The Extension of Statistical Entropy Analysis to Chemical Compounds
Abstract
:1. Introduction
2. Description of the Investigated System and Data
3. Extension of the Statistical Entropy Analysis (eSEA)
3.1. Estimation of the Statistical Entropy of the Input (H_{IN})
SEA  eSEA 

$${\mathrm{H}}_{\mathrm{I}\mathrm{N},\mathrm{N}}({\mathrm{m}}_{\mathrm{i}},{\mathrm{c}}_{\mathrm{i}\mathrm{N}})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{m}}_{\mathrm{i}}*{\mathrm{c}}_{\mathrm{i}\mathrm{N}}*{\mathrm{log}}_{2}({\mathrm{c}}_{\mathrm{i}\mathrm{N}})}$$

$${\mathrm{H}}_{\mathrm{I}\mathrm{N}}({\mathrm{m}}_{\mathrm{i}},{\mathrm{c}}_{\mathrm{i}\mathrm{m}})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{k}}{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{r}}{\mathrm{m}}_{\mathrm{i}}*{\mathrm{c}}_{\mathrm{i}\mathrm{m}}*{\mathrm{log}}_{2}({\mathrm{c}}_{\mathrm{i}\mathrm{m}})}}$$

Entropy is a function of the total nitrogen load (N) in the different input flows  Entropy is a function of the load of all nitrogen compounds (m) that appear in the different input flows 
3.2. Estimation of the Statistical Entropy of the Output (H_{OUT})
SEA  eSEA 

$${\mathrm{H}}_{\text{OUT},\mathrm{N}}=({\mathrm{m}}_{\mathrm{i}}^{\prime},{\mathrm{c}}_{\mathrm{i}\mathrm{N}}^{\prime})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{l}}{\mathrm{m}}_{\mathrm{i}}^{\prime}*{\mathrm{c}}_{\mathrm{i}\mathrm{N}}^{\prime}*{\mathrm{log}}_{2}({\mathrm{c}}_{\mathrm{i}\mathrm{N}}^{\prime})}$$

$${\mathrm{H}}_{\mathrm{O}\mathrm{U}\mathrm{T}}({\mathrm{m}}_{\mathrm{i}\mathrm{m}}^{\prime},{\mathrm{c}}_{\mathrm{i}\mathrm{m}}^{\prime})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{l}}{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{s}}{\mathrm{m}}_{\mathrm{i}\mathrm{m}}^{\prime}*{\mathrm{c}}_{\mathrm{i}\mathrm{m}}^{\prime}*{\mathrm{log}}_{2}({\mathrm{c}}_{\mathrm{i}\mathrm{m}}^{\prime})}}$$

Entropy for the output depends on total nitrogen loads (N) in the different output flows  Entropy depends on the load of all nitrogen compounds (m) that appear in the different output flows 
CALCULATION OF THE DILUTING TERMS  
$${\mathrm{m}}_{\mathrm{i}}^{\prime}={\mathrm{m}}_{\mathrm{i}}*\frac{{\mathrm{c}}_{\mathrm{i}\mathrm{N}}{\mathrm{c}}_{\mathrm{i}\mathrm{N},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}{{\mathrm{c}}_{\mathrm{i}\mathrm{N},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}*100+{\mathrm{m}}_{\mathrm{i}}$$

$${\mathrm{m}}_{\mathrm{i}\mathrm{m}}^{\prime}={\mathrm{m}}_{\mathrm{i}}*\frac{{\mathrm{c}}_{\mathrm{i}\mathrm{m}}{\mathrm{c}}_{\mathrm{i}\mathrm{m},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}{{\mathrm{c}}_{\mathrm{i}\mathrm{m},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}*100+{\mathrm{m}}_{\mathrm{i}}$$

$${\mathrm{c}}_{\mathrm{i}\mathrm{N}}^{\prime}=\frac{{\mathrm{c}}_{\mathrm{i}\mathrm{N}}*{\mathrm{c}}_{\mathrm{i}\mathrm{N},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}{{\mathrm{c}}_{\mathrm{i}\mathrm{N}}0.99*{\mathrm{c}}_{\mathrm{i}\mathrm{N},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}*0.01$$

$${\mathrm{c}}_{\mathrm{i}\mathrm{m}}^{\prime}=\frac{{\mathrm{c}}_{\mathrm{i}\mathrm{m}}*{\mathrm{c}}_{\mathrm{i}\mathrm{m},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}{{\mathrm{c}}_{\mathrm{i}\mathrm{m}}0.99*{\mathrm{c}}_{\mathrm{i}\mathrm{m},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}}}*0.01$$

