On the calculation of thermodynamic parameters in sorption calorimetric experiments

The sorption calorimetry method developed by Wadsö and co-workers is one of the most successful methods for studying the enthalpy of vapor sorption by solids and gels. A unique feature of this method is a simultaneous measurement of the water sorption isotherm and the sorption enthalpy. The accuracy of the enthalpy measurements in sorption calorimetric experiments can be affected by diffusion of water vapour through the injection channel tube and potentially through small leaks in the sorption cell. At high water activities this leads to an apparent drift of the measured enthalpies towards endothermic values. In this work we propose an improvement of the enthalpy calculation method, that eliminates these effects and substantially improves the accuracy of the enthalpy measurements. The new method is successfully tested on previously published sorption calorimetric data and can be recommended for use in future experiments.


Introduction
Accurate measurements of heat released or absorbed in water sorption processes is an experimental challenge.These experiments require non-standard calorimetric equipment, they are time consuming and may be influenced by various experimental errors.Working with different ranges of water activities may require different experimental methods and procedures.One of the most successful methods for studying heat effects of sorption of water and other substances is the sorption calorimetry method developed by Wadsö and co-workers [1,2].A unique feature of this method is a simultaneous measurement of the water sorption isotherm and the sorption enthalpy with a very high resolution.Even though the main application area of the method is sorption of water, sorption of organic vapours can also be studied [3,4].During the past two decades, the method was successfully applied for studies of cellulose [5], carbohydrate polymers [6,7], proteins [8], surfactants [9,10], lipids [11,12], skin [13] and nanomaterials [14].The results obtained by the sorption calorimetric method helped to resolve several complex research problems, such as the driving forces of phase transitions in surfactants and lipids [15].The papers cited above represent only a fraction of studies done using the sorption calorimetry and the full review of its applications is outside of the scope of this article.
The method of sorption calorimetry is, however, not free from experimental problems and has certain limitations.For example, at high water activities the signals become very low, which substantially decreases the accuracy of the enthalpy measurements [2].In particular, the sorption enthalpy values measured at high water activities tend to be higher than expected.As a partial solution to this problem, a desorption calorimetric method was developed [16], where water content is scanned in the opposite direction: towards the dry material.The use of desorption is, however, not a universal solution to the problem.In many cases materials show sorption-desorption hysteresis i.e. the results obtained during desorption are expected to be different from results obtained during sorption.
In this work, the main source of errors in calculations of heat of sorption is identified and corrected calculation procedures are proposed.

