Henry's Law: A Retrospective - Journal of Chemical Education (ACS

Research: Science and Education

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Henry’s Law: A Retrospective Robert M. Rosenberg* Department of Chemistry, Northwestern University, Evanston, IL 60208-3113; *[email protected] Warner L. Peticolas Department of Chemistry, University of Oregon, Eugene, OR 97403-1253

In 1803, William Henry published in the Philosophical Transactions of the Royal Society of London his article on the solubility of gases in water (1). His experiments established the law, now known as Henry’s law, that the solubility of a gas is proportional to the pressure of the gas. Now that we are observing the 200th anniversary of that article, it seems worthwhile to consider the substantial evolution of that law over the intervening 200 years to include changes in the treatment of the solubility of gases, application to the vapor–liquid equilibrium of a binary liquid solution, application to the thermodynamics of a binary solid solution, and the use of Henry’s law to define the standard state of the solute in liquid and solid solutions. We also want to review the question: Is there sufficient experimental evidence to demonstrate that Henry’s law applies in any specific system at finite concentration or whether it is only a limiting law valid at zero concentration of solute? Early History In the first 73 years of its existence, Henry’s law was applied only to the effect of pressure on the solubility of a gas in a liquid, or as Henry put it, “the influence of pressure in accomplishing this strong impregnation”, referring to the solubility of carbon dioxide in water. Henry included no equations in his article, only tables of data. If he had included an equation, it would probably have been (in our notation), (1)

X2 = k p2

where f2 is the fugacity of the vapor. Henry’s Law and the Solubility of Gases A good recent study of the applicability of Henry’s law to the solubility of gases is Giacobbe’s measurement of the solubility of CO2 in acetone (2). Giacobbe concluded that Henry’s law is followed for the solubility of CO2 in acetone because he obtained a straight line passing through 0 in a plot of fugacity against mole fraction; when pressure was plotted against mole fraction, a linear fit was not as good as a quadratic fit. His data are shown in a plot of fugacity against mole fraction in Figure 1. The straight line in Figure 1 is from a least-squares fit of fugacity against mole fraction of CO2 in the solution. An f-test (3) showed that the addition of a quadratic term to the polynomial did not improve the fit. If the data in Figure 1 fit a straight line through zero, a sensitive test is to plot f兾X against X; such a plot should be a horizontal line. The results of such a plot are seen in Figure 2. The first five points were not included in the least-squares fit, because they varied systematically and not randomly from the line derived from the last six points. We assume that there was a small

10

65

8

60

4

50

45

2

0 0.00

55

XCO2

6

(2)

X 2 = k f2

fCO2 / atm

fCO2 / atm

where X2 is the mole fraction of the dissolved gas and p2 is

the partial pressure of the gas above the liquid. In fact, the determination of Henry’s law constants for the solubility of gases in liquids is still being investigated (2). If we want to take into account the nonideality of the gas at higher pressures, we can write eq 1 as,

0.05

0.10

0.15

0.20

40 0.00

0.05

0.15

0.20

XCO2

XCO2 Figure 1. A plot of the fugacity of CO2 in equilibrium with a solution in acetone at 30 C. Data from Giacobbe, ref 2: experimental points, ---- linear fit, slope = 51.3 ± 0.2.

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0.10

•

Figure 2. A plot of f/X for CO2 in acetone. Data from Giacobbe, ref 2: experimental points, ---- weighted linear fit of last six points; y = 51.5 ± 0.3 − (0.02 ± 2)x.

Vol. 81 No. 11 November 2004

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systematic error in the most dilute solutions. The linear regression for f兾X must be weighted because the ratio no longer can be assumed to have the same weight for all points (4). The slope of the line in Figure 2 is equal to zero within experimental error, which is consistent with the linearity of the line in Figure 1. Also, the slope of the line in Figure 1 is equal, within experimental error, with the intercept in Figure 2. Thus, the experiment is consistent with Henry’s law. Henry’s Law and Solutes in Dilute Solutions J. Willard Gibbs (5) was the first to relate the chemical potential of a solute in condensed phase dilute solution to the concentration in the way that we now do. He concluded that the chemical potential of the solute in dilute solution is proportional to the natural logarithm of the concentration of the solute. In our current notation, we also take into account the standard chemical potential, as in eq 3, µ 2 ( cond ) = µ 2° ( cond ) + R T ln X 2 ( cond )

