J. Chem. Thermodynamics 2001, 33, 755–764
doi:10.1006/jcht.2000.0776
Available online at http://www.idealibrary.com on
Determination of Henry’s constant using a
photoacoustic sensor
Jörn Ueberfeld,a Hugo Zbinden, Nicolas Gisin,
Université de Genève, Group of Applied Physics, Rue de l’Ecole de
Médecine 20, 1211 Genève, Switzerland
and Jean-Paul Pellaux
Orbisphère Laboratories SA, Rue du Puits-Godet 20, 2000 Neuchâtel,
Switzerland
We present a simple method for measuring Henry’s constant kH of ethanol using photoacoustic spectroscopy. At T = 298.1 K the measured value for kH is
(0.877 ± 0.039) kPa · kg · mol−1 . Our data show that Henry’s law is valid at ethanol
molalities between 0.1 mol · kg−1 and 1.4 mol · kg−1 . The temperature dependence of
Henry’s constant was carefully examined by measuring the ethanol vapour pressure of six
different aqueous solutions between T = 273.1 K and T = 298.1 K. By analysing the gas
phase concentration and applying Henry’s law, an ethanol molality of 0.864 mol · kg−1 in
the liquid phase can be measured with an error of ±0.038 mol · kg−1 . The detection limit
of the photoacoustic sensor is a gaseous ethanol pressure of 10−3 kPa. Ethanol molality
c 2001 Academic Press
changes as low as 1.10−3 mol · kg−1 can be measured. KEYWORDS: Henry’s constant; photoacoustic spectroscopy; sensor; ethanol
1. Introduction
Henry’s constant kH describes the distribution of a volatile solute between two gaseous and
liquid phases. Consequently, changes in ethanol concentration in liquids can be determined
by measuring the gas phase concentration and a knowledge of kH . One application area in
which Henry’s law plays an important role is the beverage industry, where the variation of
ethanol concentration in beer and yeast cultures is a critical parameter, but where sample
taking from these liquids has to be avoided in order not to disturb the brewing or cultivation
process. The calibration of an ethanol sensor which analyses the gas phase over solutions
relies on accurate Henry’s constants.
a To whom correspondence should be addressed (E-mail: [email protected]).
0021–9614/01/070755 + 10 $35.00/0
c 2001 Academic Press
756
J. Ueberfeld et al.
Henry’s constants for ethanol have been determined by several research groups (for
a compilation of Henry’s constants see reference 1 and references cited therein). The
values range from 0.439 kPa · kg · mol−1 to 0.824 kPa · kg · mol−1 at T = 298.15 K.
The temperature behaviour of kH was investigated experimentally by one group (2) and
calculated from activity coefficients by another group. (3,4) However, the experimental
determination was restricted to only two temperatures. The accuracy of the calculated data
was reported with a temperature error of ±3 K. As the literature values are not accurate
for our purpose, we have determined a more accurate Henry’s constant for ethanol at
T = 298.1 K. Another objective of our work was to verify if Henry’s law was applicable to
the molality range for which our sensor was developed. Furthermore, we have investigated
the temperature behaviour of kH in the temperature range 273.1 K to 298.1 K.
The working principle of our sensor is based on photoacoustic spectroscopy, a powerful technique for measuring vapour phase concentrations of organic and inorganic compounds. (5,6)
(GAS + LIQUID) EQUILIBRIUM AND HENRY’S CONSTANT
In diluted real solutions the partial pressure pA of a volatile solute A is proportional to its
molality m A in the liquid. The constant of proportionality is Henry’s constant kH :
kH = pA /m A .
(1)
Like any other equilibrium constant, kH depends on temperature. Under restrictive
conditions such as nonvolatility of the solvent and small solubility, (7) the van’t Hoff
equation can be used to model the temperature dependence of Henry’s constant at a
constant pressure as follows: (1–3,8)
∂ ln
(kH /Pa · kg · mol−1 ) = 1tr H/RT 2 ,
(2)
∂T
where 1tr H is the enthalpy of transfer of one mole of solute from the solution to the
vapour phase. This transfer includes “demixing” of the gaseous species and vaporization.
Assuming that 1tr H is constant over the temperature range of interest (a valid assumption
over the narrow temperature range regarded in our work), equation (2) can be integrated to
give: (1–3,8)
kH,T2 = kH,T1 · exp{−1tr H/R · (1/T2 − 1/T1 )}.
