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Onsager Heat of Transport Measured at the n-Heptanol Liquid−Vapor Interface
Abstract
The Onsager heat of transport Q* has been determined by measuring the effect of a temperature gradient at the interface on the apparent vapor pressure of the liquid. Results for n-heptanol extend to pressures that are low enough for the separation between the liquid and the warmer surface above it to be less than one mean free path. At low pressure, the dependence of ΔP on ΔT is found to be linear, in contrast to the curves obtained when the separation is greater than a mean free path. As with aniline, the heat of transport is negative and its magnitude approaches the latent heat of vaporization at low pressures. The results are discussed in relation to three anomalous effects that have been predicted or observed during steady-state evaporation and also in relation to the problem of determining rates of air−sea exchange for atmospheric gases.
... Experiments done more recently by the research group of Professor Leon Phillips at Canterbury University involve a liquid-vapour interface and result in much larger values of Q*. Measurements of the heat of transport at the interface have been carried out with aniline, n-heptanol, sulfuric acid, glycerol-water mixtures and most recently water and ice [23,27282930. The Q* values were all found to be negative, and tended towards the latent heat of vaporisation in magnitude, as n λ , the number of mean free paths in the vapour gap, tended towards zero [23, 29]. ...
... Measurements of the heat of transport at the interface have been carried out with aniline, n-heptanol, sulfuric acid, glycerol-water mixtures and most recently water and ice [23,27282930. The Q* values were all found to be negative, and tended towards the latent heat of vaporisation in magnitude, as n λ , the number of mean free paths in the vapour gap, tended towards zero [23, 29]. The vapour gap itself is the gap between the liquid surface and the upper plate of the cell, as shall be described in detail in Section 2.Figure 1 shows the dependence of the magnitude of Q* for n-heptanol on n λ [31]. ...
... After initial experiments with aniline [23] had established the existence of large, negative heats of transport at the liquid-vapour interface, n-heptanol was studied because it had a low triple point vapour pressure. The low triple point vapour pressure allowed for the measurement of Q* values for δ values less than one mean free path, so measurements of P versus ΔT for n-heptanol could be performed at very low pressures [29] and values of |Q*| amounting to more than 80% of the heat of vaporisation were obtained. Next sulfuric acid-water and the glycerol-water mixture were chosen in order to see how binary mixtures compared to the one-component systems. ...
The Onsager heat of transport for p-tert-butyltoluene was measured, as part of a series of preliminary experiments towards the determination of the importance of temperature gradients on the air-sea flux of carbon dioxide. The results presented in this thesis imply that the temperature gradient is a major contributor to the magnitude of the air-sea flux. The heat of transport has been measured for the p-tert-butyltoluene system by measuring stationary-state pressure changes for known temperature differences on the vapour side of the interface. At the pressure ranges used the number of mean free paths was always outside the Knudsen zone, but the values of Q* were approximately 100 % of the latent heat of vaporisation. Departures from linearity of plots of P against ΔT are attributed to temperature jumps at the surface of the dry upper plate. Both the results taken for p-tert-butyltoluene and the earlier results for water from this laboratory fit to a Type III BET isotherm, where the c parameter is not constant. They also reveal the importance of the temperature gradient in determining the value of the thermal accommodation coefficient, and provide a new method of measuring thermal accommodation coefficients for a variety of surfaces and vapours
... The Dufour and Soret effects are contained within the cross-coefficients r qi and l qi that both can be expressed in terms of the heats of transfer. Many methods are presented in the literature [13][14][15][16][17][18][19][20] to measure or predict the heat of transfer in liquid solutions, solids, liquid-vapor interfaces, and so forth. ...
... The heat of transfer by one component not only depends on its own concentration but also on the other concentrations, as also follows from eq 4. It demonstrates that in a multicomponent system different components interact with each other, which makes the system more complex than the binary system. The calculated heats of transfer here are comparable to the values obtained by Mill et al. 19 for a liquid mixture (in the work by Mill et al., the definition of the heats of transfer are a factor R different from ours). ...
