# Morphology Effect on the Kinetic Parameters and Surface Thermodynamic Properties of Ag3PO4 Micro-/Nanocrystals

Authors: Zai-Yin Huang; Xing-Xing Li; Zuo-Jiao Liu; Liang-Ming He; Xue-Cai Tan
Journal: Journal of Nanomaterials
 (2015)

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## Abstract

Considerable effort has been exerted using theoretical calculations to determine solid surface energies. Nanomaterials with high surface energy depending on morphology and size exhibit enhanced reactivity. Thus, investigating the effects of morphology, size, and nanostructure on the surface energies and kinetics of nanomaterials is important. This study determined the surface energies of silver phosphate (Ag3PO4) micro-/nanocrystals and their kinetic parameters when reacting with HNO3 by using microcalorimetry. This study also discussed rationally combined thermochemical cycle, transition state theory, basic theory of chemical thermodynamics with thermokinetic principle, morphology dependence of reaction kinetics, and surface thermodynamic properties. Results show that the molar surface enthalpy, molar surface entropy, molar surface Gibbs free energy, and molar surface energy of cubic Ag3PO4 micro-/nanocrystals are larger than those of rhombic dodecahedral Ag3PO4 micro-/nanocrystals. Compared with rhombic dodecahedral Ag3PO4, cubic Ag3PO4 with high surface energy exhibits higher reaction rate and lower activation energy, activation Gibbs free energy, activation enthalpy, and activation entropy. These results indicate that cubic Ag3PO4 micro-/nanocrystals can overcome small energy barrier faster than rhombic dodecahedral Ag3PO4 micro-/nanocrystals and thus require lower activation energy.


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## Body

## 1. Introduction

Nanomaterials that exhibit high specific surface effect differ from massive materials in terms of physical and chemical properties [1, 2]. Du et al. [3] explained that the specific surface effect, surface heat capacity, and specific surface energy of nanomaterials cannot be neglected. The overall thermodynamic property of nanoparticles comprises surface and bulk phases [3–9]. Surface thermodynamic properties are the intuitive expression of the special structure-activity relationship of nanomaterial surface, which significantly affects many physical/chemical reactions, including chemical thermodynamics [3, 4], chemical kinetics [5], catalysis [10–13], sense [11], adsorption [14], phase transition [15], and electrochemistry of nanomaterials [16]. Theoretical calculation results showed that nanomaterials with different sizes and facets have different surface energies [17–19] and that the reactivity of nanomaterials depends on surface energy [11, 12]. Studying the surface thermodynamic property of nanomaterials and the structure-function relationship between reaction dynamics and size, morphology, and structure is valuable to understand the nature of chemical reactions.Calorimetry [7, 8, 20, 21], contact angle [22], Young modulus [23, 24], balance crystal shape [3, 25], zero creep, and field-emission microscopic method [26] are common methods of measuring solid surface energy. Hulett [27, 28] reported that the surface energies of BaSO4 and CaSO4·2H2O obtained using the Ostwald-Freundlich formula are within 1000–3000 mJ·m−1. However, Tang et al. [29–32] discovered that results obtained using the Ostwald-Freundlich formula for nanoparticles with far higher solubility than common crystals contradict with experimental facts. Thus, Tang et al. [31, 32] measured the surface energy of inorganic insoluble salt indirectly through crystal growth or dissolution kinetics. Methods commonly used in Young modulus (e.g., tearing) and other methods involving compression, solubilization, high-temperature dissolution, or lowering of melting point produce high and even applied stress-covered surface energies. Hence, theoretical calculation is preferred to measure solid surface energy [23]. Nevertheless, theoretical calculation often deviates from the practical ideal model and from many hypotheses. Real surfaces with abundant atom ladders and unsaturated bonds, as well as those with unstable thermodynamics, tend to absorb water [7], gas molecules, and surfactant [11, 12]; undergo surface atom reconstruction and aggregation; or even form a protective film layer. The surface energies of these complicated real surfaces are extremely difficult to theoretically calculate. Experimental measurement still faces several challenges. Different experimental methods provide significantly different values for the surface energies of the same material [33]; researchers using the same experimental method also obtain varying results [34, 35]. A universal method to determine surface energy has yet to be developed. Developing a scientific and universal experimental method to measure the surface energy of nanomaterials is a pressing need in the scientific endeavors on solid surface and in other disciplines.The visible-light-driven Ag3PO4 photocatalyst has become popular since its introduction in 2010 [36, 37]. Scientists calculated the surface energies of different facets of Ag3PO4 on the basis of the relationship between included angle of crystal faces and Miller index [38]. Other researchers performed the same calculation by using density functional theory [39]. Studying the structure-function relationship between the surface energy of micro-/nanomaterial Ag3PO4 and size, morphology, and structure is valuable to understand the natures of chemical reactions. However, the surface energy of this material was rarely calculated using experimental methods. The present study determined the surface energies of Ag3PO4 micro-/nanocrystals with cubic and rhombic dodecahedral morphologies and their kinetic parameters when reacting with HNO3 by using microcalorimetry. This study also discussed rationally combined thermochemical cycle, transition state theory, basic theory of chemical thermodynamics with thermokinetic principle, morphology dependence of reaction kinetics, and surface thermodynamic properties.


