Environmental Management Avoid Common Pitfalls When Using Henry’s Law Francis L. Smith Wilcrest Consulting Here is practical guidance on the use of Associates, LLC Henry’s law in engineering calculations and Allan H. Harvey how to account for the temperature dependence National Institute of of the Henry’s constant — which is often ignored Standards and Technology but which can be crucial for accurate process design. enry’s law and the Henry’s constant are widely used sure, and the solvent may be a liquid, a vapor, a supercriti- in chemical and environmental engineering. Unfor- cal fluid, or even a solid. Most applications, however, Htunately, many people do not appreciate that the involve a liquid solvent. Furthermore, it is usually conven- Henry’s “constant” is not a true constant, but instead has a ient to define Hi at a pressure equal to the solvent’s vapor significant non-linear temperature dependence. Ignoring this pressure. (Reference 1 explains how to use these values at temperature dependence can lead to serious inaccuracies. higher pressures.) When restricted to the solvent’s vapor- This article discusses the temperature dependence of the pressure curve, Hi becomes a function of temperature only. Henry’s constant and suggests ways of estimating this For environmental work at pressures near ambient, this dependence when data are limited or not available. It also “restricted” Henry’s constant is adequate. (The pressure briefly discusses other common pitfalls to be avoided when dependence is ignored in the remainder of this article.) using Henry’s law. The focus is on dilute aqueous systems, More-complex methods covering wide ranges of tempera- such as those found in environmental applications, but much tures and pressures, such as those encountered in geochem- of the information is also appropriate for other applications. istry, may be found elsewhere (4–6). Since real systems never reach the limit of infinite dilu- Thermodynamic basics tion, most applications of Henry’s law involve the use of CEP published a series of articles by Carroll (1–3) cov- Eq. 1 without the infinite-dilution limit. The assumption ering many aspects of Henry’s law and its uses in chemical that fi /xi remains constant for non-zero values of xi is rea- engineering. This article will not repeat the material pre- sonable for many environmental applications, where the sented there, but will summarize a few key concepts that goal is to describe the solubility or volatility of small are relevant to this discussion. amounts of contaminants in water. In a given solvent, the Henry’s constant for solute i, Hi, With the proportionality between fi and xi established by is defined as an infinite-dilution limit: Henry’s law for one phase (e.g., an aqueous phase being purified), one needs only a model for fi in the other phase in fi Hi = lim (1) order to design a separation process based on phase equilib- xi →0 x i rium. For steam or air stripping, where the coexisting phase where fi and xi are the fugacity and the mole fraction of the is a vapor, an ideal-gas assumption is often sufficient, or solute, respectively. (Variables other than mole fraction are corrections for nonideality may be added. If the coexisting sometimes used to describe the amount of solute; this will phase is a liquid (for example, in liquid-liquid extraction), be discussed later.) Raoult’s law is the simplest assumption; this may be cor- Equation 1 may be applied at any temperature or pres- rected by a liquid-phase activity model as appropriate. CEP September 2007 www.aiche.org/cep 33 Environmental Management sat Nomenclature aqueous phase, and pi is the vapor pressure of pure com- ponent i at the temperature of the solution. A comprehensive Ci = any of various measures of composition in the collection of mutual solubility data for hydrocarbons with liquid phase (various units may be used) water and seawater is available in Ref. 7. ∆Cp,sol = heat capacity of solution, J/mol-K f = fugacity of solute i, MPa For solutes with large or complete miscibility with the i solvent (such as lower alcohols with water), the determina- Hi = Henry’s constant for solute i, MPa Hi* = the “solubility form” of the Henry’s constant for tion of Henry’s constant must again come from Eq. 1, but solute i, MPa–1 full analysis of the phase equilibrium is typically required. ∆Hsol = enthalpy of solution, J/mol This usually involves fitting a liquid-activity model to ∆Hvap = enthalpy of vaporization, J/mol experimental data and extrapolating the solute fugacity back sat pi = vapor pressure of pure component i at the temper- to infinite dilution with the model. Equation 1 ultimately ature of