The surface temperature and surface mole fractions are calculated for a catalytic hydrogen oxidation reaction over a Pt/Al2O3 catalyst pellet. The thermodynamics of irreversible processes was used in order to ensure the correct introduction of coupled heat and mass transfer. Two pathways, one using the 4 x 4 resistivity matrix and the other using a simplified effective conductivity matrix, were proven to yield equivalent results. By using expressions for the thermal diffusion coefficients, heats of transfer, and the Maxwell-Stefan diffusion coefficients given in the literature, available experimental data could be reproduced. The Dufour effect was found to be negligible for the prediction of the surface temperature. Neglecting the Soret effect would increase the predicted value of the surface temperature significantly-more than 30 K out of an average of about 400 K. It is found that the reaction rate can be used to predict the surface temperature.
... These resistance coe¢ cients can be expressed in terms of heats of transport for the individual e¤ects. Many methods are presented in the literature [43,44,45,46,47,48,49,50] to measure or predict the heat of transport in liquid solutions, solids, liquid-vapour interfaces, etc. Here we shall use an expression for the heat of transfer analogous to the one by Taylor and Krishna for the continuous case [31], see also [23]. ...
In this dissertation we study the surface temperature excess in heterogeneous catalysis. For heterogeneous reactions, such as gas-solid catalytic reactions, the reactions take place at the interfaces between the two phases: the gas and the solid catalyst. Large amount of reaction heats are released at the interface for strong exothermic reactions, which makes it possible that the catalyst surface temperatures are higher than those in the gas phases due to poor thermal conductivities of gases. These are very often the case for transport limited reactions based on the literature and the temperature differences can be a few hundred degrees. It is often found in kinetic studies that Arrhenius plots are gradually curved when measured catalyst temperatures or gas temperatures are applied. It seems that the activation energy changes gradually with increasing temperature. However, according to Arrhenius, the activation energies should be constant. Therefore, there is a possibility that the real reaction temperature in a 2-D reacting surface differs from the measured catalyst temperature or gas temperature and that the real reaction temperature Tr should be used in the Arrhenius plot. Using irreversible thermodynamics three distinct temperatures, gas temperature, solid catalyst surface temperature and reaction temperature in a 2-dimensional Gibbs surface, are modelled for two oxidation reactions. They are one transport limited reaction: hydrogen oxidation, and one kinetically controlled reactions: CO oxidation, respectively
The effect of a surface temperature gradient on the water vapour pressure over glycerol–water mixtures has been measured for glycerol concentrations in the range from 80% to 94.5% by weight. Graphs of ΔP versus ΔT are linear, for both positive and negative ΔT, at thicknesses of the vapour layer up to at least eight mean free paths. Compound formation, which for sulfuric-acid/water caused the measured heat of transport to decrease markedly at high acid concentrations, has no discernible effect on the measured heat of transport for glycerol–water mixtures.
The effect of a surface temperature gradient on the water vapour pressure over aqueous sulfuric acid has been measured for acid concentrations in the range from 33% to 80% by weight. Graphs of [Delta]P versus [Delta]T are linear, for both positive and negative [Delta]T, at thicknesses of the vapour layer up to at least 10 mean free paths. At high acid concentrations, the rate of evaporation is limited by the rate of dissociation of sulfuric acid monohydrate, with the result that the effect of the temperature gradient, and hence the measured heat of transport, is greatly reduced.
Recent measurements of the Onsager heat of transport Q∗ for ammonia at the surface of water have found Q∗ to be constant for gas pressures over the range 10–900 Torr. This result cannot be understood in terms of a molecular-scale model for the origin of the heat of transport at a gas–liquid interface, which predicts that Q∗ should decrease with increasing pressure, and must be regarded as a direct consequence of Onsager’s reciprocal relation for coupled heat and matter fluxes.
The ultimate magnitude of global warming is likely to be limited mainly by the rate of absorption of CO2 by the oceans. Model calculations of this limit need to take account of the effect of the temperature gradient at the ocean surface on the effective saturation pressure of CO2 above that surface. We present a description of this and other unfamiliar phenomena in terms of Onsager’s irreversible thermodynamics.