(3)

where µ2 is the standard chemical potential, component 2 is the solute, and (cond) refers to the condensed phase. If we consider the equality of the chemical potentials of solute in solution and the solute vapor, µ 2 ( cond ) = µ 2 (g) = µ2° ( g ) + R T ln

p2 P °

(4)

where P is the standard pressure, usually 1 bar, we obtain Henry’s law in the form (5)

p2 = KH X2 where, KH =

p2 = P ° e X2

µ 2° ( cond ) − µ2° ( g ) RT

(6)

is the Henry’s law constant, which is characteristic of a par-

ticular solute in a particular solvent. Notice that KH is the inverse of k in eq 1. Max Planck later derived Henry’s law from the thermodynamic condition of equilibrium for a gas and an ideal solution (6), but only in the context of the solubility of a gas in a liquid. Ostwald, in an English translation of the portions of his book, Lehrbuch der Allgemeine Chemie (7), which pertain to solutions, limits the application of Henry’s law to solutions of gases in liquids, as described by Henry (1) and successors such as Bunsen (8), and Dalton (9). He makes no mention of the applicability of Henry’s law to the solute in dilute solutions. By 1912, however, in the 3rd edition of his Outlines of General Chemistry, Ostwald clearly applies Henry’s law to the vapor pressure of a solute in dilute solution (10). Thus, around the beginning of the 20th century, about a hundred years after Henry published his article, the law was applied to the vapor pressure of a solute in dilute solution, interchanging the dependent and independent variables. Lewis and Randall (11) pointed out that Henry’s law is a mathematical necessity at infinite dilution, since the slope of a curve of y as a function of x is necessarily y兾x near x = 0. We can make the application of Henry’s law to binary liquid solutions more concrete with an example, a solution of chloroform and acetone at 35 C (12). The data of Röck and Schröder are shown in Figure 3. It would appear that the points for both components approach the Raoult’s law lines as the respective mole fractions approach 1. A more rigorous test would be a plot of p兾X against X, which should show a horizontal asymptote as x approaches 1. Such a plot is shown in Figure 4. It would appear that the curves for both components approach a horizontal asymptote as the respective mole fractions approach 1, implying that Raoult’s law is followed for both as the mole fraction approaches 1. However, if we look at the p兾X curves as the mole fractions approach zero, we do not see a horizontal asymptote, implying that Henry’s law is not followed for either component as the mole fraction approaches 0. George Scatchard stated in one of his investigations of vapor–liquid equilibrium (13) that “It is well to note

400

400 350

350

300

p / torr X

p / torr

300

250 200 150

250 200

100 150

50

100

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

Figure 3. A plot of p against X for solutions of chloroform and acetone at 35 C. Data from ref 12: acetone, chloroform, ---Raoult’s law lines, - - - Henry’s law lines.


0.4

0.6

0.8

1.0

Xacetone

Xacetone

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142

130

•

Figure 4. A plot of p/X against X for solutions of chloroform and acetone at 35 oC. Data from ref 12: acetone., chloroform, ---lines for extrapolation to Henry’s law constant, - - - Raoult’s law lines.


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that this system does not obey Henry’s law so rigidly that the curve for p兾X has a horizontal asymptote at X = 0.... That such horizontal asymptotes are rare exceptions in non-ideal mixtures is shown best by freezing point measurements...”. Since Henry’s law for the solute implies Raoult’s law for the solvent, by application of the Gibbs–Duhem relation, and Raoult’s law for the solvent implies Henry’s law for the solute from the Gibbs–Duhem relation (14), it is clear that if one is only a limiting law, so is the other. It may be that both are exact only in the limit of infinite dilution. The Henry’s law lines in Figure 3 were plotted using the values of the Henry’s law constants found from Figure 4. Since there is no horizontal asymptote as X approaches 0, Henry’s law does not apply at the dilute concentrations measured. Thus Henry’s law should be expressed as a limiting law for the solutions studied in this work: lim