## 2. Experiments

### 2.1. Materials

Analytical-grade sodium dihydrogen phosphate dihydrate (NaH2PO4·2H2O), disodium phosphate dodecahydrate (NaH2PO4·12H2O), silver nitrate (AgNO3), ammonium hydroxide (NH3·H2O), potassium chloride (KCl), and nitric acid were purchased from Sinopharm Chemical Reagent Co. Ltd. and used without further purification. Deionized water with a resistivity of 18.2 MΩ·cm was used in all experiments.


### 2.2. Characterization

The morphology of the sample was examined under a field-emission scanning electron microscope (Zeiss SUPRA 55 Sapphire, Germany). The X-ray diffraction (XRD) pattern was recorded on an X-ray powder diffractometer (Philips PW 1710 with Cu Kα radiation, λ = 1.5406 Å, Holland). Trace amounts of the sample were measured on a XPE analytic balance (Mettler Toledo, Switzerland). Calorimetric experiments were performed using a microcalorimeter (RD496L, Mianyang CP Thermal Analysis Instrument Co., Ltd., China) under constant temperature and pressure.


### 2.3. Calorimetric Experiment

Rhombic dodecahedrons, cubes, and bulk Ag3PO4 were prepared by a simple ion-exchange method at room temperature [40]. The microcalorimeter was calibrated by Joule effect, and its calorimetric constant was (69.91±0.56) μV·mW−1 at 298.15 K. The dissolution enthalpy of KCl in deionized water (1 : 1110, mKCl/mde-water) was (17.792±0.029) kJ·mol−1. This value agrees with the previously published value of (17.524±0.028) [41]. This agreement indicates that the calorimetric system is accurate and reliable.A small glass tube containing 1.0 mL of 0.36 M HNO3 solution was placed above a 15 mL glass tube charged with 1.500 mg of Ag3PO4 samples (bulk or the obtained nanocubes). Simultaneously with the establishment of equilibrium, the small glass tube with HNO3 solution was pushed down. The in situ thermodynamic and kinetic information for this reaction was recorded using the microcalorimeter.


### 2.4. Establishment of Chemical Reaction Kinetic Models for Cubic and Rhombic Dodecahedron Ag3PO4 Micro-/Nanocrystals

The specific surface area and specific surface energy of the reactant increase after being refined; thus, the mean molar energy of the refined reactant is higher than that of the corresponding bulk reactant. If the reactant particle size is insignificant to the mean energy of the activated molecules, then the difference between the mean molecular activation energy of 1 M of nanoparticles and mean energy of 1 mol super-refined reactant is the chemical activation energy of the nanomaterial [42]. Figure 1 shows the transition state theory [8, 42, 43]. In the same chemical reaction, the reactant experiences the same transition state to the final state. Therefore, the apparent activation energy of nanoparticles Ea is the difference between the activation energy of corresponding bulk material [Eabulk] and the molar surface energy of nanoparticles (EmS):(1)Eanano=Eabulk-EmS.Figure 1
Schematic of relationship between surface energy and apparent activation energy.If the dispersion phase in heterogeneous reaction has only one reactant and others belong to the continuous phase, then the relationship between surface energy and apparent activation energy for cubic nanoparticles without inner bores can be expressed as(2)Eanano=Eabulk-EmS=Eabulk-4σMρl,where σ, M, ρ, and l are the surface tension, molar mass, density, and particle size (length of cube edge) of the cubic nanoparticle reactant, respectively. Equation (2) provides that the apparent activation energy in the chemical reaction of the nanomaterials is proportional to the particle size of the reactant.If the heterogeneous reaction follows Arrhenius Law, substituting (2) into it yields the Arrhenius equation of the nanocube:(3)k=Aexp⁡-EabRT+4σMRTρl.Similarly, the Arrhenius equation of dodecahedron is obtained:(4)k=Aexp⁡-EabRT+6σMRTρl,where T is the reaction temperature, k is the reaction rate constant, and A is the preexponential factor.Substituting the logarithm on both sides of (4), we obtain the following:Cube:(5)ln⁡k=ln⁡A-EabRT+4σMρl.Dodecahedron:(6)ln⁡k=ln⁡A-EabRT+6σMρl.Therefore, when the particle size is larger than 10 nm, the surface tension slightly changes and can be viewed as a constant [43]. On the basis of (5) and (6), the logarithm of the reaction rate constant is inversely proportional to the particle size of the reactant.


### 2.5. Acquisition of Dynamic Parameters of Ag3PO4 Reacting with HNO3

The thermodynamic equation of reversible chemical reaction under constant temperature and pressure can be expressed as [44] (7)ln⁡1H∞dHidt=ln⁡k+nln⁡1-HiH∞,where H∞ is the enthalpy change during the whole reaction and may be directly obtained by microcalorimetry, dHi/dt is the enthalpy change rate, k (s−1) is the reaction rate constant expressed by conversion rate, and Hi is the enthalpy change at reaction time t. k can be calculated from the linear regression of thermodynamic data:(8)ln⁡k=ln⁡A-EaRT(9)ΔG≠θ=RTln⁡RTNAhk(10)ln⁡kT=ln⁡kBh+ΔS≠θR-ΔH≠θRT,where NA is Avogadro’s constant, kB is Boltzmann’s constant, h is Planck’s constant, and R is the molar gas constant. The diagram of 1/T was drawn with ln⁡k. Ea and A were calculated using (8). ΔG≠θ, ΔH≠θ, and ΔS≠θ were calculated from (9) and (10).