The Onsager heat of transport Q∗ at the liquid–vapour interface of water is constant at −24.3±3.6kJmol−1 (two standard deviations) over the pressure range from 4 to 9.5Torr. This result is consistent with a model in which Q∗ is determined by the temperature gradient across the Knudsen layer adjacent to the liquid, but not with one based on perturbed temperature profiles from temperature-jump theory, which predicts that the measured heat of transport should vary inversely with pressure. It implies that the Onsager heat of transport should be significant for gas–liquid exchange processes even at atmospheric pressure.
The Onsager heat of transport at a gas-liquid interface is obtained as a consequence of the requirement of detailed balance for the fluxes of gas molecules arriving at and departing from the liquid surface, together with the assumption that the effect of the temperature gradient on the free energy barrier for evaporation is given by DeltaG(vap)(1) = DeltaG(vap) - etaDeltaTDeltaS(vap), where DeltaT is the temperature, difference across a thin layer of vapour adjacent to the surface of the liquid and eta is a transfer coefficient.
The Onsager heat of transport, Q *, for
nitrous oxide at the surface of water has been measured by using
high-resolution diode-laser spectroscopy to monitor the partial pressure
of N2O as a function of the temperature difference across a
5-mm vapour gap above the water surface. The metal surface above the
vapour gap was conditioned with ammonia prior to the measurements, to
enable efficient thermal accommodation at that surface, which allowed
the temperature difference to be as large as 15 K. We find
Q * = -6.6 ± 0.85 kJ
mol-1, so that
Q */ RT is
-2.9 at T = 275.15 K. These
experiments are a necessary preliminary to a planned series of
measurements of Q * for CO2 at
the surface of simulated sea water.
The Onsager heat of transport Q * for
ammonia at the surface of water was measured at ammonia partial
pressures from 12 to 875 Torr and found to be independent of pressure
over that range, with an average value of -7.7 ± 2.8 kJ
mol-1 (95% confidence limits). Plots of
Δ P versus
Δ T initially showed a change of
slope, or “knee”, at Δ T ~
0.5 °C. After the stainless-steel Onsager cell had been conditioned
by exposure to several hundred Torr of ammonia, plots of
Δ P versus
Δ T were linear up to
Δ T ~ 6 °C. The conditioning is
reversible, and is attributed to the effect of chemisorbed ammonia on
thermal accommodation coefficients at the metal surface. The results are
discussed in relation to the free-energy barrier to evaporation at the
surface of a liquid, and in relation to field measurements of the rates
of air-sea exchange of greenhouse gases.
Topics discussed in this article include: the structure of a gas–liquid interface; the nature of the small-scale motions of a liquid surface; a brief summary of some of the important results of beam experiments; the use of statistical mathematics to predict the efficiency of energy transfer during collisions of fast, light atoms with liquid surfaces, to account for the marked difference in energy transfer efficiency between ordinary liquids and liquid metals, and to predict the distribution of scattering angles for fast neon atoms striking a liquid surface; the application of Onsager's non-equilibrium thermodynamics to a gas–liquid interface; explanations for some paradoxical phenomena involving evaporation and condensation; experimental measurements of the Onsager heat of transport at the gas–liquid interface, and the significance of non-equilibrium thermodynamics in relation to air–sea exchange of carbon dioxide and global warming.
The Onsager heat of transport Q * for carbon
dioxide at the surface of water has been determined by measuring the
total gas pressure as a function of the temperature difference across a
6-mm vapour gap above the water surface, and subtracting previously
measured water-vapour pressures. The metal surface above the
vapour gap was conditioned with carbon dioxide prior to the
measurements, to enable efficient thermal accommodation for
CO2 molecules striking that surface, so that plots of
ΔP against ΔT were linear for ΔT values up to at least
6 K. For liquid surface temperatures ranging from 0 to 5°C, we find
Q * = -6.9 ±1.7 kJ
mol-1. Within experimental error, the results were
independent of pH over the range from 6.50 to 9.04, and of
CO2 partial pressure between 13 and 46 Torr. The factor
Q * / RT is
-3, which implies that the fractional temperature gradient is
at least three times as important as the
fractional partial-pressure gradient in controlling the magnitude and
direction of the steady-state CO2 flux through a water
surface.