Calculation methods
A sorption calorimetric cell consists of two chambers connected by a tube, see Fig. 1.An initially dry sample is placed into the sorption chamber and pure water is injected into the vaporisation chamber.The injection is performed using a thin cannula that is inserted through a long and thin tube on top of the two-chamber cell.The thermal powers P V and P S are measured at the vaporisation and sorption chambers respectively and the partial molar enthalpy of mixing of water H m w is calculated according to the following equation: where H V w is the enthalpy of vaporisation of pure water, which is a constant at a given temperature.In this and the following equations, for the power values the same sign convention as for the enthalpies is used (exothermic values are negative).Even though the key process here is sorption, the results are usually presented in terms of enthalpy of mixing since the latter better characterises deviations from ideal behaviour, while enthalpy of sorption is normally strongly exothermic.
The water activity can be calculated from the same calorimetric data using the following simple equation: where P V;max is the maximum thermal power observed when the difference between water activities in the two chambers is 1.0.A more complex and accurate equation that considers the bulk flow of vapour [17] was also used in a number of studies.Eqs. ( 1) and ( 2) are derived based on assumption that all water evaporated in the vaporisation chamber reaches the sample and absorbs there.This is, however, an approximation because due to the complex structure of the calorimetric cell, small leakages of water vapor are unavoidable.Normally they are orders of magnitude smaller than the flux of vapor through the tube connecting the two chambers and are ignored in enthalpy calculations.Still, at high water activities when the thermal powers are low, they may introduce a substantial error in the calculations of enthalpy.Below we will show that taking them into account significantly increases accuracy of enthalpy calculations.
In Fig. 1, a schematic image of a sorption calorimetric cell is shown.The sample is placed in the top chamber (sorption chamber) and water is injected through a long cannula (omitted for clarity in the Figure ) into the vaporisation chamber.Water evaporates from the vaporisation chamber, diffuses through the tube and is absorbed by the sample in the sorption chamber.The sorption chamber is connected to the atmosphere through a very narrow long tube, and a part of water vapor may diffuse through this tube into the atmosphere.The total mass balance of vapor diffusion can be written in the following way: where n w is the number of mols of water, t is time, subscripts V, S and L stand for vaporisation, sorption and leakage, respectively.The loss of water vapor due to leakage includes the diffusion through the thin long tube and other losses (e.g.due to diffusion through micro cracks or due to imperfect sealing of the parts of the calorimetric cell by Teflon rings etc.).In some variants of the calorimetric cell, the sample is placed in the bottom cell, while water is injected into the top cell.The mass balance (Eq.( 3)) is, however, valid for both cases.In sorption calorimetric experiments, the derivatives dn w =dt are calculated from the thermal powers.For example, the amount of water evaporated from the vaporisation chamber is: Analogously, for sorption chamber one can write: where H S w is the partial molar enthalpy of sorption of water.The loss of water due to leakage can also be written in the similar way: although the power P L associated with it is not directly measured in the sorption experiments.Then Eq. ( 3) can be re-written as follows: Since our goal is to calculate the sorption (and then mixing) enthalpy, this equation can be rearranged: Or in terms of enthalpy of mixing: It is easy to see that in the absence of leakage, the Eq. ( 1) is retrieved.
Since P L is not measured in the sorption experiments, it should be calculated (or estimated) using other approaches.In particular, the leakage that is always present in the considered experiments, is the diffusion of water through the long narrow tube used as a guide for the injection cannula as shown in Fig 1 .Knowing its dimensions one can calculate the flux of water vapor using Fick's law.For water vapor diffusion it can be written in the following form: Where A is the cross-section area, D w is the diffusion coefficient, C 0 w is concentration of water in saturated vapor, a w is water activity and z is the spatial coordinate.Assuming steady state conditions and using Eq. ( 6), one arrives at: It is convenient to introduce P L;max -the maximum power achieved when Da w ¼ 1:0.Then where a S w and a atm w are water activities in the sorption chamber and atmosphere respectively.Finally, the equation for calculation of the partial molar enthalpy of mixing of water is: Analogously, subtracting from P V in Eq. ( 2) the contribution due to leakage P V one arrives at the corrected equation for water activity: a w ¼ P V;max À P V À P L;max a atm w P V;max À P L;max ð14Þ

Calculations of enthalpy and water activity
Let us firstly consider the case when the water vapor leakage occurs only through the long narrow tube (the injection channel) on top of the calorimetric cell (Fig. 1).Since its dimensions A and Dz are known, one can calculate the maximum thermal power due to leakage: where p w is the partial pressure of water, R and T are the gas constant and the temperature in K respectively (used to convert the concentration of water in the vapor phase to the partial pressure of water).For 25 °C, the parameters available from literature can be used for the calculation: H V w = 44.0kJÁmol À1 , D w = 0.260 cm 2 Ás À1 [18], p w = 3169 Pa [19].For a tube of the length of 27 cm and inner radius of 0.4 mm, calculation using Eq.(15) gives P L;max value of 2.7 mW.
Using this value, one can illustrate a typical error arising from ignoring the contribution due to leakage.In Fig. 2 this is shown for two different sizes of tubes connecting the vaporisation and sorption chambers that can be characterised by two different P V;max values: 100 and 900 mW.For calculations of water activity the correction is negligible for the large tube, while for the 100 mW tube it can reach almost 2%.At high humidities, the corrected value is higher than non-corrected, i.e. the leakage decreases the water activity value if it is calculated using Eq. ( 2).In the case of low water activities the effect is opposite, however, one should keep in mind that a similar error can be present in calibration experiments for determination of P V;max .
The effect of the proposed correction on the calculation of the partial molar enthalpy of mixing of water is even more pronounced.Even for the larger tube (that in general ensures lower sensitivity to the effect of leakages) the effect is strong at high water activities -at least several kJÁmol À1 .At low water activities the effect is, however, negligible, and equals 0 when the water activity in the sample coincides with the relative humidity in the atmosphere.At the lowest water activities the corrected value is higher than the non-corrected (in Fig. 2 the absolute value of the difference is shown for the sake of using the logarithmic coordinates).
In the sorption calorimetric literature there are plenty of examples of unexpected rise of the hydration enthalpy at higher water activities or water contents.It is, however, expected that in majority of cases the partial molar enthalpy of mixing of water should approach zero values at high water contents.The calculations presented here readily explain this observation.Moreover, using Eq. ( 13) one can re-process the experimental data on thermal powers P V and P S obtained in previous studies to obtain more accurate enthalpy values.The baselines for such calculations should be taken from a blank experiment with an empty calorimetric cell b) a) Fig. 2. The difference between corrected and non-corrected values of water activity calculated using Eqs.( 2) and ( 14) respectively (a).The absolute value of the difference between results of H m w calculation using Eqs.( 1) and ( 13) (b).RH of the atmosphere is assumer to be 30%.The legend shows P V;max value of the tube connecting the two chambers.without water injection.In Fig. 3 this is illustrated for the data on the hydration enthalpy of acid hydrolysed starch (maltodextrin) [6].In the original data (red curve) a clear ascending trend at high water contents is observed.In the enthalpy data calculated using Eq. ( 13) (blue curve) this trend is eliminated.