### 2.6. Theoretical Derivation of Molar Surface Gibbs Free Energy, Molar Surface Enthalpy, Molar Surface Entropy, and Molar Surface Energy

The molar Gibbs free energy of chemical reaction of nanosystem consists of bulk phase (ΔrGmB) and surface phase (ΔrGmS) [8, 42]:(11)ΔrGmnano=ΔrGmB+ΔrGmS.The molar Gibbs free energy of the bulk chemical reaction nearly exhibits bulk phase. The bulk phase of the nanosystem is similar to that of bulk:(12)ΔrGmB=ΔrGmbulk,where ΔrGmbulk is the molar Gibbs free energy of the same chemical reaction bulk material.Therefore, the molar Gibbs free energy difference between the nanosystem and the bulk lies in the molar surface Gibbs free energy of the nanomaterial (excessive Gibbs free energy compared with bulk material). Substituting (12) into (11), we obtain(13)ΔrGmS=ΔrGmnano-ΔrGmbulk.On the basis of (13), the thermochemical cycles of nano- and bulk Ag3PO4 were designed (Figure 2). The thermodynamic functions of nano- and bulk Ag3PO4 reactions with HNO3 were tested. The thermodynamic function of nano-Ag3PO4 conversion into the bulk one was calculated on the basis of their difference.Figure 2
Thermochemical cycle of nano- and bulk Ag3PO4 reacting with HNO3.On the basis of (12), the chemical reaction for the molar surface Gibbs of Ag3PO4 micro-/nanocrystals with a net reaction of Ag3PO4 (nano) → Ag3PO4 (bulk) is(14)ΔrGmθ=ΔrGmθAg3PO4,nano-ΔrGmθAg3PO4,bulk=ΔrGmSAg3PO4,nano=-GmSAg3PO4,nano,where ΔGmθAg3PO4,nano and ΔGmθAg3PO4,bulk are the standard molar Gibbs free energies of the reactions of Ag3PO4 micro-/nanocrystals and bulk Ag3PO4 with HNO3, respectively; ΔGmSAg3PO4,nano and GmSAg3PO4,nano are the molar reaction surface Gibbs free energy and molar surface Gibbs free energy, respectively.In accordance with transition state theory, the relationship between reaction rate constants of Ag3PO4 micro-/nanocrystals reaction system and bulk Ag3PO4 reaction system and the Gibbs free energy can be expressed as [8, 45](15)ΔrGmθAg3PO4,nano-ΔrGmθAg3PO4,bulk=ΔrG≠θbulk-ΔrG≠θnano=RTln⁡kbulk-ln⁡knano,where ΔrG≠θ and k are the activation Gibbs free energy and rate constant of Ag3PO4 chemical reaction.On the basis of (14) and (15), (16)GmSAg3PO4,nano=-ΔrGmSAg3PO4,nano=-RTln⁡kbulk-ln⁡knano.Similarly,ΔH≠θ can be calculated from (10). Molar surface enthalpy can be deduced from transition state theory: (17)ΔrHmS=ΔrHmθnano-ΔrHmθbulk=ΔrH≠θbulk-ΔrH≠θnanoΔrHmS=HmAg3PO4,nano-HmAg3PO4,bulk=-HmSAg3PO4,where ΔrHmS, ΔrHmθ, ΔrH≠θ, and HmS are the molar surface reaction enthalpy, molar reaction enthalpy, molar activation enthalpy, and molar surface enthalpy of Ag3PO4 chemical reaction, respectively.Similarly, molar surface entropy can be deduced from transition state theory:(18)ΔrSmS=ΔrSmθnano-ΔrSmθbulk=ΔrS≠θbulk-ΔrS≠θnanoΔrSmS=SmAg3PO4,nano-SmAg3PO4,bulk=-SmSAg3PO4,where ΔrSmS, ΔrSmθ, ΔrS≠θ, and SmS are the molar surface reaction entropy, molar reaction entropy, molar activation entropy, and molar surface entropy of Ag3PO4 chemical reaction, respectively.Apparent activation energy refers to the total energy needed for the material to overcome activation. The essential difference between Ag3PO4 micro-/nanocrystals and bulk Ag3PO4 is the high specific surface effect of the surface phase. After the same transition state to the final state in the same chemical reaction (Figure 1), the surface energy of the nanosystem surface phase (EmS) is the energy difference between the nano-reaction system and the bulk reaction system. The molar surface energy in the manuscript cited reference [8] which deduced in our published paper. Its correct form is [8]. Consider(19)ΔEmS=Enano-Ebulk=Eabulk-EananoΔrEmS=EmAg3PO4,nano-EmAg3PO4,bulk=-EmSAg3PO4,where ΔEmS, Ea, and Em are the molar surface reaction energy, activation energy, and molar surface energy of Ag3PO4 chemical reaction, respectively.


## 2.1. Materials

Analytical-grade sodium dihydrogen phosphate dihydrate (NaH2PO4·2H2O), disodium phosphate dodecahydrate (NaH2PO4·12H2O), silver nitrate (AgNO3), ammonium hydroxide (NH3·H2O), potassium chloride (KCl), and nitric acid were purchased from Sinopharm Chemical Reagent Co. Ltd. and used without further purification. Deionized water with a resistivity of 18.2 MΩ·cm was used in all experiments.