(3)
By measuring kH at different temperatures, the enthalpy of transfer can be calculated.
The static equilibrium approach, which is one method for determining Henry’s constant,
is based on the direct measurement of gaseous and/or aqueous concentrations. There
are several variations of this technique which have been reviewed by Staudinger and
Roberts. (3) The main experimental challenge is to reach equilibrium between the two
phases and to keep it during the sampling process. When the gaseous concentration has
to be measured, sampling can cause considerable losses due to sorption from the gaseous
phase. For (water + ethanol), kH at T = 298.15 K has been determined by various research
groups but with insufficient accuracy. (2,4) Here we present a new static equilibrium method
for the determination of Henry’s constant. It is based on the measurement of gaseous
Determination of Henry’s constant
757
ethanol concentrations using photoacoustic spectroscopy. This technique avoids sorption
losses during sampling.
PHOTOACOUSTIC SPECTROSCOPY
The principle of photoacoustic (or optoacoustic) spectroscopy relies on the generation of
pressure waves by selective absorption of modulated light: chopped light of a suitable
wavelength is absorbed by the analyte and converted into heat. As a result, a change in the
pressure occurs in the measurement cell. This soundwave is detected with a microphone.
Depending on parameters such as the geometry of the measurement cell and the analyte,
gas concentrations in the low p.p.b. range as well as concentrations of nonvolatile solutes
can be detected. (9,10)
In what follows we refer to the voltage readout of the microphone as the photoacoustic
signal S. This voltage is proportional to the absorbed energy E a which can be calculated
using the Beer–Lambert law:
E 0 − E a = E 0 · exp(−α · c · l),
(4)
where E 0 is the energy of the incoming light pulse, α is the absorption coefficient, c the
analyte concentration, and l is the interaction length of the light beam with the sample. By
analogy with equation (4), the dependence of the photoacoustic signal S(C2 H5 OH) on the
ethanol partial pressure p(C2 H5 OH) can be described by: (9)
S(C2 H5 OH) = S0 · [1 − exp{−a · p(C2 H5 OH)}],
(5)
where p(C2 H5 OH) is the partial pressure of ethanol and a is a constant that contains
information about the ethanol absorption coefficient and the geometry of the measurement
cell. The quantity S0 is the photoacoustic signal for the (hypothetical) case where all
incoming light is absorbed.
2. Experimental
A schematic diagram of the instrument is shown in figure 1. The photoacoustic signal
caused by ethanol vapour was measured according to the following procedure: 30 cm3 of
an ethanol solution were placed in a thermostatted three-necked round bottom flask. To
transfer the ethanol vapour to the photoacoustic sensor, nitrogen (mole fraction purity =
0.9998, Carbagas SA) was bubbled at a flow rate of 30 cm3 min−1 through the liquid.
To obtain small bubbles, the nitrogen was introduced with a Pasteur pipette. As the
photoacoustic signal was independent of the gas flow rate up to 90 cm3 min−1 , the gas
phase was assumed to be in equilibrium with the solution. The gas stream was guided
into the photoacoustic cell by a Tefzel tube (inner diameter, 4 mm; length, 30 cm)
fitted with swagelock connections. To prevent adsorption or condensation of ethanol
on the inner tube walls, the tube was heated to T = 353 K. As no change in the
photoacoustic signal was observed, ethanol condensation was ruled out. The temperature
of the measurement cell was 323 K, well above the sample solution temperatures.
Condensation inside the measurement cell was therefore ruled out. All measurements
were carried out using a commercial photoacoustic ethanol analyser from Orbisphère
758
J. Ueberfeld et al.
PC
11
9,10
13
nitrogen containing
ethanol vapour
nitrogen
2
3
12
7, 8
1
(ethanol + water)
or
pure ethanol
6
5
4
photoacoustic sensor
FIGURE 1. Schematic diagram of the experimental setup: 1, thermostatted bath; 2, measurement cell;
3, reference cell; 4, infrared lamp; 5, mechanical chopper; 6, optical filter (9.4 µm); 7 and 8, ZnSe
windows; 9 and 10, microphones; 11, signal amplification unit; 12, inlet valve; 13, outlet valve.
Laboratories SA, Neuchâtel. The nonresonant photoacoustic system consists of a brass
block containing the measurement and reference cells. Both cells are equipped with a ZnSe
window and a microphone. The cell containing the reference gas is completely closed, but
the measurement cell is provided with inlet and outlet valves. The opening and closing
times of these valves are regulated by the internal software of the sensor.