Estimation of the correction parameters from the sorption power baseline
The exact geometric parameters of the tubes, cracks, holes and pores through which the leakage can occur is, however, in general case not known.In addition to diffusion of water through the injection channel tube discussed above, additional flows of vapor might be present and it is difficult to eliminate them.It is, however, possible to correct the experimental data to compensate for the influence of these artefacts using the equations presented above.For that, the P L;max value has to be known for each experiment.An algorithm for its calculation is proposed below.
Experimentalists working with sorption calorimetry often experience that the baseline recorded from the sorption chamber before the injection very slowly decays to a stable level, and this level does not match the baseline observed in blank experiments with an empty cell.The reason for that is the absorption of the water vapor that diffuses from the atmosphere towards the sample through the injection channel, small pores and cracks etc.. Analogously to the equations presented above, for the situation when the evaporation chamber is empty one can write: Rearranging this equation, one arrives at the following expression for calculation of the maximal thermal power associated with the leakage: In this equation, P S is the thermal power released from the sorption chamber in case of steady-state diffusion of vapor.As it was mentioned above, achieving a stable baseline level before water injection is a challenge.This is illustrated in Fig. 4, where the base-line of the experiment with acid hydrolysed starch is shown.To obtain the baseline level corresponding to the steady state diffusion, this data can be approximated by an exponential decay: where P BL is the baseline level at the steady state, a and k are constants and t is time.To obtain the parameters of this decay, we used a non-linear least square fitting implemented in MATLAB (The MathWorks, Inc.).For the data shown in Fig. 4, the value obtained in the fitting procedure was 1.1 mW.For calculations using Eq. ( 17), one also needs to know H m w at the conditions when the baseline was recorded, i.e. at zero water content.Fortunately, its value is not sensitive to the discussed correction (see the left part of the Fig. 3) and thus can be calculated using Eq.(1).Calculation using H m w = À18 kJÁmol À1 and a S w = 0, gives the P L;max value of 2.6 mW, in excellent agreement with the calculations presented in the Section 3.1.This result indicates that the only source of water vapor diffusion in this experiment was the thin injection channel shown in the upper part of Fig. 1.

Conclusions
The enthalpy measurements in sorption calorimetric experiments can be affected by diffusion of water vapour through the injection channel tube and potentially through small leaks in the sorption cell The accuracy of the enthalpy measurements can be substantially improved by introducing corrections that take into account the additional vapour flow When the geometrical parameters of the leaks are not known, they can be calculated from the course of the calorimetric baseline recorded before the water injection CRediT authorship contribution statement Vitaly Kocherbitov: Conceptualization, Methodology, Software, Validation, Investigation, Data curation.Fig. 3.The partial molar enthalpy of mixing of water for acid hydrolyzed starch (maltodextrin).The red curve shows the data by Carlstedt et al. [6], the blue curve shows the enthalpy calculated using Eq. ( 13).(For interpretation of the references to colour in this Fig.legend, the reader is referred to the web version of this article.)Fig. 4. A part of the sorption power P S baseline measured in the experiment with acid hydrolysed starch (maltodextrin) before the injection of water.The signal is exothermic (raw calorimetric data), the black curve is the fit with Eq. ( 18).

Fig. 1 .
Fig. 1.A two-chamber sorption calorimetric cell.The arrows show direction of water diffusion.The subscripts V, S and L stand for vaporisation, sorption and leakage respectively.