On the calculation of thermodynamic parameters in sorption
calorimetric experiments
Vitaly Kocherbitov
a,b,⇑ aBiomedical Science, Faculty of Health and Society, Malmö University, SE-205 06 Malmö, Sweden
b
Biofilms – Research Center for Biointerfaces, Malmö University, SE-205 06 Malmö, Sweden
a r t i c l e i n f o
Article history: Received 19 June 2020
Received in revised form 5 August 2020 Accepted 6 August 2020
Available online 8 August 2020 Keywords:
Sorption calorimetry Isothermal calorimetry Water sorption Mixing enthalpy
Partial molar enthalpy of mixing
a b s t r a c t
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 val-ues. 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 suc-cessfully tested on previously published sorption calorimetric data and can be recommended for use in future experiments.
Ó 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Accurate measurements of heat released or absorbed in water sorption processes is an experimental challenge. These experi-ments 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 dif-ferent experimental methods and procedures. One of the most suc-cessful 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, sorp-tion 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 sub-stantially 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 solu-tion to this problem, a desorpsolu-tion calorimetric method was devel-oped [16], where water content is scanned in the opposite direction: towards the dry material. The use of desorption is, how-ever, not a universal solution to the problem. In many cases mate-rials 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.
2. Calculation methods
A sorption calorimetric cell consists of two chambers connected by a tube, seeFig. 1. An initially dry sample is placed into the sorp-tion chamber and pure water is injected into the vaporisasorp-tion chamber. The injection is performed using a thin cannula that is inserted through a long and thin tube on top of the two-chamber
https://doi.org/10.1016/j.jct.2020.106264
0021-9614/Ó 2020 The Author(s). Published by Elsevier Ltd.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). ⇑ Address: Biomedical Science, Faculty of Health and Society, Malmö University,
SE-205 06 Malmö, Sweden.
E-mail address:Vitaly.Kocherbitov@mau.se
Contents lists available atScienceDirect
J. Chem. Thermodynamics
cell. The thermal powers PV
and PSare measured at the
vaporisa-tion and sorpvaporisa-tion chambers respectively and the partial molar enthalpy of mixing of water Hmwis calculated according to the
fol-lowing equation: Hmw¼ H V w 1þ PS PV ! ð1Þ where HVwis the enthalpy of vaporisation of pure water, which is a
constant at a given temperature. In this and the following equa-tions, 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 nor-mally strongly exothermic.
The water activity can be calculated from the same calorimetric data using the following simple equation:
aw¼ 1 P
V
PV;max ð2Þ
where PV;maxis 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 magni-tude 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 cham-ber) and water is injected through a long cannula (omitted for clar-ity 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:
dnw dt V ¼ dnw dt S þ dnw dt L ð3Þ where nwis 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 dnw=dt are calculated from the thermal powers. For
example, the amount of water evaporated from the vaporisation chamber is: dnw dt V ¼P V HV w ð4Þ Analogously, for sorption chamber one can write:
dnw dt S ¼P S HSw ð5Þ where HSwis the partial molar enthalpy of sorption of water. The loss
of water due to leakage can also be written in the similar way: dnw dt L ¼P L HV w ð6Þ although the power PLassociated with it is not directly measured in
the sorption experiments.
Then Eq.(3)can be re-written as follows: PV HVw ¼P S HSw þP L HVw ð7Þ Since our goal is to calculate the sorption (and then mixing) enthalpy, this equation can be rearranged:
HSw¼
HV w P
S
PV PL ð8Þ
Or in terms of enthalpy of mixing: Hmw¼ H V w 1þ PS PV PL ! ð9Þ It is easy to see that in the absence of leakage, the Eq. (1)is retrieved.
Since PLis 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 inFig 1. Knowing its 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.
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: dnw dt ¼ ADwC 0 w daw dz ð10Þ
Where A is the cross-section area, Dwis the diffusion coefficient, C0w
is concentration of water in saturated vapor, awis water activity and
z is the spatial coordinate. Assuming steady state conditions and using Eq.(6), one arrives at:
PL HVw ¼ A
D
zDwC 0 wD
aw ð11ÞIt is convenient to introduce PL;max - the maximum power achieved whenDaw¼ 1:0. Then
PL¼ PL;max aS w aatmw
ð12Þ where aS
wand aatmw 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:
Hmw¼ H V w 1þ PS PV PL;max aS w aatmw ! ð13Þ Analogously, subtracting from PVin Eq.(2)the contribution due
to leakage PV one arrives at the corrected equation for water
activity: aw¼P
V;max PV PL;maxaatm w
PV;max PL;max ð14Þ
3. Results and discussion
3.1. 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: PL;max¼ HV w A
D
zDw pw RT ð15Þwhere pwis the partial pressure of water, R and T are the gas
con-stant 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: HV
w= 44.0 kJmol1, Dw= 0.260 cm2s1
[18], pw= 3169 Pa[19]. For a tube of the length of 27 cm and inner
radius of 0.4 mm, calculation using Eq.(15)gives PL;max value of 2.7mW.
Using this value, one can illustrate a typical error arising from ignoring the contribution due to leakage. InFig. 2this is shown for two different sizes of tubes connecting the vaporisation and sorption chambers that can be characterised by two different PV;max values: 100 and 900mW. For calculations of water activity the correction is negligible for the large tube, while for the 100mW tube it can reach almost 2%. At high humidities, the cor-rected value is higher than non-corcor-rected, 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 cal-ibration experiments for determination of PV;max.
The effect of the proposed correction on the calculation of the partial molar enthalpy of mixing of water is even more pro-nounced. 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 kJmol1. 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 (inFig. 2the absolute value of the difference is shown for the sake of using the logarithmic coordinates).