## 2.2. Characterization

The morphology of the sample was examined under a field-emission scanning electron microscope (Zeiss SUPRA 55 Sapphire, Germany). The X-ray diffraction (XRD) pattern was recorded on an X-ray powder diffractometer (Philips PW 1710 with Cu Kα radiation, λ = 1.5406 Å, Holland). Trace amounts of the sample were measured on a XPE analytic balance (Mettler Toledo, Switzerland). Calorimetric experiments were performed using a microcalorimeter (RD496L, Mianyang CP Thermal Analysis Instrument Co., Ltd., China) under constant temperature and pressure.


## 2.3. Calorimetric Experiment

Rhombic dodecahedrons, cubes, and bulk Ag3PO4 were prepared by a simple ion-exchange method at room temperature [40]. The microcalorimeter was calibrated by Joule effect, and its calorimetric constant was (69.91±0.56) μV·mW−1 at 298.15 K. The dissolution enthalpy of KCl in deionized water (1 : 1110, mKCl/mde-water) was (17.792±0.029) kJ·mol−1. This value agrees with the previously published value of (17.524±0.028) [41]. This agreement indicates that the calorimetric system is accurate and reliable.A small glass tube containing 1.0 mL of 0.36 M HNO3 solution was placed above a 15 mL glass tube charged with 1.500 mg of Ag3PO4 samples (bulk or the obtained nanocubes). Simultaneously with the establishment of equilibrium, the small glass tube with HNO3 solution was pushed down. The in situ thermodynamic and kinetic information for this reaction was recorded using the microcalorimeter.


## 2.4. Establishment of Chemical Reaction Kinetic Models for Cubic and Rhombic Dodecahedron Ag3PO4 Micro-/Nanocrystals

The specific surface area and specific surface energy of the reactant increase after being refined; thus, the mean molar energy of the refined reactant is higher than that of the corresponding bulk reactant. If the reactant particle size is insignificant to the mean energy of the activated molecules, then the difference between the mean molecular activation energy of 1 M of nanoparticles and mean energy of 1 mol super-refined reactant is the chemical activation energy of the nanomaterial [42]. Figure 1 shows the transition state theory [8, 42, 43]. In the same chemical reaction, the reactant experiences the same transition state to the final state. Therefore, the apparent activation energy of nanoparticles Ea is the difference between the activation energy of corresponding bulk material [Eabulk] and the molar surface energy of nanoparticles (EmS):(1)Eanano=Eabulk-EmS.Figure 1
Schematic of relationship between surface energy and apparent activation energy.If the dispersion phase in heterogeneous reaction has only one reactant and others belong to the continuous phase, then the relationship between surface energy and apparent activation energy for cubic nanoparticles without inner bores can be expressed as(2)Eanano=Eabulk-EmS=Eabulk-4σMρl,where σ, M, ρ, and l are the surface tension, molar mass, density, and particle size (length of cube edge) of the cubic nanoparticle reactant, respectively. Equation (2) provides that the apparent activation energy in the chemical reaction of the nanomaterials is proportional to the particle size of the reactant.If the heterogeneous reaction follows Arrhenius Law, substituting (2) into it yields the Arrhenius equation of the nanocube:(3)k=Aexp⁡-EabRT+4σMRTρl.Similarly, the Arrhenius equation of dodecahedron is obtained:(4)k=Aexp⁡-EabRT+6σMRTρl,where T is the reaction temperature, k is the reaction rate constant, and A is the preexponential factor.Substituting the logarithm on both sides of (4), we obtain the following:Cube:(5)ln⁡k=ln⁡A-EabRT+4σMρl.Dodecahedron:(6)ln⁡k=ln⁡A-EabRT+6σMρl.Therefore, when the particle size is larger than 10 nm, the surface tension slightly changes and can be viewed as a constant [43]. On the basis of (5) and (6), the logarithm of the reaction rate constant is inversely proportional to the particle size of the reactant.


## 2.5. Acquisition of Dynamic Parameters of Ag3PO4 Reacting with HNO3

The thermodynamic equation of reversible chemical reaction under constant temperature and pressure can be expressed as [44] (7)ln⁡1H∞dHidt=ln⁡k+nln⁡1-HiH∞,where H∞ is the enthalpy change during the whole reaction and may be directly obtained by microcalorimetry, dHi/dt is the enthalpy change rate, k (s−1) is the reaction rate constant expressed by conversion rate, and Hi is the enthalpy change at reaction time t. k can be calculated from the linear regression of thermodynamic data:(8)ln⁡k=ln⁡A-EaRT(9)ΔG≠θ=RTln⁡RTNAhk(10)ln⁡kT=ln⁡kBh+ΔS≠θR-ΔH≠θRT,where NA is Avogadro’s constant, kB is Boltzmann’s constant, h is Planck’s constant, and R is the molar gas constant. The diagram of 1/T was drawn with ln⁡k. Ea and A were calculated using (8). ΔG≠θ, ΔH≠θ, and ΔS≠θ were calculated from (9) and (10).


