# Fluctuation Theory in Chemical Kinetics

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## Abstract

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## 1. Introduction

## 2. Reaction Mechanisms and Kinetic Rates

#### 2.1. Chemical Equilibrium

#### 2.2. Fluctuation Theory

## 3. Chemical Equilibrium Formation

#### 3.1. Flow Components

#### 3.2. Concentration Capacity

#### 3.3. Concentration Correlation

#### 3.4. Equilibrium Stability

#### 3.4.1. Local Stability

#### 3.4.2. Global Stability

## 4. Discussion of the Results

#### 4.1. Chemical Reactions with Orders $(-1,-1)$

#### 4.2. Chemical Reactions with Orders $(-1,\frac{1}{2})$

#### 4.3. Chemical Reactions with Orders $(-1,2)$

#### 4.4. Chemical Reactions with Orders $(\frac{1}{2},\frac{1}{2})$

#### 4.5. Chemical Reactions with Orders $(\frac{1}{2},2)$

#### 4.6. Chemical Reactions with Orders $(2,2)$

## 5. Verification of the Model

#### 5.1. An Inorganic Equilibrium Formation

#### 5.2. An Organic Equilibrium Formation

#### 5.3. A Catalytic Oxidation of $CO-{H}_{2}$ Mixtures

## 6. Future Scope of the Work

#### 6.1. Proton Donor-Acceptor Equilibrium

#### 6.2. Fall of the Proton

#### 6.3. Buffer Capacity

#### 6.4. Acid Raining

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The fluctuation quantities as a function of the chemical concentrations $\{{c}_{1},{c}_{2}\}$ for $m=-1$ and $n=-1$ plotted in the interval $-50\le {c}_{1},{c}_{2}\le 50$ against (

**a**) the rate $r({c}_{1},{c}_{2})$, (

**b**) flow ${r}_{1}({c}_{1},{c}_{2})$, (

**c**) flow ${r}_{2}({c}_{1},{c}_{2})$, (

**d**) capacity ${r}_{11}({c}_{1},{c}_{2})$, (

**e**) correlation ${r}_{12}({c}_{1},{c}_{2})$, (

**f**) capacity ${r}_{22}({c}_{1},{c}_{2})$, (

**g**) trace $tr\left(H\right)$ and (

**h**) determinant $\Delta ({c}_{1},{c}_{2})$. (Note: with ${c}_{1}$, ${c}_{2}$ measured in mol/L on X and Y axis respectively, the other quantities on Z-axis are measured in the units as: $r\to $ mol/L/s; ${r}_{1},{r}_{2}\to $ s${}^{-1}$; ${r}_{11},{r}_{12},{r}_{22},tr\left(H\right)\to $ mol${}^{-1}$ L s${}^{-1}$; $\Delta \to $ mol${}^{-2}$ L${}^{2}$ s${}^{-2}$).

**Figure 2.**The fluctuation quantities as a function of the chemical concentrations $\{{c}_{1},{c}_{2}\}$ for $m=-1$ and $n=\frac{1}{2}$ plotted in the interval $-50\le {c}_{1},{c}_{2}\le 50$ against (

**a**) the rate $r({c}_{1},{c}_{2})$, (

**b**) flow ${r}_{1}({c}_{1},{c}_{2})$, (

**c**) flow ${r}_{2}({c}_{1},{c}_{2})$, (

**d**) capacity ${r}_{11}({c}_{1},{c}_{2})$, (

**e**) correlation ${r}_{12}({c}_{1},{c}_{2})$, (

**f**) capacity ${r}_{22}({c}_{1},{c}_{2})$, (

**g**) trace $tr\left(H\right)$ and (

**h**) determinant $\Delta ({c}_{1},{c}_{2})$. (Note: with ${c}_{1}$, ${c}_{2}$ measured in mol/L on X and Y axis respectively, the other quantities on Z-axis are measured in the units as: $r\to $ mol/L/s; ${r}_{1},{r}_{2}\to $ s${}^{-1}$; ${r}_{11},{r}_{12},{r}_{22},tr\left(H\right)\to $ mol${}^{-1}$ L s${}^{-1}$; $\Delta \to $ mol${}^{-2}$ L${}^{2}$ s${}^{-2}$).

**Figure 3.**The fluctuation quantities as a function of the chemical concentrations $\{{c}_{1},{c}_{2}\}$ for $m=-1$ and $n=2$ plotted in the interval $-50<{c}_{1},{c}_{2}<50$ against (

**a**) the rate $r({c}_{1},{c}_{2})$, (

**b**) flow ${r}_{1}({c}_{1},{c}_{2})$, (

**c**) flow ${r}_{2}({c}_{1},{c}_{2})$, (

**d**) capacity ${r}_{11}({c}_{1},{c}_{2})$, (

**e**) correlation ${r}_{12}({c}_{1},{c}_{2})$, (

**f**) capacity ${r}_{22}({c}_{1},{c}_{2})$ and (

**g**) trace $tr\left(H\right)$. (Note: with ${c}_{1}$, ${c}_{2}$ measured in mol/L on X and Y axis respectively, the other quantities on Z-axis are measured in the units as: $r\to $ mol/L/s; ${r}_{1},{r}_{2}\to $ s${}^{-1}$; ${r}_{11},{r}_{12},{r}_{22},tr\left(H\right)\to $ mol${}^{-1}$ L s${}^{-1}$; $\Delta \phantom{\rule{3.33333pt}{0ex}}\to $ mol${}^{-2}$ L${}^{2}$ s${}^{-2}$).