X → 0

p = KH X

(7)

Since Henry’s law is only a limiting law for the solutions studied in this work, it is clear that Raoult’s law is also only a limiting law for these solutions as shown in eq 8:

p lim = P ° X → 1 X

(8)

Is Henry’s Law Only a Limiting Law for Binary Solutions? It may be that we were unable to obtain a horizontal asymptote in plots of p兾X against X for the vapor pressures of chloroform–acetone solutions because there were not data at sufficiently low values of X to demonstrate the validity of Henry’s law. Certainly, many examples exist in the literature in which the authors claim that they have demonstrated that Henry’s law is followed in the system studied (2, 15, 16), usually by plotting pressure against mole fraction and seeing

a straight line. We have seen that Giacobbe (2) correctly concluded that solutions of CO2 in acetone follow Henry’s law at pressures below 10 atmospheres. In 1995 Koga (17) published an article on the vapor pressures of dilute aqueous t-butyl alcohol, with the subtitle: “How Dilute Is the Henry’s Law Region?” He concluded that, at a lower limit of 0.000068 mole fraction of t-butyl alcohol, Henry’s law is still not followed. Figure 5 shows a plot of p兾X for t-butyl alcohol as a function of X from his data. The most dilute points were omitted from the linear fit, as they were by the original authors, because they are less reliable, and we assume that there were systematic errors at very low mole fraction. In a later article, Westh, Haynes, and Koga (18) published a study of the vapor pressures of dilute solutions of nhexane in cyclohexane, and dilute solutions of cyclohexane in n-hexane, in which they used a quantity ∆ f兾X. ∆f is defined as, ∆ f = f tot − f c° yclohexane

(

)

° = K H − f cyclohexane X n -hexane

(9)

where ftot = KHXn-hexane + f cyclohexane (1 − Xn-hexane), which is correct only if Henry’s law is followed. The authors state “the values (of f兾X) ... appear constant”, and concluded that Henry’s law is followed. We plotted f兾X against X as shown in Figure 6 and did a weighted linear fit to the data. We did find a horizontal asymptote, as shown in Figure 6, where the three most dilute points are not shown because they were considered less reliable by the authors. If Henry’s law were followed, an even more rigorous criterion is that the excess partial molar enthalpy of the solute, H Em2, derived from the enthalpy of mixing, should also be constant. The excess partial molar enthalpy of mixing is the derivative with respect to the number of moles of solute of the excess molar enthalpy of the solution. The excess molar enthalpy of solution is the difference between the enthalpy of mixing and the ideal enthalpy of mixing, which is zero (19). (The proof

80

650

78

∆f / torr Xn-hexane

p TBA / torr XTBA

600

550

76

74

500

72

70 0.0

0.5

1.0

1.5

2.0

0

5

ⴚ3

Xn-hexane / 10

XTBA / 10

Figure 5. A plot of p/X for t-butyl alcohol in water at 25 C. Data from ref 17: experimental points, ---- weighted linear fit, y = 533.0 − (2100 ± 400)x.

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10

15

ⴚ3

Figure 6. A plot of f/X against X for n-hexane in cyclohexane at 25 C. Data from ref 18: experimental points, ---- weighted linear fit, y = 73.9 ± 0.3 + (19 ± 30)x.