## 2.6. Theoretical Derivation of Molar Surface Gibbs Free Energy, Molar Surface Enthalpy, Molar Surface Entropy, and Molar Surface Energy

The molar Gibbs free energy of chemical reaction of nanosystem consists of bulk phase (ΔrGmB) and surface phase (ΔrGmS) [8, 42]:(11)ΔrGmnano=ΔrGmB+ΔrGmS.The molar Gibbs free energy of the bulk chemical reaction nearly exhibits bulk phase. The bulk phase of the nanosystem is similar to that of bulk:(12)ΔrGmB=ΔrGmbulk,where ΔrGmbulk is the molar Gibbs free energy of the same chemical reaction bulk material.Therefore, the molar Gibbs free energy difference between the nanosystem and the bulk lies in the molar surface Gibbs free energy of the nanomaterial (excessive Gibbs free energy compared with bulk material). Substituting (12) into (11), we obtain(13)ΔrGmS=ΔrGmnano-ΔrGmbulk.On the basis of (13), the thermochemical cycles of nano- and bulk Ag3PO4 were designed (Figure 2). The thermodynamic functions of nano- and bulk Ag3PO4 reactions with HNO3 were tested. The thermodynamic function of nano-Ag3PO4 conversion into the bulk one was calculated on the basis of their difference.Figure 2
Thermochemical cycle of nano- and bulk Ag3PO4 reacting with HNO3.On the basis of (12), the chemical reaction for the molar surface Gibbs of Ag3PO4 micro-/nanocrystals with a net reaction of Ag3PO4 (nano) → Ag3PO4 (bulk) is(14)ΔrGmθ=ΔrGmθAg3PO4,nano-ΔrGmθAg3PO4,bulk=ΔrGmSAg3PO4,nano=-GmSAg3PO4,nano,where ΔGmθAg3PO4,nano and ΔGmθAg3PO4,bulk are the standard molar Gibbs free energies of the reactions of Ag3PO4 micro-/nanocrystals and bulk Ag3PO4 with HNO3, respectively; ΔGmSAg3PO4,nano and GmSAg3PO4,nano are the molar reaction surface Gibbs free energy and molar surface Gibbs free energy, respectively.In accordance with transition state theory, the relationship between reaction rate constants of Ag3PO4 micro-/nanocrystals reaction system and bulk Ag3PO4 reaction system and the Gibbs free energy can be expressed as [8, 45](15)ΔrGmθAg3PO4,nano-ΔrGmθAg3PO4,bulk=ΔrG≠θbulk-ΔrG≠θnano=RTln⁡kbulk-ln⁡knano,where ΔrG≠θ and k are the activation Gibbs free energy and rate constant of Ag3PO4 chemical reaction.On the basis of (14) and (15), (16)GmSAg3PO4,nano=-ΔrGmSAg3PO4,nano=-RTln⁡kbulk-ln⁡knano.Similarly,ΔH≠θ can be calculated from (10). Molar surface enthalpy can be deduced from transition state theory: (17)ΔrHmS=ΔrHmθnano-ΔrHmθbulk=ΔrH≠θbulk-ΔrH≠θnanoΔrHmS=HmAg3PO4,nano-HmAg3PO4,bulk=-HmSAg3PO4,where ΔrHmS, ΔrHmθ, ΔrH≠θ, and HmS are the molar surface reaction enthalpy, molar reaction enthalpy, molar activation enthalpy, and molar surface enthalpy of Ag3PO4 chemical reaction, respectively.Similarly, molar surface entropy can be deduced from transition state theory:(18)ΔrSmS=ΔrSmθnano-ΔrSmθbulk=ΔrS≠θbulk-ΔrS≠θnanoΔrSmS=SmAg3PO4,nano-SmAg3PO4,bulk=-SmSAg3PO4,where ΔrSmS, ΔrSmθ, ΔrS≠θ, and SmS are the molar surface reaction entropy, molar reaction entropy, molar activation entropy, and molar surface entropy of Ag3PO4 chemical reaction, respectively.Apparent activation energy refers to the total energy needed for the material to overcome activation. The essential difference between Ag3PO4 micro-/nanocrystals and bulk Ag3PO4 is the high specific surface effect of the surface phase. After the same transition state to the final state in the same chemical reaction (Figure 1), the surface energy of the nanosystem surface phase (EmS) is the energy difference between the nano-reaction system and the bulk reaction system. The molar surface energy in the manuscript cited reference [8] which deduced in our published paper. Its correct form is [8]. Consider(19)ΔEmS=Enano-Ebulk=Eabulk-EananoΔrEmS=EmAg3PO4,nano-EmAg3PO4,bulk=-EmSAg3PO4,where ΔEmS, Ea, and Em are the molar surface reaction energy, activation energy, and molar surface energy of Ag3PO4 chemical reaction, respectively.


## 3. Results and Discussion

### 3.1. Product Characterization

Figures3(a)–3(c) show the SEM images of Ag3PO4 micro-/nanocrystals. Cubic Ag3PO4 has six (100) faces, clear and sharp edges and angles, and smooth surfaces; the mean particle size is (695.9±100.2) nm. Figure 3(b) is the rhombic dodecahedral Ag3PO4 with 12 rhombus (110) faces. White spots are scattered on the surfaces; the mean particle size is (647.1±91.8) nm. Figure 3(c) shows the SEM image of the irregularly shaped Ag3PO4 that forms the bulk of the substance; its mean particle size is (6.9 ± 3.9) μm.Figure 3
SEM images of cubic (a), rhombic dodecahedral (b), and bulk (c) Ag3PO4 micro-/nanocrystals.
(a)
(b)
(c)Figure4 shows the XRD pattern of the bulk, rhombic dodecahedral, and cubic Ag3PO4. All diffraction peaks in Figure 4 agree with those of Ag3PO4 with the standard calorie JPCDS (06-0505). No other impurity peak is observed. Moreover, the full width at half maximum of all diffraction peaks is narrow, indicating purity and good crystallinity of the prepared samples.Figure 4
XRD patterns of bulk (a), rhombic dodecahedral (b), and cubic (c) Ag3PO4 micro-/nanocrystals.