3.3. Estimation of the Maximum Statistical Entropy of the Output (H_{max})
SEA  eSEA 

$${\mathrm{H}}_{\mathrm{max}}={\mathrm{log}}_{2}\left({\mathrm{m}}_{\mathrm{i},\mathrm{max}}^{\prime}\right)={\mathrm{log}}_{2}\left(\frac{1}{\mathrm{min}({\mathrm{c}}_{\mathrm{N},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}})}*100\right)$$

$${\mathrm{H}}_{\mathrm{max}}={\mathrm{log}}_{2}\left({\mathrm{m}}_{\mathrm{i}\mathrm{m},\mathrm{max}}^{\prime}\right)={\mathrm{log}}_{2}\left(\frac{1}{\mathrm{min}({\mathrm{c}}_{\mathrm{m},\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{g}})}*100\right)$$

With min(c_{N,geog})=1E05 kgN/kg ^{(1)}  With min(c_{m,geog})_{.}=7E11 kgN/kg ^{(2)} 
3.4. Estimation of the Concentrating Power/Diluting Extent, ΔH
3.5. SEA vs. eSEA
4. Application of the eSEA: A Numerical Example
(9)  (10)  (11)  (12,13)  
${\dot{\mathrm{M}}}_{\text{i}}$  c_{im}  c_{im,geog}  ${\dot{\mathrm{X}}}_{\text{im}}$  m_{i}  X_{im}  ∑X_{im}  
kg/ha/yr  kgN/kg  kgN/kg  kgN/a  kg/kgN  kgN/kgN  kgN/kgN  
INPUT  Fertilizer  211  4.74  1  
NH_{4}NO_{3}  0.13  28.20  0.63  
CO(NH_{2})_{2}  0.01  1.80  0.04  
Seeds  25  0.56  
N_{org}  0.02  0.50  0.01  
Deposition  571  12.84  
NO_{3}^{−}  0.01  4.80  0.11  
NH_{4}^{+}  0.01  3.20  0.07  
Nfixation  8  0.76  6.00  0.18  0.13  
OUTPUT  Product  2300  51.68  1  
N_{org}  0.01  23.00  0.52  
Offgas  11  0.24  
N_{2}  0.7553  0.7553  8.19  0.18  
NH_{3}  0.16  4.1 E9  1.75  0.04  
N_{2}O  0.14  2.0 E8  1.50  0.03  
NO_{x}  0.02  2.5 E10  0.20  4.5 E03  
Surface water  42  0.94  
NO_{3}^{−}  2.0 E02  3.0 E06  0.85  0.02  
NH_{4}^{+}  2.4 E03  1.2 E06  0.10  2.3 E03  
N_{org}  1.2 E03  1.0 E10  0.05  1.1 E03  
Groundwater  373  8.37  
NO_{3}^{−}  2.0 E02  1.0 E05  7.52  0.17  
NH_{4}^{+}  2.4 E03  3.5 E07  0.90  0.02  
N_{org}  1.2 E03  7.0 E11  0.45  0.01 
(19)  (21)  (15), (17)  (23)  (24)  
m'_{im}  c'_{im}  H_{IN} / H_{OUT}  H_{IN,rel} / H_{OUT,rel}  H_{max}  ΔH  
[]  []  []  []  []  [%]  
INPUT  Fertilizer  3.52  0.09  40.38  229  
NH_{4}NO_{3}  
CO(NH_{2})_{2}  
Seeds  
N_{org}  
Deposition  
NO_{3}^{}  
NH_{4}^{+}  
Nfixation  
OUTPUT  Product  11.60  0.29  
N_{org}  
Offgas  
N_{2}  2.4 E01  7.553 E01  
NH_{3}  9.6 E+08  4.1 E11  
N_{2}O  1.7 E+08  2.0 E10  
NO_{x}  1.8 E+08  2.0 E12  
Surface water  
NO_{3}^{−}  6.4 E+05  3.0 E08  
NH_{4}^{+}  1.9 E+05  1.2 E08  
N_{org}  1.1 E+09  1.0 E12  
Groundwater  
NO_{3}^{−}  1.7 E+06  1.0 E07  
NH_{4}^{+}  5.8 E+06  3.5 E09  
N_{org}  1.4 E+10  7.0 E13 
5. Conclusions
Acknowledgments
List of all indices and parameters
H  Statistical entropy according to Shannon’s definition 
ΔH  Change in statistical entropy representing the concentrating power (ΔH < 0) or the diluting extent (ΔH > 0) for nitrogen 
