99123–0.9988, 0.99614–0.99954, and 0.99844–0.99984 for NBB, FSM, 4-IPP, and RhB, respectively. Overall, the correlation coefficients are in the following order: NBB > RhB > 4-IPP > FSM. Since the correlation coefficients for the investigated pollutants differ so small, no conclusions can be drawn from them. For all the contaminants examined, the chi-square test values are low. The chi-square value is a sole numeral calculated by summing the difference between the experimental results and what the general rate law model would predict if there were no difference. If there is no discrepancy between the predicted and experimental results, the chi-square value is null. The greater the difference, the higher the chi-square value will be. The lower chi-square values are determined for RhB, followed by 4-IPP, then NBB, and finally FSM. The variances in the chi-square test results are negligible and cannot be used to make judgments. These findings indicate that the experimental results of the sonolytic destruction of the investigated contaminants in water can be adequately described by the general rate law model, i.e., the pseudo-nth order equation.

4.2. Impact of sonication frequency

In order to simplify the paper and to focus on its main objective, which is the application of the general rate law model to the sonolytic decomposition of nonvolatile organic contaminants, only the results obtained for FSM were simulated and analyzed to investigate the impacts of ultrasonication frequency and intensity on its sono-oxidation in aqueous phase. Additionally, the contaminants investigated in the present work showed a similar behavior with respect to the influence of sonication frequency and intensity on their sonolytic decomposition.

The ultrasound intensity was kept at 4.3 W/cm² for the three sonication frequencies of 580, 860, and 1140 kHz during the sono-degradation experimentations of FSM in aqueous solution at 10 mg/L. The effect of ultrasonication frequency on the sono-decomposition of FSM, as well as the predicted curves determined using the general rate law model, i.e., the pseudo-nth order equation, were illustrated in Fig. 6. The parameters of the general rate law model, as well as the chi-square test values and the correlation coefficients obtained at various sonication frequencies, were tabulated in Table 3. In the investigated ultrasonication frequency range, the degradation yield increases as the sonication frequency decreases. As shown in in Fig. 5 and Table 3, higher correlation coefficients and lower nonlinear chi-square test values were obtained, and it can be concluded that the experimental results of the sonochemical destruction of FSM in aqueous media at the investigated sonication frequency interval can be predicted excellently with the general rate law model.

Fig. 6.

Experimental and predicted results for the sonochemical degradation kinetics of FSM in water at different ultrasonic frequencies by the general rate law model (conditions: volume: 300 mL, initial concentrations: 10 mg/L, acoustic intensity: 4.3 W/cm², temperature: 25 °C, pH: 5.2 (natural)).

Table 3.

General rate law model parameters obtained for sonolytic degradation of furosemide by nonlinear regression technique for different ultrasonic frequencies (conditions: volume: 300 mL, initial concentration: 10 mg/L, acoustic intensity: 4.3 W/cm² (FSM), temperature: 25 °C, pH: 5.2 (natural)).

	Ultrasonic frequency (kHz)
	585	860	1140
n	0.93448	1.29542	1.26635
kn (mg1−n/L1−n·min)	0.13978	0.03450	0.02722
χ2	0.00819	0.02826	0.04368
R	0.99966	0.99869	0.99784

After successfully applying the general rate law model to the kinetic results (Fig. 5 and Table 3), it was determined that the pseudo-nth order (n) of the sono-oxidation reaction was between 0.93448 and 1.29542. The ultrasonic oxidation reaction has a fractional order, the order is often non-integer, which frequently denotes a complicated sono-degradation reaction mechanism. As the sonication frequency increased from 585 to 860 and 1140 kHz, the pseudo-nth order rate constants diminished from 0.13978 to 0.03450 and 0.02722 mg¹⁻ⁿ/L¹⁻ⁿ·min, respectively.

4.3. Impact of sonication intensity

The influence of ultrasonic intensity in the interval of 0.83–4.3 W/cm² on the sonolytic decomposition of 10 mg/L FSM in aqueous solution was investigated for a sonication frequency of 585 kHz, and the findings were given in Fig. 7. The general rate law model was applied to the experimental data of FSM sono-decomposition at the three tested intensities, and the generated curves are superimposed with the experimental results in Fig. 6. The determined parameters of the general rate law model as well as the nonlinear chi-square test values and the correlation coefficients were consigned in Table 4. The general rate law model ideally fits the experimental results of FSM sono-decomposition in water regardless the intensity, which is reflected by the lower chi-square test values and higher correlation coefficient values.

Fig. 7.

Experimental and predicted results for the sonochemical degradation kinetics of FSM in water at different ultrasonic intensities by the general rate law model (conditions: volume: 300 mL, initial concentrations: 10 mg/L, frequency: 585 kHz, temperature: 25 °C, pH: 5.2 (natural)).

The pseudo-nth order (n) of the sonolytic degradation reaction was determined to be in the interval of 0.91310–1.08738 by successfully applying the general rate law model to the kinetic results (Table 4). The ultrasonic decomposition reaction of FSM has a fractional order, the order is often not an integer, which frequently denotes a complex sono-oxidation reaction mechanism. As the ultrasonic intensity increased from 0.83 to 2.64 and 4.3 W/cm², the pseudo-nth order rate constants increased from 0.02340 to 0.06221 and 0.13978 mg¹⁻ⁿ/L¹⁻ⁿ·min, respectively.

5. Conclusion

In this paper, the general rate law model has been applied for the first time to the ultrasonic destruction of diverse nonvolatile organic pollutants in water. It is not necessary to specify the order of sono-oxidation kinetics as pseudo-first or pseudo-second, as usually done in sonolytic degradation research. In reality, the general rate law equation must predict the true kinetics order of the sono-decomposition process without any predetermined assumptions or constraints. The sonolytic decomposition reaction has a fractional order, the order is frequently not an integer, which often indicates a complex sono-degradation reaction mechanism. The degradation mechanism of NBB and RhB does not change with the initial substrate concentration and they are ultrasonically degraded by hydroxyl radicals both in the bulk liquid solution and at the liquid/bubble interfacial layer. As the initial contaminant concentration changes, the decomposition mechanism of FSM and 4-IPP changes. At low initial contaminant concentrations, these pollutants are oxidized mainly in the bulk liquid solution and at the interfacial shell of the acoustic cavitation bubbles by reaction with hydroxyl radicals. At high initial substrate concentrations, FSM and 4-IPP are degraded by thermal destruction in the liquid/bubble interfacial layer and by hydroxyl radicals both in the bulk liquid solution and at the liquid/bubble interfacial layer.

Overall, the general rate law model should be used, without preconceptions or limitations, to assess the actual kinetics order of the sonolytic degradation process. The general rate law model has been applied to wastewater remediation by sonolysis in this study, which also highlights the potential implications of using this model in environmental remediation and water treatment using advanced oxidation technologies.

CRediT authorship contribution statement

Oualid Hamdaoui: Investigation, Conceptualization, Methodology, Formal analysis, Project administration, Supervision, Visualization, Writing – original draft, Writing – review & editing. Abdulaziz Alghyamah: Visualization, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.