the solution, MPa R = molar gas constant = 8.314472 J/mol-K reduces to: T = temperature, K inf sat Hpiii= γ (3) Tc = critical temperature of the solute, K inf T0, T1 = initial and final temperatures for an extrapolation where γi is the infinite-dilution value for the activity coef- of Hi, K ficient of solute i in the solvent. xi = mole fraction of solute i An engineer who cannot find a tabulated Henry’s con- aq xi = mole fraction of the organic solute in the aqueous stant is not necessarily stymied. If good phase-equilibrium phase org data exist for the solute-solvent pair at the temperature of xw = mole fraction of water dissolved in the organic phase interest, a reliable value of Henry’s constant can be derived. Greek Letters Temperature dependence inf γi = infinite-dilution value for the activity coefficient of solute i in the solvent, dimensionless As stated earlier, the Henry’s “constant” is not constant with respect to temperature. Figure 1 illustrates some typi- cal behavior of Hi(T) for several solutes in water. (Other Where do Henry’s constants come from? aspects of Figure 1 will be discussed later.) The values for A review of the many data sources for Henry’s con- CO2 are taken from the correlation of Fernández-Prini, et al. stants is beyond the scope of this article. However, it is (8), while those for other solutes come from the correlating helpful to understand how these data are obtained. Henry’s equations used by de Hemptinne, et al. (9). These are not constants cannot be measured simply by placing a meter necessarily the best values in the range plotted, but only the into a solution. Rather, they are derived indirectly from qualitative behavior is important for this discussion. experiment via Eq. 1. The Henry’s constant typically increases with tempera- For a sparingly soluble gaseous solute, the determina- ture at low temperatures, reaches a maximum, and then tion of Hi is straightforward from Eq. 1 (although the decreases at higher temperatures. The temperature at which experiments may be difficult). Measurements of the solu- the maximum occurs depends on the specific solute-solvent bility xi at low gas partial pressures yield Hi directly (per- pair. As a rule of thumb, the maximum tends to increase haps after small corrections for gas nonideality to improve with increasing solute critical temperature for a given the value of fi). solvent and with increasing solvent critical temperature for For a liquid solute that is nearly immiscible with the sol- a given solute. vent, such as toluene with water, use of Eq. 1 is again Clearly, the use of a Henry’s constant that was derived at straightforward. For the case where component i forms an 25°C at a different temperature could lead to serious design organic-rich phase that is nearly immiscible with an aque- errors. Even a variation as small as 10 K can cause the ous phase, the solute fugacity in the solute-rich phase Henry’s constant to change by a factor of two, which could (which must be equal to that in the solvent-rich phase) can have a serious impact on many designs. be approximated by Raoult’s law: In some cases, reliable values for Henry’s constants are available over the entire temperature range of interest. For org sat (1 − xpw i example, for common gases in water, the correlation of H = ) 2 i aq ( ) Fernández-Prini, et al. (8) extends from 0°C to the critical xi point of water. In other cases, solubility data are available org where xw is the mole fraction of water in the organic so that Henry’s constants can be derived by the methods aq phase, xi is the mole fraction of the organic solute in the described in the previous section. But in many cases, only a 34 www.aiche.org/cep September 2007 CEP 100,000 zontal axis has units of degrees Celsius, it is actu- n-Octane ally linear in inverse absolute temperature (but reversed, so that temperature increases along the 10,000 axis from left to right). Methyl-cyclohexane As expected from the Clausius-Clapeyron equa- sat 1,000 tion, log10pi is nearly linear in inverse tempera- CO2 inf Tc = 30.9°C ture. The temperature dependence of γi is mod- est near ambient temperature, so that the tempera- Toluene 100 ture variation of Hi is primarily a consequence of sat inf the variation in pi . At higher temperatures, γi Henryʼs Constant, MPa Henryʼs varies much more rapidly, so that the temperature 10 variation of Hi becomes more strongly dependent sat inf on the combined variations of both pi and γi . 1 The behavior noted in Figure 2 suggests an 20 40 60 80 100 120 140 160 180 extrapolation procedure. Especially for small Temperature, °C temperature intervals near ambient conditions, inf I Figure 1.