^{1}

^{*}

^{2}

^{2}

^{3}

Activity coefficients at infinite dilution, γ ∞ i, were calculated for 12 solutes, with organic solutes including linear alcohols (methanol, ethanol, propanol), linear alkanes (heptane, octane), benzene, toluene, cyclohexane, 1, 2-dichloroethane, trichloroethylene, acetonitrile and carbon tetrachloride. The values of γ ∞ i were determined via either thermodynamic or artificial neural network modelling at different temperatures. A comparison between extracted results from these two methods confirmed that experimental and predicted results are roughly the same. The accuracy of predicted results proves this model is fully compatible with a wide range of solutes, and it can readily be used as an alternative to conventional gas-liquid chromatography for the measurements of activity coefficient at infinite dilution.

The measurements of activity coefficient at infinite dilution (γ∞) are crucially important for either theoretical or practicing chemistry. This parameter describes the behavior of a solute completely surrounded by solvent molecules. Activity coefficients at infinite dilution have been widely used for determining quantity of solutes’ volatility and also made information about intermolecular energy between solvent and solute [

Several methods were developed for the measurement of γ∞ such as dilutor technique (DT) [

It is important to have a simple method to estimate all property distributions from known bulk properties. Artificial Neural Networks (ANN) has been widely applied to an extensive range of chemical engineering such as process modeling, optimization and PVT behavior over the last 20 years. In the mathematical algorithm of ANN, it is possible to relate input and output parameters without requiring prior knowledge of relationships between the process parameters [

In this work, values of γ∞ (the activity coefficients at infinite dilution) for 12 compounds in the following di-cationic ionic liquid with three phase loadings (10%, 15% and 20%) have been determined at various temperatures 308, 313, 318 and 323 K. Regarding the importance of activity coefficient at infinite dilution in thermodynamic and separation processes, a growing need for gaining activity coefficient in a simple and fast way has been felt. Therefore, an artificial neural network (ANN) model has been developed to predict the measures of γ∞ for an extensive range of solutes.

All solvents were distilled from standard drying agents before use. All used Ionic Liquids were synthesized in CCERI [

Structures of di-cationic ionic liquids C_{12}(mPy)_{2}(NTf_{2})_{2} is shown in

All solutes including linear alcohols (methanol, ethanol, propanol), linear alkanes (heptane, octane), benzene, toluene, cyclohexane, 1, 2-dichloroethane, trichloroethylene, acetonitrile and carbon tetra chloride, were supplied from MERCK.

Gas chromatography experiments were performed using a Varian CP-3800 gas chromatograph equipped with a heated 1041 injector and a thermal conductivity detector (TCD). The injector and detector temperatures were kept constant at 473 K during all experiments. The flow rate of helium was adjusted to obtain adequate retention times. The dead time was determined by injection of air with each solute. A personal computer equipped software as used for recording detector signals and corresponding chromatograms were obtained by Galaxie software.

Column packing, containing from 10%, 15% and 20% of stationary phase (IL) on Chromosorb W-AW (80 - 100 mesh), was prepared using the rotary evaporator technique. After evaporation of the dichloromethane under vacuum, the support was equilibrated at 323 K for 18 hours. The solid support material with the stationary phase was filled in a stainless steel column with an inner diameter of 3 mm and a length of 1 m. The weight of the packing material was calculated from the weights of the packed and empty column. A volume of the headspace vapor of samples of 0.1 - 0.5 micro liter was introduced to be in infinite dilution conditions. No differences in retention times t_{r} were found by injecting individual pure components or their mixtures. The measurements were carried out at temperatures between 308 and 323 K. At a given temperature, each experiment was repeated at least three times to verify the reproducibility. The difference of the retention times of the three measurements was ordinarily reproducible within (0.01 to 0.1) min.

Under aforementioned condition, the retention data for 12 solutes in 3 gas-chromatography columns with different phase load (10%, 15%, and 20%) and in different temperature (308, 313, and 318 K) have been obtained and used for calculating of activity coefficients at infinite dilution.