Modulated light of wavelength 9.4 µm, corresponding to the ethanol C–O stretching
vibration is obtained by chopping the light of an infrared lamp (4 W, Aritron SA,
Switzerland) at a frequency of 10 Hz and passing it through an interference filter (NB 9400,
FWHM 200 nm, Spectragon AB, Däby, Sweden). An infrared lamp was used as the light
source because of cost considerations. However, as noise increases proportionally to the
source power, the use of a CO2 laser would not have significantly increased the signal-tonoise ratio.
In a typical measurement cycle the measurement cell was purged for 10 s with the
carrier gas containing ethanol vapour. The inlet and outlet valves were then closed and the
photoacoustic signal was measured 20 times within 15 s. The average value was calculated,
corrected for temperature and light intensity fluctuations by dividing it with the reference
signal, and finally transferred to a personal computer via a serial port. This measurement
cycle was repeated 20 times at each temperature.
Determination of Henry’s constant
759
CALIBRATION OF THE PHOTOACOUSTIC SENSOR
To determine the dependence of the photoacoustic signal on ethanol partial pressure
p(C2 H5 OH) six aqueous ethanol solutions were thermostatted at different temperatures,
and after a 30 min equilibration time the photoacoustic signal was determined. The
solution temperature was measured with a mercury column thermometer. The error in the
temperature reading was ±0.1 K.
As already mentioned, the dependence of S(C2 H5 OH) on p(C2 H5 OH) can be described
with equation (5). Division of S(C2 H5 OH) and S0 by the reference signal gives the
normalized photoacoustic signals s(C2 H5 OH) and s0 , which are independent of changes
in light intensity and temperature. Substitution of p(C2 H5 OH) in equation (5) with
m(C2 H5 OH) · kH {equation (1)} and taking into account the temperature dependence of
kH {equation (3)} leads to the following expression:
s(C2 H5 OH) = s0 · [1 − exp −[w · m(C2 H5 OH)/ exp[B · {(1/T ) − (1/298.1)}]]], (6)
where
w = a · kH (298.1 K),
(7)
B = 1tr H/R.
(8)
and
The parameters s0 , B, and w have been obtained by fitting the measured s(C2 H5 OH) values
to equation (6). The measured photoacoustic signal s contains an offset soff :
s = s(C2 H5 OH) + soff ,
(9)
caused by electronic noise, the window noise, and light absorption by water vapour. The
window noise is caused by light absorbed by the ZnSe window and subsequent emission
of acoustic waves. As with s(C2 H5 OH), the offset is corrected for temperature and light
fluctuations, even though the electronic noise does not depend on fluctuations of the
incoming light intensity. However, the error introduced was considered negligible.
The photoacoustic signal caused by the water vapour, s(H2 O), depends on temperature.
Hence, the offset soff was recorded at each temperature. For this purpose the flask was
filled with deionized water and the photoacoustic signal was recorded according to the
given procedure.
DETERMINATION OF HENRY’S CONSTANT
Henry’s constant at T = 298.1 K kH (298.1 K) was determined by using equation (7). The
parameter a was obtained by expanding the exponential part of equation (5) as a Taylor
series: (11)
s(C2 H5 OH) = s0 · a · p(C2 H5 OH) − s0 · a 2 · p 2 (C2 H5 OH)/2+ · · · .
(10)
For small ethanol vapour pressures (p(C2 H5 OH) <100 kPa) the higher order terms become
negligible and s(C2 H5 OH) depends linearly on p(C2 H5 OH). As one already knows s0 ,
a can be calculated from the gradient of the linear part of the s(C2 H5 OH) against
p(C2 H5 OH) curve.