### 3.2. Effect of Morphology on the Chemical Reaction Rate Constant of Ag3PO4 Micro-/Nanocrystals

Linear regression was performed using the original thermodynamics data obtained from (7), and the reaction rate constants of Ag3PO4 and HNO3 under varying temperatures were obtained (Table 1).Table 1
Reaction rate constant of Ag3PO4 micro-/nanocrystals with HNO3.
T (K)
298.15
303.15
308.15
313.15
318.15
k
⋅
10
-
3  (s−1)
Cube
3.37
3.53
3.77
3.99
4.17
Rhombic dodecahedra
3.25
3.44
3.64
3.86
4.08
Bulk
2.95
3.25
3.48
3.71
3.98The curves shown in Figure5 were drawn on the basis of the reaction rate constants of Ag3PO4 with HNO3 and the reciprocal of temperature in Table 1.Figure 5
Effect of morphology on the reaction rate constant of Ag3PO4 micro-/nanocrystals with HNO3.Figure5 shows that the reaction rate is proportional to temperature when the particle size is fixed. Compared with bulk Ag3PO4, the super-refined materials have significantly more particles of the surface phase. Particles of the surface phase account for a large proportion of the total particles. Atoms of the surface phase have uneven stresses, unsaturated force field, and dangling bonds, which lead to high surface energy. This result explains the faster reaction rate of Ag3PO4 micro-/nanocrystals than bulk Ag3PO4. As the temperature increases, the surface turbulence and surface energy of Ag3PO4 micro-/nanocrystals increase; consequently, the chemical reaction increases. The effect of morphology on the reaction rate shows that cubic Ag3PO4 has higher reaction rate than dodecahedral Ag3PO4.


### 3.3. Effect of Morphology on the Activation Energy, Activation Gibbs Free Energy, Activation Enthalpy, and Activation Entropy of Ag3PO4 Micro-/Nanocrystals

Linear regression of logarithmic reaction rate constant and temperature reciprocal (slope and intercept in Figure4) was performed using (8), and the Ea of the Ag3PO4 micro-/nanocrystals reaction was obtained. The ΔG≠θ of the Ag3PO4 micro-/nanocrystals reaction was calculated using (9). The ΔH≠θ and ΔS≠θ of the Ag3PO4 micro-/nanocrystals reaction were calculated using (10). Results are listed in Table 2.Table 2
Activation energy, activation Gibbs free energy, activation enthalpy, and activation entropy of Ag3PO4 micro-/nanocrystals.
Ag3PO4  Micro-/nanocrystals
T (K)
Δ
G
≠
θ (kJ⋅mol−1)
E
a (kJ⋅mol−1)
Δ
H
≠
θ  (kJ⋅mol−1)
Δ
S
≠
θ  (J⋅K−1⋅mol−1)
Cube
298.15
87.132
8.653
6.093
−271.831
303.15
88.518
308.15
89.852
313.15
91.204
318.15
92.585
Rhombic dodecahedra
298.15
87.222
8.990
6.430
−270.987
303.15
88.583
308.15
89.941
313.15
91.290
318.15
92.643
Bulk
298.15
87.462
11.548
8.988
−263.123
303.15
88.726
308.15
90.056
313.15
91.393
318.15
92.708As shown in Table1, the activation energy, activation Gibbs free energy (as shown in Figure 6), activation enthalpy, and activation entropy of Ag3PO4 micro-/nanocrystals decrease with decreasing particle size. Surface atoms are in metastable state because of the high specific surface effect of micro-/nanomaterials. The nanosystem has higher potential energy than the bulk system because of high specific surface effect. Transition state theory states that the nanosystem has to overcome smaller energy barrier than the bulk system to reach the same transition state. Therefore, smaller particles require less activation energy. The effect of morphology on Ea, ΔG≠θ, ΔH≠θ, and ΔS≠θ demonstrates that cubic Ag3PO4 overcomes smaller energy barrier than dodecahedral Ag3PO4 in the same reaction. Thus, cubic Ag3PO4 requires less activation energy than dodecahedral Ag3PO4.Figure 6
Morphology effect on the activation Gibbs free energy of Ag3PO4 micro-/nanocrystals.


### 3.4. Effect of Morphology on the Surface Gibbs Free Energy of Ag3PO4 Micro-/Nanocrystals

Combining (14)–(16), we calculated the surface Gibbs free energy of Ag3PO4 micro-/nanocrystals on the basis of the activation Gibbs free energy listed in Table 3.Table 3
Surface Gibbs free energy of Ag3PO4 micro-/nanocrystals.
T (K)
298.15
303.15
308.15
313.15
318.15
G
S (kJ⋅mol−1)
Cube
0.330
0.208
0.205
0.189
0.123
Rhombic dodecahedra
0.240
0.143
0.115
0.103
0.066The molar surface Gibbs free energy of Ag3PO4 micro-/nanocrystals (GmS) under different temperatures is shown in Figure 7. Cubic Ag3PO4 has higher GmS than rhombic dodecahedral Ag3PO4. Both conditions are inversely proportional to temperature. The uneven stress on the atoms of the surface phase intensifies with increasing reaction temperature. The thermal motion of nanoparticles also intensifies with the increase in unsaturated force field and dangling bonds. The widening particle space weakens their interaction and decreases the surface tension of nanomaterials, thereby reducing the surface Gibbs free energy.Figure 7
Morphology effect on surface Gibbs free energy of Ag3PO4 micro-/nanocrystals.