[1] Recent laboratory measurements have shown that the stationary-state vapor pressures of aniline and n-heptanol are enhanced by the application of a positive temperature gradient in the vapor, that the vapor pressure of water over 50 percent sulfuric acid is enhanced by a positive temperature gradient and diminished by a negative temperature gradient, and that values of the Onsager heat of transport derived from these measurements are of similar magnitude to the latent heats of vaporization. When the vapor gap over which the temperature difference is applied is less than about 0.8 mean free paths, the liquid behaves as though its surface were at the temperature close to that of the gas on the other side of the Knudsen layer. These results are discussed in relation to field measurements of the rate of air-sea exchange of carbon dioxide.
The non-linearity of plots of ΔP versus ΔT in measurements of the Onsager heat of transport at a gas–liquid interface was previously attributed to a difference in thermal accommodation coefficients between dry and wetted states of the solid surface at which ΔT was applied. This implies that the accommodation coefficient should be a measure of the fractional coverage of the upper surface, as described by Brunauer–Emmett–Teller (BET) theory. A BET treatment of our accumulated ΔP versus ΔT data has given values for the average number of layers of adsorbed liquid in the wetted area for aniline, n-heptanol, water, and p-tert-butyltoluene.
Recent work on the gas-liquid interface is discussed with emphasis on theoretical studies of the small-scale motion of the capillary-wave zone, experimental and theoretical studies of the ultra-small-scale surface roughness of liquids, experimental measurements of the Onsager heat of transport at a gas-liquid interface, the likely origin of the heat of transport at a molecular level, and the resolution of several types of paradoxical behavior that have been observed or predicted to occur at a liquid-vapor interface.
The Onsager heat of transport Q* has been measured for water vapour at the surface of water, supercooled water, and ice, over the temperature range -8 to +10 degrees C. For liquid water, Q* is constant at -24.7 +/- 3.6 kJ mol(-1) (two standard deviations) over the pressure range 4-9.5 Torr. Provided the ice is suitably aged, the |Q*| values are very similar for water and ice, a result which is consistent with the presence of a liquid-like layer at the surface of ice. The values are slightly larger for ice, in proportion to the ratio of the heat of sublimation of ice to the heat of vaporization of the liquid. Departures from linearity of plots of P against DeltaT are attributed to temperature jumps at the surface of the dry upper plate. Hence jump coefficients and thermal accommodation coefficients have been derived as a function of temperature for collisions of water molecules with type-304 stainless steel.
In this chapter we attempt to present a brief introduction to the subject of air-sea gas exchange. First the basic equations governing such exchange are given, then a review of some models proposed to describe the gas transfer process. Following this, experimental approaches through both laboratory (principally using wind/water tunnels) and field measurements are summarised. Finally, we present what seems to us to be the best current synthesis of the wind tunnel and field results for the prediction of gas exchange rates across the sea surface.
Recent measurements of the temperature profile across the interface of an evaporating liquid are in strong disagreement with the predictions from classical kinetic theory or nonequilibrium thermodynamics. However, these previous measurements in the vapor were made within a minimum of 27 mean free paths of the interface. Since classical kinetic theory indicates that sharp changes in the temperature can occur near the interface of an evaporating liquid, a series of experiments were performed to determine if the disagreement could be resolved by measurements of the temperature closer to the interface. The measurements reported herein were performed as close as one mean free path of the interface of an evaporating liquid. The results indicate that it is the higher-energy molecules that escape the liquid during evaporation. Their temperature is greater than that in the liquid phase at the interface and as a result there is a discontinuity in temperature across the interface that is much larger in magnitude (up to 7.8 °C in our experiments) and in the opposite direction to that predicted by classical kinetic theory or nonequilibrium thermodynamics. The measurements reported herein support the previous ones.