Equation (1) suggested by Everett and Cruickshank et al. [

L n γ i ∞ = L n ( n R T V n P s ) − P s B 11 − V s R T + 2 B 12 − V ∞ R T J P ο (1)

where n is the mole number of the stationary phase component inside the column, R is the ideal gas constant, T is the temperature of the oven, V^{N} is the standardized retention volume of the solute, P˚ is the column outlet pressure (equal to atmospheric pressure), V^{S} the saturated liquid molar volume of the solute at T and V^{∞} is the partial molar volume of the solute at infinite dilution in the solvent. B^{11} the second Virial coefficient of the solute in the gaseous state at temperature T, B^{12} the mutual Virial coefficient between the solute 1 and the carrier gas helium 2 and P^{S} is the probe vapor pressure at temperature T. The second and third terms in Equation (1) are correction terms that result from the non-ideality of the mobile gaseous phase. The molar volume of the solute V^{S} was determined from experimental densities, and the partial molar volumes of the solutes at infinite dilution V^{∞} were assumed to be equal to V^{S}. The vapor pressure values were calculated using the Antoine equation [^{N}, can be calculated with the following relationship:

V N = J U ο t ′ r (2)

The adjusted retention time, t ′ r calculated from the difference between the retention times of a solute and that of air. U_{0}, the flow rate of the carrier gas, measured at the room temperature. The factor J corrects for the influence of the pressure drop along the column. Among of J relies on the pressure at the column outlet and inlet. This factor is defined by Equation (3).

J = 3 2 ( p i − p o ) 2 − 1 ( p i − p o ) 3 − 1 (3)

The values of B^{11} and B^{12} were calculated using the McGlashan and Potter [

The critical properties of the pure component ( T c 11 and V c 11 ) was extracted from the literature [

Activity coefficients at infinite dilution of various types of solutes were computed in the di-cationic stationary phase with different phase load (10%, 15%, and 20%) in four temperatures (308, 313, 318, and 323 K). The obtained results of activity coefficients at infinite dilution for 12 solutes are presented in

An artificial neural network was applied to model the system in order to predict activity coefficient of dilute solution for lots of chemical compounds. 144 data sets were used for training and testing. 70% of these data have been used for training, test data and validate data used the equal percentage of 15.

One of the most popular and commonly used networks is the multilayer perceptron network (MLP). The MLP configuration has gained a widespread use in static regression applications [