In the sorption calorimetric literature there are plenty of exam-ples of unexpected rise of the hydration enthalpy at higher water activities or water contents. It is, however, expected that in major-ity of cases the partial molar enthalpy of mixing of water should approach zero values at high water contents. The calculations pre-sented here readily explain this observation. Moreover, using Eq.
(13)one can re-process the experimental data on thermal powers PV and PS 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 Hm
wcalculation using Eqs.(1) and (13)(b). RH of the
atmosphere is assumer to be 30%. The legend shows PV;max value of the tube
without water injection. InFig. 3this 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.
3.2. 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 injec-tion channel tube discussed above, addiinjec-tional flows of vapor might be present and it is difficult to eliminate them. It is, however, pos-sible to correct the experimental data to compensate for the influ-ence of these artefacts using the equations presented above. For that, the PL;maxvalue has to be known for each experiment. An algo-rithm for its calculation is proposed below.
Experimentalists working with sorption calorimetry often expe-rience 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.. Analo-gously to the equations presented above, for the situation when the evaporation chamber is empty one can write:
PS HSw ¼ P L HVw ð16Þ Rearranging this equation, one arrives at the following expres-sion for calculation of the maximal thermal power associated with the leakage: PL;max¼ P S HV w Hm w H V w aS w aatmw ð17Þ
In this equation, PSis the thermal power released from the sorp-tion 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 inFig. 4, where the
base-line of the experiment with acid hydrolysed starch is shown. To obtain the baseline level corresponding to the steady state diffu-sion, this data can be approximated by an exponential decay:
P ¼ PBLþ a exp ktð Þ ð18Þ
where PBLis the baseline level at the steady state, a and k are
con-stants 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 inFig. 4, the value obtained in the fitting procedure was 1.1 mW. For calculations using Eq.
(17), one also needs to know Hmwat 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 theFig. 3) and thus can be calculated using Eq.(1). Calculation using Hm
w =
18 kJmol1
and aS
w= 0, gives the PL;maxvalue of 2.6mW, in
excel-lent 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 ofFig. 1.
4. Conclusions
The enthalpy measurements in sorption calorimetric experi-ments 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 substan-tially 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 base-line 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 PSbaseline 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).
Declaration of Competing Interest
The authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The author acknowledges the Gustaf Th Olsson foundation for financial support.
References
[1]I. Wadsö, L. Wadsö, A new method for determination of vapour sorption isotherms using a twin double microcalorimeter, Thermochim. Acta 271 (1996) 179–187.
[2]L. Wadsö, N. Markova, A method to simultaneously determine sorption isotherms and sorption enthalpies with a double twin microcalorimeter, Rev. Sci. Instrum. 73 (7) (2002) 2743–2754.
[3]N. Markova, E. Sparr, L. Wadsö, On application of an isothermal sorption microcalorimeter, Thermochim. Acta 374 (2) (2001) 93–104.
[4]S. Bjorklund, V. Kocherbitov, Alcohols react with MCM-41 at room temperature and chemically modify mesoporous silica, Sci. Rep. 7 (2017) 11.
[5]V. Kocherbitov et al., Hydration of microcrystalline cellulose and milled cellulose studied by sorption calorimetry, J. Phys. Chem. B 112 (12) (2008) 3728–3734.
[6]J. Carlstedt et al., Hydration and the phase diagram of acid hydrolyzed potato starch, Carbohydr. Polym. 112 (2014) 569–577.
[7]V. Kocherbitov et al., Hydration of a natural polyelectrolyte xanthan gum: comparison with non-ionic carbohydrates, Carbohydr. Polym. 82 (2) (2010) 284–290.
[8]V. Kocherbitov, T. Arnebrant, O. Söderman, Lysozyme–water interactions studied by sorption calorimetry, J. Phys. Chem. B 108 (49) (2004) 19036– 19042.
[9]V. Kocherbitov, O. Söderman, Effect of oligomerization of counterions on water activity in aqueous cationic surfactant systems, J. Phys. Chem. B 120 (28) (2016) 6961–6968.
[10] V. Kocherbitov, O. Söderman, Hydration of dimethyldodecylamine-N-oxide: Enthalpy and entropy driven processes, J. Phys. Chem. B 110 (27) (2006) 13649–13655.
[11]E. Sparr et al., The effect of bacteriorhodopsin, detergent and hydration on the cubic-to-lamellar phase transition in the monoolein–distearoyl phosphatidyl glycerol–water system, Biochim. Biophys. Acta 1665 (1-2) (2004) 156–166. [12]N. Markova et al., A calorimetric study of phospholipid hydration.
Simultaneous monitoring of enthalpy and free energy, J. Phys. Chem. B 104 (33) (2000) 8053–8060.
[13]C.L. Silva et al., Stratum corneum hydration: phase transformations and mobility in stratum corneum, extracted lipids and isolated corneocytes, Biochim. Biophys. Acta 1768 (11) (2007) 2647–2659.
[14]V. Kocherbitov, V. Alfredsson, Assessment of porosities of SBA-15 and MCM-41 using water sorption calorimetry, Langmuir 27 (7) (2011) 3889–3897. [15]V. Kocherbitov, Driving forces of phase transitions in surfactant and lipid
systems, J. Phys. Chem. B 109 (13) (2005) 6430–6435.
[16]V. Kocherbitov, L. Wadsö, A desorption calorimetric method for use at high water activities, Thermochim. Acta 411 (1) (2004) 31–36.
[17]V. Kocherbitov, A new formula for accurate calculation of water activity in sorption calorimetric experiments, Thermochim. Acta 414 (1) (2004) 43–45. [18] E.L. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University
Press, 1997.
[19] L. Haar, J. Gallagher, G. Kell, Nbs/Nrc Steam Tables, Taylor & Francis, 1984.