Recent measurements of the temperature at the interface of an evaporating liquid have been found to be in conflict with the predictions of classical kinetic theory. Under 20 different experimental conditions for water evaporation, the temperature in the vapor at the interface was measured to be greater than that in the liquid at the interface and this relation between the interfacial temperatures is the opposite to that predicted from classical kinetic theory. When these same data were used to examine the statistical rate theory (SRT) expression for the liquid evaporation rate, almost complete agreement was found. This theoretical approach is based on the transition probability concept, as defined in quantum mechanics, a hypothesis that assumes the rate of exchange between the possible quantum mechanical states of an isolated system that are within the energy uncertainty has the same value, and the Boltzmann definition of entropy. To determine whether the SRT expression for the evaporation rate also describes the liquid-vapor phase transition for liquids other than water, two hydrocarbons have been examined. The agreement between the predictions from SRT and the measurements is equally as good. These results raise the question of whether a quantum mechanical description is essential to describe the condition existing at the interface of an evaporating liquid.
In a series of experiments, a temperature discontinuity has been found to exist at the surface of an evaporating water droplet. Statistical rate theory has been used to predict the pressure in the vapor to within the experimental uncertainty during each of the experiments. While the qualitative trend of the D 2 law is observed to be consistent with the measurements, it underpredicts the measured rate of evaporation by 21%–37%. When the temperature discontinuity is taken into account in the D 2 law, the difference between the predicted and measured values is at most 7%. The results suggest that the rate limiting process in the experiments is not diffusion in the gas phase, as is assumed in the D 2 law, but is the interface kinetics. © 2002 American Institute of Physics.
Steady-state evaporation and condensation experiments have been conducted with water under conditions where buoyancy-driven convection is not present. The temperature profile in each phase has been measured. At the interface, independently of the direction of the phase change, a temperature discontinuity has been found to exist in which the interfacial vapor temperature is greater than that in the liquid. In a thin layer immediately below the interface the temperature is uniform in a layer ( approximately 0.5 mm) and below that the temperature profile is linear, indicating thermal conduction. The uniform temperature layer indicates a mixing process occurs near the interface that could result from surface-tension driven (Marangoni-Bénard) convection and/or from "energy partitioning" that is necessary to account for the measured temperature discontinuity near the interface. When the measured interfacial properties are used with the expression for the phase change rate that is obtained from statistical rate theory, it is found that the predictions are in close agreement with the measurements.
We report the condensation of aniline vapour upon a surface which is significantly warmer than the liquid from which the vapour originated, and show that this apparently paradoxical behaviour is predictable on the basis of our recent measurements of the Onsager heat of transport at the aniline liquid–vapour interface. In the previous experiments, which for experimental reasons were restricted to positive temperature gradients, we measured the increase in vapour pressure produced by the applied temperature gradient. The present results imply that a negative temperature gradient at the surface produces a similar decrease in the vapour pressure.
The thin thermal layer at the surface of the ocean can lead to an underestimate of CO2 solubility. We re-evaluate the effect of the cool thermal skin and present a high-resolution seasonal estimate of its effect on the air-sea flux of CO2. We compare air-sea flux estimates derived using both a mean wind field and a more realistic Rayleigh distribution of the wind field. Using the mean monthly wind stress and a linear relationship between wind speed and the gas exchange coefficient of CO2 (Tans et al., 1990), we estimate that excluding the southern ocean, the surface skin correction increases the air-sea flux of carbon by 0.48 Gt yr-1. -Authors
The motion of a vapor gas in contact with interphase surfaces is studied on the basis of kinetic theory of gases. The mass and energy fluxes are determined from the condition of the vapor far away from the interfaces. The result for a vapor gas between two interfaces shows that the temperature gradient in the vapor is opposite in sign to the externally maintained temperature difference.
Equations are derived for the steady-state heat and matter fluxes through the boundary layer at a general phase interface; their range of validity is discussed, and the importance of the timescale of the observations is emphasized. Where only diffusive matter fluxes and conductive heat fluxes are involved, the results derived from irreversible thermodynamics remain valid for large departures from equilibrium. The theory includes the case where the location of the boundary layer is marked by an obstruction such as a membrane: the properties of the obstruction control the magnitudes of the fluxes but do not affect the thermodynamic forces. The theory is applied to transport processes at a plane liquid surface and at the surface of an aerosol droplet.