Solutes | T/K | 10 | 15 | 20 |
---|---|---|---|---|

Benzene | 308 | 0.2653 | 0.3249 | 0.2644 |

313 | 0.2778 | 0.3322 | 0.2597 | |

318 | 0.2751 | 0.4048 | 0.2744 | |

323 | 0.3020 | 0.4219 | 0.3101 | |

Methanol | 308 | 0.3905 | 0.4311 | 0.4185 |

313 | 0.3656 | 0.4528 | 0.3859 | |

318 | 0.3571 | 0.4696 | 0.4092 | |

323 | 0.4426 | 0.5885 | 0.4454 | |

Ethanol | 308 | 0.4968 | 0.5988 | 0.5385 |

313 | 0.4626 | 0.5990 | 0.4967 | |

318 | 0.4373 | 0.6220 | 0.5130 | |

323 | 0.5625 | 0.7060 | 0.5372 | |

Propanol | 308 | 0.6000 | 0.7150 | 0.6612 |

313 | 0.6142 | 0.7967 | 0.6564 | |

318 | 0.5975 | 0.6890 | 0.6201 | |

323 | 0.6716 | 0.8444 | 0.6473 | |

Acetonitrile | 308 | 0.1529 | 0.1838 | 0.1596 |

313 | 0.1594 | 0.1968 | 0.1565 | |

318 | 0.1620 | 0.2021 | 0.1591 | |

323 | 0.1991 | 0.2310 | 0.1796 | |

Cyclohexane | 308 | 2.4003 | 2.6037 | 2.3272 |

313 | 2.3132 | 2.7101 | 2.2169 | |

318 | 2.2900 | 2.7948 | 2.2222 | |

323 | 2.3168 | 3.9487 | 2.4903 | |

Toluene | 308 | 0.3538 | 0.4473 | 0.3446 |

313 | 0.3732 | 0.4436 | 0.3468 | |

318 | 0.3726 | 0.4538 | 0.3705 | |

323 | 0.3753 | 0.4306 | 0.4171 | |

Heptane | 308 | 4.5788 | 5.5034 | 5.0667 |

313 | 5.3570 | 5.5809 | 4.7640 | |

318 | 4.6549 | 5.5048 | 4.8457 | |

323 | 5.7666 | 5.8925 | 5.6490 | |

Octane | 308 | 7.0796 | 7.9705 | 7.6836 |

313 | 5.9958 | 7.4999 | 6.6670 | |

318 | 5.7462 | 7.3969 | 6.8222 | |

323 | 6.3740 | 9.2261 | 9.1810 | |

Dichloroethane | 308 | 0.1622 | 0.2034 | 0.1712 |

313 | 0.2052 | 0.2481 | 0.1670 | |

318 | 0.2362 | 0.2919 | 0.1976 | |

323 | 0.3322 | 0.4202 | 0.2465 | |

Carbontetrachloride | 308 | 0.4863 | 0.5602 | 0.4735 |

313 | 0.5864 | 0.6787 | 0.4642 | |
---|---|---|---|---|

318 | 0.7309 | 0.8205 | 0.5208 | |

323 | 0.8610 | 1.1592 | 0.6568 | |

Trichloroethylene | 308 | 0.2722 | 0.3176 | 0.2680 |

313 | 0.3210 | 0.3880 | 0.2615 | |

318 | 0.3823 | 0.4747 | 0.2560 | |

323 | 0.5137 | 0.6638 | 0.3002 |

non-linear function with sufficient number of neurons and more hidden layers may result in over-fitting, therefore, in this work, one hidden layer was applied as displayed in

MSE = 1 N ∑ i = 1 N e i 2 (4)

where e_{i} is the differences between experimental and predicted data.

In order to improve the model generalization and prevent over-fitting, the number of neurons has to be chosen so that the number of internal parameters in the network does not exceed the number of training data sets [

n t o t = ( n i + 1 ) × n h 1 + ( n h 1 + 1 ) × n h 2 + ⋯ + ( n h n + 1 ) × n o (5)

where, n t o t is the total number of network parameters, n o is the number of outputs and n h i is the number of the neurons in the ith hidden layer. In this work, the maximum number of neurons that can be used in hidden layer in this system to prevent over-fitting was calculated to be seven. Thus, the choice of neuron number was limited in the range of 1 - 7 neurons for the hidden layer. At the second step, the network with the neuron number of the last step was used to find a training method that leads to minimum MSE of the network. If the network MSE was less than the desirable MSE the third step was started. Otherwise, the last two steps were repeated till the desirable MSE value was reached.

^{1}Hesian updating methods of Broyden, Fletcher, Goldfarb, Shanno (BFGS).

The applied training methods consist of Bayesian Regularization (BR), BFGS^{1} Quasi-Newton (BFG), Resilient Backpropagation (RP), Scaled Conjugate Gradient (SCG), Conjugate Gradient with Powell/Beale Restarts (CGB), Levenberg-Marquardt (LM), Fletcher-Powell Conjugate Gradient (CGF), Polak-Ribiére Conjugate Gradient (CGP), One Step Secant (OSS), Variable Learning Rate Gradient Descent (GDX), Gradient Descent with Momentum (GDM), and Gradient Descent (GD).

At the third step, the selected training method was applied to train the network using a number of neurons (1 - 8). Each of these trainings was repeated 1000 time and the means of MSEs for the repeated trainings were recorded.