760
J. Ueberfeld et al.
TABLE 1. Photoacoustic signal s(C2 H5 OH) for different (ethanol + water) of
ethanol molality m at temperatures between 273.1 K and 298.1 K. The experimental
error for s(C2 H5 OH) was estimated to be ±2 · 10−4 . The offset soff was determined
by purging the measurement cell with nitrogen saturated with water vapour
T /K
m/(mol · kg−1 )
soff
0.086
0.172
0.345
0.518
0.864
1.385
s(C2 H5 OH)
273.1
0.0024
0.0021
0.0048
0.0090
0.0129
0.0225
0.0369
278.1
0.0025
0.0035
0.0076
0.0148
0.0211
0.0364
0.0579
283.1
0.0025
0.0060
0.0125
0.0241
0.0349
0.0586
0.0916
288.1
0.0027
0.0094
0.0192
0.0371
0.0532
0.0880
0.1364
293.1
0.0031
0.0148
0.0296
0.0583
0.0837
0.1351
0.2051
298.1
0.0037
0.0231
0.0447
0.0875
0.1275
0.2016
0.2963
Small ethanol vapour pressures were generated by cooling pure ethanol down to
temperatures between 241.8 K and 254.6 K. The measurement procedure was the same
as for the sensor calibration. The vapour pressure p(C2 H5 OH) of the cooled ethanol was
calculated by using equation (11):
p(C2 H5 OH) = 2.852 · 106 · p0 · exp(−5197 K/T ).
(11)
Here p0 is the standard pressure (101.325 kPa) and T is the temperature. Equation (11)
is an adapted form of the formula given in reference 12. The error in p(C2 H5 OH) was
estimated to be ±1 per cent, assuming a temperature error of ±0.1 K and that the last digit
of the two numbers in equation (11) is uncertain. The offset soff , which depends on the
electronic noise and the window noise, was measured by purging the measurement cell
with nitrogen.
3. Results and discussion
The first step consisted in determining the photoacoustic signal s of the vapour over six
different ethanol solutions. This was done according to the above given procedure at
different temperatures between 273.1 K and 298.1 K. The photoacoustic signal s(C2 H5 OH)
was obtained according to equation (9). Table 1 gives the values of s(C2 H5 OH) and soff .
To prevent adsorption of gaseous ethanol on the wall of the Tefzel tube, the tube was
heated to T = 353 K during the measurement involving the 0.518 mol · kg−1 solution.
No change in the photoacoustic signal s was observed, i.e., no adsorption or condensation
took place on the wall of the tube. To check for ethanol depletion from the solution, the
photoacoustic signal s from the 0.518 mol · kg−1 solution at T = 298.1 K was recorded
continuously for 2 h. It remained stable for the whole period. To obtain s0 , B, and w, the
s(C2 H5 OH) values were fitted to equation (6). Figure 2 shows the measurement values and
the surface represented by equation (6). Table 2 gives the values and standard deviations
761
0.3
0.3
0.2
0.2
0.1
0.1
0.0
s (C2H5 OH)
s (C2H5 OH)
Determination of Henry’s constant
0.0
95
2
0
29
85
2
T/ K
0
28
5
27
0
27
1.4
1.2
1.0
0.8
m
0.6
0.4
0.2
–1 )
. kg
ol
/(m
•
FIGURE 2. Dependence of the photoacoustic signal s(C2 H5 OH) ( ) on ethanol molality m and
temperature T . The mesh corresponds to the surface represented by equation (6).
TABLE 2. Values and standard deviations of the parameters s0 , B, and w obtained
from the s(C2 H5 OH) values of table 1 by fitting s(C2 H5 OH) to equation (6). The
parameters a and kH (298.1 K) were determined from the linear domain of the
p(C2 H5 OH) against s(C2 H5 OH) curve
σ
Parameter
s0
0.806
σ (s0 )
B/K
7605
σ (B)/K
w/kg · mol−1
0.33
σ (w)/(kg · mol−1 )
0.01
a/kPa−1
0.375
σ (a)/kPa−1
0.003
0.877
σ {kH (298.1 K)}/(kPa · kg · mol−1 )
0.039
kH (298.1 K)/(kPa · kg · mol−1 )
0.002
41
σ of s0 , B, and w. In a second step the Henry’s constant at T = 298.1 K kH (298.1 K) is
determined from the gradient a of the linear domain of the p(C2 H5 OH) against s(C2 H5 OH)
curve. Figure 3 shows the linear dependence of s(C2 H5 OH) on small ethanol vapour
pressures p(C2 H5 OH).
The offset soff was determined by purging the measurement cell with pure nitrogen. The
values for a and kH (298.1 K) are given in table 2.