**Figure 4.**The fluctuation quantities as a function of the chemical concentrations $\{{c}_{1},{c}_{2}\}$ for $m=\frac{1}{2}$ and $n=\frac{1}{2}$ plotted in the interval $-50<{c}_{1},{c}_{2}<50$ against (

**a**) the rate $r({c}_{1},{c}_{2})$, (

**b**) flow ${r}_{1}({c}_{1},{c}_{2})$, (

**c**) flow ${r}_{2}({c}_{1},{c}_{2})$, (

**d**) capacity ${r}_{11}({c}_{1},{c}_{2})$, (

**e**) correlation ${r}_{12}({c}_{1},{c}_{2})$, (

**f**) capacity ${r}_{22}({c}_{1},{c}_{2})$, and (

**g**) trace $tr\left(H\right)$. (Note: with ${c}_{1}$, ${c}_{2}$ measured in mol/L on X and Y axis respectively, the other quantities on Z-axis are measured in the units as: $r\to $ mol/L/s; ${r}_{1},{r}_{2}\to $ s${}^{-1}$; ${r}_{11},{r}_{12},{r}_{22},tr\left(H\right)\to $ mol${}^{-1}$ L s${}^{-1}$; $\Delta \phantom{\rule{3.33333pt}{0ex}}\to $ mol${}^{-2}$ L${}^{2}$ s${}^{-2}$).

**Figure 5.**The fluctuation quantities as a function of the chemical concentrations $\{{c}_{1},{c}_{2}\}$ for $m=\frac{1}{2}$ and $n=2$ plotted in the interval $-50<{c}_{1},{c}_{2}<50$ against (

**a**) the rate $r({c}_{1},{c}_{2})$, (

**b**) flow ${r}_{1}({c}_{1},{c}_{2})$, (

**c**) flow ${r}_{2}({c}_{1},{c}_{2})$, (

**d**) capacity ${r}_{11}({c}_{1},{c}_{2})$, (

**e**) correlation ${r}_{12}({c}_{1},{c}_{2})$, (

**f**) capacity ${r}_{22}({c}_{1},{c}_{2})$, (

**g**) trace $tr\left(H\right)$ and (

**h**) determinant $\Delta ({c}_{1},{c}_{2})$. (Note: with ${c}_{1}$, ${c}_{2}$ measured in mol/L on X and Y axis respectively, the other quantities on Z-axis are measured in the units as: $r\to $ mol/L/s; ${r}_{1},{r}_{2}\to $ s${}^{-1}$; ${r}_{11},{r}_{12},{r}_{22},tr\left(H\right)\to $ mol${}^{-1}$ L s${}^{-1}$; $\Delta \to $ mol${}^{-2}$ L${}^{2}$ s${}^{-2}$).

**Figure 6.**The fluctuation quantities as a function of the chemical concentrations $\{{c}_{1},{c}_{2}\}$ for $m=2$ and $n=2$ plotted in the interval $-50<{c}_{1},{c}_{2}<50$ against (

**a**) the rate $r({c}_{1},{c}_{2})$, (

**b**) flow ${r}_{1}({c}_{1},{c}_{2})$, (

**c**) flow ${r}_{2}({c}_{1},{c}_{2})$, (

**d**) capacity ${r}_{11}({c}_{1},{c}_{2})$ (remains the same for the capacity ${r}_{22}({c}_{1},{c}_{2})$ with the replacement of ${c}_{1}$ by ${c}_{2}$, (

**e**) correlation ${r}_{12}({c}_{1},{c}_{2})$, (

**f**) trace $tr\left(H\right)$ and (

**g**) determinant $\Delta ({c}_{1},{c}_{2})$. (Note: with ${c}_{1}$, ${c}_{2}$ measured in mol/L on X and Y axis respectively, the other quantities on Z-axis are measured in the units as: $r\to $ mol/L/s; ${r}_{1},{r}_{2}\to $ s${}^{-1}$; ${r}_{11},{r}_{12},{r}_{22},tr\left(H\right)\to $ mol${}^{-1}$ L s${}^{-1}$; $\Delta \to $ mol${}^{-2}$ L${}^{2}$ s${}^{-2}$).

**Table 1.**A comparative analysis of the stability of the equilibrium as a function of the rate r with respect to the orders m and n of the reactants.

SN | m | n | $\mathbb{O}$ = m + n | Δ | ${\mathit{r}}_{11}$ |
---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 0 |

2 | 1 | 0 | 1 | 0 | 0 |

3 | 0 | 1 | 0 | 0 | 0 |

4 | 1 | 1 | 2 | - | - |

5 | 1 | 2 | 3 | - | - |

6 | 2 | 1 | 3 | - | - |

7 | 2 | 2 | 4 | - | - |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

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**MDPI and ACS Style**

Tiwari, B.N.; Kishore, S.C.; Marina, N.; Bellucci, S. Fluctuation Theory in Chemical Kinetics. *Condens. Matter* **2018**, *3*, 49.
https://doi.org/10.3390/condmat3040049

**AMA Style**

Tiwari BN, Kishore SC, Marina N, Bellucci S. Fluctuation Theory in Chemical Kinetics. *Condensed Matter*. 2018; 3(4):49.
https://doi.org/10.3390/condmat3040049

**Chicago/Turabian Style**

Tiwari, Bhupendra Nath, S. Chandra Kishore, Ninoslav Marina, and Stefano Bellucci. 2018. "Fluctuation Theory in Chemical Kinetics" *Condensed Matter* 3, no. 4: 49.
https://doi.org/10.3390/condmat3040049