### 3.5. Effect of Morphology on the Molar Surface Enthalpy, Molar Surface Entropy, and Molar Surface Energy of Ag3PO4 Micro-/Nanocrystals

TheHmS, SmS, and EmS of Ag3PO4 micro-/nanocrystals were calculated using (17), (18), and (19), respectively.Table4 shows that cubic Ag3PO4 micro-/nanocrystals have higher HmS, SmS, and EmS than rhombic dodecahedral Ag3PO4. This result agrees with that on surface Gibbs free energy. Molar surface energy is the sum of the kinetic energy, potential energy, and chemical energy of surface phase particles. After super-refinement of the material, atoms of the surface phase suffer from uneven stress, display unsaturated force field, and possess dangling bonds because of strong specific surface effect. Consequently, the interaction of nanoparticles is enhanced. This phenomenon explains why micro-/nanomaterials have high kinetic energy and potential energy.Table 4
Morphology effect on the surface enthalpy, surface entropy, and surface energy of Ag3PO4 micro-/nanocrystals.
Surface energies
Ag3PO4 micro-/nanocrystals
Cube
Rhombic dodecahedron
H
m
S (kJ⋅mol−1)
2.895
2.558
S
m
S (kJ⋅mol−1)
8.708
7.864
E
m
S (kJ⋅mol−1)
2.895
2.558


## 3.1. Product Characterization

Figures3(a)–3(c) show the SEM images of Ag3PO4 micro-/nanocrystals. Cubic Ag3PO4 has six (100) faces, clear and sharp edges and angles, and smooth surfaces; the mean particle size is (695.9±100.2) nm. Figure 3(b) is the rhombic dodecahedral Ag3PO4 with 12 rhombus (110) faces. White spots are scattered on the surfaces; the mean particle size is (647.1±91.8) nm. Figure 3(c) shows the SEM image of the irregularly shaped Ag3PO4 that forms the bulk of the substance; its mean particle size is (6.9 ± 3.9) μm.Figure 3
SEM images of cubic (a), rhombic dodecahedral (b), and bulk (c) Ag3PO4 micro-/nanocrystals.
(a)
(b)
(c)Figure4 shows the XRD pattern of the bulk, rhombic dodecahedral, and cubic Ag3PO4. All diffraction peaks in Figure 4 agree with those of Ag3PO4 with the standard calorie JPCDS (06-0505). No other impurity peak is observed. Moreover, the full width at half maximum of all diffraction peaks is narrow, indicating purity and good crystallinity of the prepared samples.Figure 4
XRD patterns of bulk (a), rhombic dodecahedral (b), and cubic (c) Ag3PO4 micro-/nanocrystals.


## 3.2. Effect of Morphology on the Chemical Reaction Rate Constant of Ag3PO4 Micro-/Nanocrystals

Linear regression was performed using the original thermodynamics data obtained from (7), and the reaction rate constants of Ag3PO4 and HNO3 under varying temperatures were obtained (Table 1).Table 1
Reaction rate constant of Ag3PO4 micro-/nanocrystals with HNO3.
T (K)
298.15
303.15
308.15
313.15
318.15
k
⋅
10
-
3  (s−1)
Cube
3.37
3.53
3.77
3.99
4.17
Rhombic dodecahedra
3.25
3.44
3.64
3.86
4.08
Bulk
2.95
3.25
3.48
3.71
3.98The curves shown in Figure5 were drawn on the basis of the reaction rate constants of Ag3PO4 with HNO3 and the reciprocal of temperature in Table 1.Figure 5
Effect of morphology on the reaction rate constant of Ag3PO4 micro-/nanocrystals with HNO3.Figure5 shows that the reaction rate is proportional to temperature when the particle size is fixed. Compared with bulk Ag3PO4, the super-refined materials have significantly more particles of the surface phase. Particles of the surface phase account for a large proportion of the total particles. Atoms of the surface phase have uneven stresses, unsaturated force field, and dangling bonds, which lead to high surface energy. This result explains the faster reaction rate of Ag3PO4 micro-/nanocrystals than bulk Ag3PO4. As the temperature increases, the surface turbulence and surface energy of Ag3PO4 micro-/nanocrystals increase; consequently, the chemical reaction increases. The effect of morphology on the reaction rate shows that cubic Ag3PO4 has higher reaction rate than dodecahedral Ag3PO4.


## 3.3. Effect of Morphology on the Activation Energy, Activation Gibbs Free Energy, Activation Enthalpy, and Activation Entropy of Ag3PO4 Micro-/Nanocrystals