Using recent laboratory and field results we explore the possibility of a cubic relationship between gas exchange and instantaneous (or short-term) wind speed, and its impact on global air-sea fluxes. The theoretical foundation for such a dependency is based on retardation of gas transfer at low to intermediate winds by surfactants, which are ubiquitous in the world's oceans, and bubble-enhanced transfer at higher winds. The proposed cubic relationship shows a weaker dependence of gas transfer at low wind speed and a significantly stronger dependence at high wind speed than previous relationships. A long-term relationship derived from such a dependence, combined with the monthly CO2 climatology of Takahashi [1997], leads to an increase in the global annual oceanic CO2 uptake from 1.4 Gigaton Cyr-1 to 2.2 Gigaton Cyr-1. Although a cubic relationship fits within global bomb-14C oceanic uptake constraints, additional checks are warranted, particularly at high wind speeds where the enhancement is most pronounced.
Recent measurements of the Onsager heat of transport at the aniline liquid–vapour interfaced revealed phenomena which are not considered by the standard gas-kinetic treatments of evaporation and condensation. Here we show that these experiments support the existence of the inverted temperature profiles that have been predicted by theoretical treatments of the two-surface problem.
An apparatus is described for the measurement of the thermo-osmotic effect of carbon dioxide, nitrogen, hydrogen and water vapour through a natural rubber membrane. The existence of the effect is demonstrated in all four cases. The experimental results are in accord with the theoretical expressions of part I concerning the rate of penetration through the membrane and the pressure ratio at the stationary state. The calculated values of the heat of transport are discussed with special reference to a molecular-kinetic theory of diffusion in quasi-crystalline lattices.
The process of thermo-osmosis is the passage of a fluid through a membrane due to a temperature gradient. Under suitable conditions it gives rise to a stationary difference of pressure. The thermo-osmosis of a gas through a membrane in which it is slightly soluble is due partly to the temperature coefficient of its solubility and partly to the existence of a thermal diffusion process inside the membrane. A theory is developed on the basis of Onsager's treatment of irreversible processes and leads to equations giving the rate of permeation and the pressure ratio at the stationary state. The magnitude of the effect depends on the algebraic sum of the heat of solution and the heat of transport within the membrane.
Molecular-dynamic studies of the behavior of the diffusion coefficient after a long time s have shown that the velocity autocorrelation function decays as s-1 for hard disks and as s-3/2 for hard spheres, at least at intermediate fluid densities. A hydrodynamic similarity solution of the decay in velocity of an initially moving volume element in an otherwise stationary compressible viscous fluid agrees with a decay of (ηs)-d/2, where η is the viscosity and d is the dimensionality of the system. The slow decay, which would lead to a divergent diffusion coefficient in two dimensions, is caused by a vortex flow pattern which has been quantitatively compared for the hydrodynamic and molecular-dynamic calculations.
The phenomena that are encapsulated in the Onsager heat of transport at a liquid−vapor interface are modeled in terms of processes occurring in the capillary-wave zone at the liquid surface. The effect of localized heat release in the outer part of the capillary-wave zone is in the right direction but is too small to account for the observations. A more successful approach considers the effect of a temperature gradient on the effective barrier height for escape from the liquid. The results imply that evaporating molecules must surmount a free-energy barrier rather than an enthalpy barrier and that the detailed dynamics of barrier crossing play an important role.
Clarifications of the interpretation of the COâ eddy flux measurements of Smith and Jones (1985) are offered in the context of criticisms by Broecker et al. Downward pumping of COâ and other atmospheric constituents by the pressurization of air entrained by breaking waves is considered to be a plausible mechanism. Apparent discrepancies between short-term eddy flux surface exchange rates and longer-term isotopic determinations can be resolved if short-term storage of dissolved gas in upper water layers is considered to influence only the former.
The heat of transport for passage of matter through a gas–liquid interface has been determined for the aniline li-quid–vapour system, by measuring stationary-state pressure differences produced by known temperature differences over a distance of 2 mm on the vapour side of the interface. For the range of pressures used, 2 mm is between 6 and 34 mean free paths. Coupling of the heat and matter fluxes is significant over the whole of this range. At the higher pressures the heat of transport is more than 20% of the heat of condensation; at the lower pressures it is more than 50%. Ó 2002 Elsevier Science B.V. All rights reserved.