In addition to MSE, correlation coefficients (R) are commonly used to verify ANN models. In this work R has also been applied as defined by Equation (6).

R = ∑ ( α i − α ¯ ) ( τ i − τ ¯ ) ∑ ( α i − α ¯ ) 2 ∑ ( τ i − τ ¯ ) 2 (6)

where, τ_{i} is the target and α_{i} is the network output and τ ¯ , α ¯ are the mean amount of the data.

The ANN model was also employed to predict activity coefficient at infinite dilution of different solutes. The procedure described in section 3 was applied to design the model. Temperature, Ionization energy, Molecular weight and stationary phase loading were chosen as the input data of network and Activity coefficient, Saturated pressure, Saturated volume, Adjusted retention time and the correction factor (J) were chosen as the output data.

Levenberg-Marquardt (LM) method was found to have the minimum error as shown in

The results of ANN model and experimental data are depicted in

Overall data | Train data | Validate data | Test data | |
---|---|---|---|---|

Trainb | 1714.857 | 1752.72 | 1070.62 | 2186.99 |

trainbfg | 63.4526 | 50.5515 | 63.4818 | 122.065 |

Trainbr | 16.33893 | 16.3389 | 1450277 | 2450.28 |

Trainc | 572.8554 | 571.264 | 793.043 | 359.9 |

traincgb | 297.3109 | 277.746 | 368.5 | 315.055 |

traincgf | 283.4068 | 276.4 | 279.52 | 319.142 |

traincgp | 270.6267 | 213.914 | 391.736 | 407.301 |

Traingd | 270.6267 | 261.347 | 350.299 | 233.134 |

traingda | 270.6267 | 266.085 | 159.743 | 402.153 |

traingdm | 270.6267 | 298.904 | 215.837 | 196.885 |

Traingd | 270.6267 | 282.871 | 165.608 | 319.99 |

Trainlm | 0.111587 | 0.08726 | 0.16761 | 0.16613 |

trainoss | 26.00845 | 28.9234 | 25.3603 | 13.4069 |

Trainr | 162.3016 | 177.837 | 172.875 | 81.1113 |

Trainrp | 80.94391 | 87.1408 | 73.3158 | 60.4043 |

Trains | 71.17467 | 74.2786 | 65.688 | 62.5527 |

trainscg | 71.17467 | 76.5718 | 54.7914 | 63.0255 |

bins is shown in

In this work, having calculated the activity coefficients at infinite dilution in three different ways, a comparison between their final results has been drawn.

Data | Number | average relative error percent | R |
---|---|---|---|

Train | 102 | 0.0873 | 0.994 |

Validate | 21 | 0.1676 | 0.988 |

Test | 21 | 0.1676 | 0.988 |

Overall | 144 | 0.1116 | 0.993 |

Hidden layer neuron | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

Optimal neural network Output Weights | |||||||

Vsat | 0.84599 | −0.64791 | 0.041857 | −6.46673 | −0.74503 | 0.065056 | −0.00501 |

Psat | −0.43605 | −0.60363 | −0.67509 | 1.42567 | −2.02872 | −0.56282 | −1.36424 |

t' | −0.62664 | 3.026805 | 1.83053 | 4.555867 | 8.531572 | 2.544383 | 3.790702 |

J | −0.20605 | 0.29563 | 0.279518 | 0.150291 | 1.031618 | 0.327164 | 0.489746 |

Γ | 1.07002 | −1.57239 | −0.27529 | −8.62001 | −1.16475 | 0.018138 | −0.37616 |

Optimal neural network Biases | |||||||

Bias | 6.290099 | −2.47087 | −0.35334 | 0.533857 | −0.31865 | 3.139325 | −0.59779 |

The first method is based on using experimental data extracted from the thermodynamic model. In the second method, data were obtained from ANN model and the third method is based on the thermodynamic model used ANN predicted data.