Linear regression of logarithmic reaction rate constant and temperature reciprocal (slope and intercept in Figure4) was performed using (8), and the Ea of the Ag3PO4 micro-/nanocrystals reaction was obtained. The ΔG≠θ of the Ag3PO4 micro-/nanocrystals reaction was calculated using (9). The ΔH≠θ and ΔS≠θ of the Ag3PO4 micro-/nanocrystals reaction were calculated using (10). Results are listed in Table 2.Table 2
Activation energy, activation Gibbs free energy, activation enthalpy, and activation entropy of Ag3PO4 micro-/nanocrystals.
Ag3PO4  Micro-/nanocrystals
T (K)
Δ
G
≠
θ (kJ⋅mol−1)
E
a (kJ⋅mol−1)
Δ
H
≠
θ  (kJ⋅mol−1)
Δ
S
≠
θ  (J⋅K−1⋅mol−1)
Cube
298.15
87.132
8.653
6.093
−271.831
303.15
88.518
308.15
89.852
313.15
91.204
318.15
92.585
Rhombic dodecahedra
298.15
87.222
8.990
6.430
−270.987
303.15
88.583
308.15
89.941
313.15
91.290
318.15
92.643
Bulk
298.15
87.462
11.548
8.988
−263.123
303.15
88.726
308.15
90.056
313.15
91.393
318.15
92.708As shown in Table1, the activation energy, activation Gibbs free energy (as shown in Figure 6), activation enthalpy, and activation entropy of Ag3PO4 micro-/nanocrystals decrease with decreasing particle size. Surface atoms are in metastable state because of the high specific surface effect of micro-/nanomaterials. The nanosystem has higher potential energy than the bulk system because of high specific surface effect. Transition state theory states that the nanosystem has to overcome smaller energy barrier than the bulk system to reach the same transition state. Therefore, smaller particles require less activation energy. The effect of morphology on Ea, ΔG≠θ, ΔH≠θ, and ΔS≠θ demonstrates that cubic Ag3PO4 overcomes smaller energy barrier than dodecahedral Ag3PO4 in the same reaction. Thus, cubic Ag3PO4 requires less activation energy than dodecahedral Ag3PO4.Figure 6
Morphology effect on the activation Gibbs free energy of Ag3PO4 micro-/nanocrystals.


## 3.4. Effect of Morphology on the Surface Gibbs Free Energy of Ag3PO4 Micro-/Nanocrystals

Combining (14)–(16), we calculated the surface Gibbs free energy of Ag3PO4 micro-/nanocrystals on the basis of the activation Gibbs free energy listed in Table 3.Table 3
Surface Gibbs free energy of Ag3PO4 micro-/nanocrystals.
T (K)
298.15
303.15
308.15
313.15
318.15
G
S (kJ⋅mol−1)
Cube
0.330
0.208
0.205
0.189
0.123
Rhombic dodecahedra
0.240
0.143
0.115
0.103
0.066The molar surface Gibbs free energy of Ag3PO4 micro-/nanocrystals (GmS) under different temperatures is shown in Figure 7. Cubic Ag3PO4 has higher GmS than rhombic dodecahedral Ag3PO4. Both conditions are inversely proportional to temperature. The uneven stress on the atoms of the surface phase intensifies with increasing reaction temperature. The thermal motion of nanoparticles also intensifies with the increase in unsaturated force field and dangling bonds. The widening particle space weakens their interaction and decreases the surface tension of nanomaterials, thereby reducing the surface Gibbs free energy.Figure 7
Morphology effect on surface Gibbs free energy of Ag3PO4 micro-/nanocrystals.


## 3.5. Effect of Morphology on the Molar Surface Enthalpy, Molar Surface Entropy, and Molar Surface Energy of Ag3PO4 Micro-/Nanocrystals

TheHmS, SmS, and EmS of Ag3PO4 micro-/nanocrystals were calculated using (17), (18), and (19), respectively.Table4 shows that cubic Ag3PO4 micro-/nanocrystals have higher HmS, SmS, and EmS than rhombic dodecahedral Ag3PO4. This result agrees with that on surface Gibbs free energy. Molar surface energy is the sum of the kinetic energy, potential energy, and chemical energy of surface phase particles. After super-refinement of the material, atoms of the surface phase suffer from uneven stress, display unsaturated force field, and possess dangling bonds because of strong specific surface effect. Consequently, the interaction of nanoparticles is enhanced. This phenomenon explains why micro-/nanomaterials have high kinetic energy and potential energy.Table 4
Morphology effect on the surface enthalpy, surface entropy, and surface energy of Ag3PO4 micro-/nanocrystals.
Surface energies
Ag3PO4 micro-/nanocrystals
Cube
Rhombic dodecahedron
H
m
S (kJ⋅mol−1)
2.895
2.558
S
m
S (kJ⋅mol−1)
8.708
7.864
E
m
S (kJ⋅mol−1)
2.895
2.558


## 4. Conclusion

This study determined the surface energies of Ag3PO4 micro-/nanocrystals and their kinetic parameters when reacting with HNO3 by using microcalorimetry. It also discussed rationally combined thermochemical cycle, transition state theory, basic theory of chemical thermodynamics with thermokinetic principle, morphology dependence of reaction kinetics, and surface thermodynamic properties. Results show that the molar surface enthalpy, molar surface entropy, molar surface Gibbs free energy, and molar surface energy of cubic Ag3PO4 micro-/nanocrystals are larger than those of rhombic dodecahedral Ag3PO4 micro-/nanocrystals. Compared with rhombic dodecahedral Ag3PO4, cubic Ag3PO4 with high surface energy exhibits higher reaction rate and lower activation energy, activation Gibbs free energy, activation enthalpy, and activation entropy. These results indicate that cubic Ag3PO4 micro-/nanocrystals possess a much higher reactivity and it is more easily activated than rhombic dodecahedral Ag3PO4 micro-/nanocrystals. This paper presents a novel facile approach to study the surface thermodynamic property of nanomaterials and the structure-function relationship between reaction dynamics and size, morphology, and structure.



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*Source: 743121-2015-11-19.xml*