Literature values for kH (298.1 K) show large differences. Burnett used the bubble
column technique and obtained a kH of 0.466 kPa · kg · mol−1 . (13) Snider et al. (2) and
Rohrschneider (14) used static equilibrium methods that yielded Henry’s constants of
0.528 kPa · kg · mol−1 and 0.439 kPa · kg · mol−1 , respectively. Other reported kH values
762
J. Ueberfeld et al.
0.12
s (C2H5 OH)
0.10
0.08
0.06
0.04
0.10
0.15
0.20
0.25
0.30
p (C2H5 OH)/kPa
0.35
0.40
FIGURE 3. Photoacoustic signal s(C2 H5 OH) against ethanol vapour pressure p(C2 H5 OH). The
ethanol vapour pressure was calculated with equation (11). —–, linear fit to obtain a {from the linear
term of equation (10)} and kH (298.1 K) {from equation (7)}.
are 0.506 kPa · kg · mol−1 and 0.632 kPa · kg · mol−1 . (3,15) Published data based on
calculations are 0.477 kPa · kg · mol−1 (at T = 293.1 K) and 0.687 kPa · kg · mol−1 . (3)
A value of 0.824 kPa · kg · mol−1 has also been reported but without specification of the
determination method. (16)
Our value for kH (298.1 K) is relatively higher compared with the literature values which
means that the measured ethanol vapour pressures were relatively high. This indicates that
sorption losses from the gaseous phase, which are a common problem in the determination
of kH using other methods, were probably avoided here.
The temperature dependence of the Henry’s constant is described by B. As B is
positive, the enthalpy of the transfer of ethanol from the aqueous solution to the gaseous
phase is also positive, i.e., the process is endothermic. The volatility increases with
increasing temperature.
The temperature dependence of kH for ethanol has already been investigated by two
groups. Snider et al. (2) measured kH at only two different temperatures (273.1 K and
298.1 K) and found a value for 1tr H/R of 6700 K. Using calculated Henry’s constants,
a 1tr H/R of 6300 K was obtained. (3) As a calculation basis, four activity coefficients
measured between T = 297 and T = 333 K with the concurrent flow technique were
used. (4) For these measurements a relatively large temperature error of ±3 K was given.
An additional experimental problem was water loss during measurements at the upper
temperature limit.
Determination of Henry’s constant
763
The 1tr H/R value obtained here is somewhat higher than the literature data. This
follows the higher vapour pressures measured at higher temperatures, which are probably
due to the fact that sorption losses from the ethanol vapour phase were avoided.
CHARACTERIZATION OF THE SENSOR
The minimum temperature that could be reached with our thermostatted bath was 241.8 K.
This gave a minimum ethanol vapour pressure of 0.134 kPa {equation (11)}. However,
the detection limit of our sensor is lower. The minimum vapour pressure difference
between two measurement points was 0.001 kPa (see figure 3). This corresponds at
T = 298.1 K (kH = 0.877 kPa · kg · mol−1 ) to a molality difference of 0.001 mol · kg−1
of dissolved ethanol.
The sensor’s response time is very short: the maximum photoacoustic signal is reached
after only one measurement cycle. The selectivity of the sensor relies on absorption
at 9.4 µm, and any other substance that absorbs at this wavelength will disturb the
measurement. The presence of such substances should therefore be avoided.
4. Conclusion
Photoacoustic spectroscopy has been used to determine the Henry’s constant kH of
ethanol. The measured data indicate that Henry’s law is valid in the molality region from
0.1 mol · kg−1 to 1.4 mol · kg−1 of liquid ethanol.
The determination of kH with a photoacoustic sensor is simpler and faster compared with
other techniques such as gas chromatography and can be easily extended to other organic or
inorganic species. The results obtained for ethanol prove that photoacoustic spectroscopy
is a powerful tool for this purpose. They also indicate that adsorption of gaseous species,
which is a common problem in gas-phase concentration measurements, can be avoided. As
the sensor analyses the gas phase above ethanol solutions, its response depends strongly on
the temperature. Higher solution temperatures cause higher photoacoustic signals. Hence,
the sampling process should guarantee the highest possible temperature. However, care
should be taken to avoid adsorption or condensation of compounds in the tubes. That would
change the composition of the gaseous phase and give incorrect kH values.
At T = 298.1 K the sensor is able to measure liquid ethanol molalities of 0.001
mol · kg−1 . This meets the demands of the beverage industry. The processes of brewing or
yeast cultivation involve the measurement of ethanol concentrations in liquids that are more
complex that aqueous ethanol solutions. As additional solutes and surfactants do change
Henry’s constant, the present data are only valid for (water + ethanol). For more complex
systems, kH has to be determined separately.
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(Received 28 February 2000; in final form 2 October 2000)
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