H_{max}  Maximum occurring statistical entropy resulting from a theoretical worst case crop farming scenario 
H_{IN}  Statistical entropy for all incoming nitrogen compounds in the region 
H_{IN,rel}  Relative value of statistical entropy (H_{IN} / H_{max}) for all incoming nitrogen compounds in the region 
H_{OUT}  Statistical entropy for all nitrogen compounds that leave the region 
H_{OUT,rel}  Relative value for statistical entropy (H_{OUT} / H_{max}) for all nitrogen compounds that leave the region 
P_{i}  Probability of an event i 
${\dot{\mathrm{M}}}_{\text{i}}$  Mass flow, e.g., rainwater in kg/ha/A 
i  Index for mass flows, e.g., rainwater 
k  Number of all input material flows i 
c_{i}  Concentration of a substance i, e.g. kg cadmium per kg massflow 
c_{im}  Nitrogen concentration of nitrogen compound m in material flow i, e.g., kg NNH_{4}^{+} per kg groundwater 
m  Index for nitrogen compounds, e.g., NO_{3}^{}, NH_{4}^{+}, N_{2} 
r  Number of different nitrogen compounds m in the input 
${\dot{\mathrm{X}}}_{\text{im}}$  Nitrogen load for nitrogen compound m in material flow i, e.g., kg NNH_{4}^{+} in kg groundwater per hectare per year 
${\text{m}}_{\text{i}}$  Specific mass for material flow i, e.g., kg groundwater per kg total nitrogen throughput of the process 
${\text{X}}_{\text{im}}$  Specific nitrogen load, e.g., kg NNH_{4}^{+} per kg total N of the output 
s  Number of different nitrogen compounds m in the output 
$\text{l}$  Number of all outgoing material flows 
${{\mathrm{m}}^{\prime}}_{\text{im}}$  Diluting mass from material flow i for nitrogen compound m 
${{\mathrm{c}}^{\prime}}_{\text{im}}$  Corresponding concentration term for nitrogen compound m in material flow i to diluting mass
${{m}^{\prime}}_{im}$ 
c_{im,geog}  Background concentration corresponding to nitrogen compound m in environmental compartment i, e.g., kg NNH_{4}^{+} per kg groundwater 
min(c_{j,geog})  Smallest occurring corresponding background concentration, e.g., NN_{org} concentration in groundwater 
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Sobańtka, A.P.; Zessner, M.; Rechberger, H. The Extension of Statistical Entropy Analysis to Chemical Compounds. Entropy 2012, 14, 24132426. https://doi.org/10.3390/e14122413
Sobańtka AP, Zessner M, Rechberger H. The Extension of Statistical Entropy Analysis to Chemical Compounds. Entropy. 2012; 14(12):24132426. https://doi.org/10.3390/e14122413
Chicago/Turabian StyleSobańtka, Alicja P., Matthias Zessner, and Helmut Rechberger. 2012. "The Extension of Statistical Entropy Analysis to Chemical Compounds" Entropy 14, no. 12: 24132426. https://doi.org/10.3390/e14122413