Calculated test data using thermodynamic model | Output | |||
---|---|---|---|---|

Predicted gama using ANN | calculated by ANN data | error of the ANN predictions | error of the thermodynamic predictions by use of ANN outputs | |

0.302 | 0.23 | 0.4615 | 0.072 | 0.1595 |

2.3168 | 2.587 | 3.5023 | 0.2702 | 1.1855 |

0.3726 | 0.5664 | 0.3205 | 0.1938 | 0.0521 |

5.357 | 5.60408 | 12.638 | 0.24708 | 7.281 |

0.2362 | 0.2451 | 0.2613 | 0.0089 | 0.0251 |

0.4863 | 0.6772 | 1.079 | 0.1909 | 0.5927 |

0.5237 | 0.6686 | 0.3 | 0.1449 | 0.2237 |

0.4219 | 0.21254 | 0.70601 | 0.20936 | 0.28411 |

0.4311 | 0.351215 | 0.4143 | 0.079885 | 0.0168 |

2.7947 | 2.8557 | 9.1918 | 0.061 | 6.3971 |

0.4306 | 0.98219 | 0.40056 | 0.55159 | 0.03004 |

5.5809 | 6.06188 | 7.2775 | 0.48098 | 1.6966 |

5.8924 | 6.3578 | 7.92725 | 0.4654 | 2.03485 |

0.6637 | 0.6582 | 0.42118 | 0.0055 | 0.24252 |

0.4967 | 0.55293 | 0.36778 | 0.05623 | 0.12892 |

0.6564 | 0.836017 | 0.26141 | 0.179617 | 0.39499 |

2.3271 | 2.7655 | 0.8657 | 0.4384 | 1.4614 |

0.3704 | 0.515176 | 0.67011 | 0.144776 | 0.29971 |

7.6835 | 6.57 | 1.82817 | 1.1135 | 5.85533 |

0.17117 | 0.27 | 0.18824 | 0.0988 | 0.0818 |

0.24652 | 0.1968 | 0.28503 | 0.04972 | 0.03851 |

Average Overall error of test data | 0.24113 | 1.353213 |

0.24 and 1.35, respectively.

The chromatographic data has been used in order to determine the values of activity coefficients at infinite dilution by either thermodynamic or ANN model. In the thermodynamic model, the values of activity coefficients at infinite dilution have been calculated for 12 solutes at different temperatures (308, 313, 318 and 323 K) in three columns with different stationary phase loadings (10%, 15% and 20%). It can be seen that the results obtained from two models come from a broadly similar direction. As a result, ANN can be efficiently used to measure the values of activity coefficients at infinite dilution in different temperatures. A great advantage associated with ANN model is that the values of activity coefficients at infinite dilution can be directly obtained through retention time (t_{r}), the saturated liquid molar volume (V^{ᶳ}), the probe vapor pressure (Pᶳ) and the ionization energy (I) at T, without getting involved in complicated thermodynamic computations. According to the strong similarity between the results of two models, the range of solutes can be expanded, and the values of activity coefficients at infinite dilution can be predicted precisely by ANN model for an extensive range of solutes according to their retention time (t_{r}), the saturated liquid molar volume (V^{ᶳ}^{)}, the probe vapor pressure (Pᶳ) and the ionization energy (I) at the wanted temperature (T). As in ANN model all the steps related to the calculation of physiochemical parameters can be skipped, ANN model can be considered as a time-saving and cost-efficient technique for determination of activity coefficients at infinite dilution, in comparison with the thermodynamic model. As it can be seen in

Yahyaee, A., Nazifi, M., Kianpour, M. and Heidar, K.T. (2018) Experimental Investigation and Modeling of Activity Coefficient at Infinite Dilution of Solutes Using Dicationic Solvent Based on Pyrrolidinium as a New Stationary Phase in Gas Chromatography. American Journal of Analytical Chemistry, 9, 257-271. https://doi.org/10.4236/ajac.2018.94020