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of the constancy of H Em2 is given in the Appendix.) The authors found that H Em2 is not constant even in the most dilute solutions studied, showing that Henry’s law is not followed even though a horizontal asymptote is found in Figure 6. The authors asked the question, how do a very small number of solute molecules influence the properties of the solution in such a way that Henry’s law is not followed; that is how do the surroundings of each solute molecule change with concentration when they are so apparently isolated. They speculate that, in the case of t-butyl alcohol in water, it is conceivable that a small number of alcohol molecules can change the hydrogen bonded network of the solvent. The problem is more difficult for n-hexane in cyclohexane. They refer to the article of Cann and Patey (20) in which the authors suggest that one solute molecule can change the global density of the solution. The question remains, does it make any practical difference whether Henry’s law is followed at finite concentrations or it is only a limiting law? The major application of Henry’s law today is the determination of the atmospheric concentration of environmentally important organic compounds. Henry’s law is usually used to calculate the gaseous concentration of a solute from the aqueous concentration (21). If the initial slope of the p兾X curve is zero within 5% or 10%, then Henry’s law can be used within experimental error for many cases. Many values of Henry’s law constants for these purposes are tabulated on the World Wide Web (22). However, if one wants to find out the concentration range in which solute–solute interactions are still important, then the nonzero slope of the p兾X curve is relevant (17, 18). Henry’s Law as a Basis for a Solute Standard State Lewis and Randall (23) first developed the notion of a standard state of a solute based on Henry’s law. For a solute that follows Henry’s law as a limiting law, the definition of activity is given by the equation, µ2 ( cond ) = µ 2° ( cond ) + R T ln a2 (cond ) lim

X 2 → 0

a2 ( cond ) X 2 ( cond )

(10)

= 1

because, when Henry’s law is followed the chemical potential is given by eq 3. The standard state is that in which a2 = 1, that is, that µ2 = µ2 when a2 = 1. However, a Henry’s law standard state is not found along the experimental vapor pressure curve, but at the extrapolation to X2 = 0 along the Henry’s law line. In Figure 3, we choose the state on the dashed Henry’s law lines for either acetone or chloroform, whichever we choose as solute, at the value of X2 = 0. Lewis and Randall (23) emphasize that they use Henry’s law as a limiting law at infinite dilution and that the standard state based on Henry’s law is a hypothetical state in which the activity is one and the partial molar volume, partial molar enthalpy, and partial molar heat capacity of the solute are the values at infinite dilution.

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In order to choose the Henry’s law standard state, we must have enough data at low values of X2 either to have a horizontal asymptote in the curve of p2兾X2 against X2, or to have a definite point of extrapolation to X2 = 0. Figure 4 shows that neither acetone nor chloroform have a horizontal asymptote, but both show definite intercepts that give the value of the Henry’s law constant KH. We can then calculate the activity of the solute with a Henry’s law standard state as follows. At equilibrium between liquid and vapor (24):

µ 2 ( cond ) = µ 2° ( cond ) + R T ln a 2 ( cond ) = µ2 (g) = µ2° ( g ) + R T ln

p2 P °

(11)

In the standard state, p2 = KH so that: µ 2° ( cond ) = µ 2 (g) = µ2° ( g ) + R T ln

KH (12) P °

If we subtract eq 12 from eq 11, we obtain:

R T ln a2 ( cond) = RT ln a 2 ( cond) =

p2 KH

p2 KH

(13)

(14)

If we want to take into account nonideality in the gas phase, we can use fugacity instead of pressure in eq 14. We can also extend the use of Henry’s law standard states to solutions in which neither component has appreciable volatility. A good example makes use of cell potential measurements of a lead amalgam cell (25), taken from the data of Hatfield and Zapponi (26). The cell potentials of a cell, Pb(amalgam, X2´);Pb(CH3COOH)2,CH3COOH; Pb(amalgam, X2)

in which both amalgam electrodes are immersed in the same solution of lead acetate and acetic acid were measured. Potentials as a function of X2 at constant X2´ can be used to calculate the activity of Pb, the solute in the amalgams with Hg as solvent (24). The activity is a function of X2. If Henry’s law applies to the solute lead in dilute solutions of mercury in the amalgam, then we express Henry’s law as a2 = K H X 2

(15)

and we test for the applicability of Henry’s law by plotting the calculated value of a2兾X2 as a function of X2. The results of such a plot are shown in Figure 7. We can see from Figure 7 that we do not have a horizontal asymptote, so that Henry’s law is not followed to the most dilute solution studied. However, we can extrapolate the line to X2 = 1 to obtain a value of the Henry’s law constant KH = 1 for lead in mercury, so that Henry’s law is a limiting law. The extrapolated point cannot be seen, since all the data are in such dilute solution. The standard state is a hypothetical state along the Henry’s law line at X2 = a2 = 1. A graph of a2 against X2 is shown in Figure 8.


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Concluding Remarks

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We have followed the evolution of Henry’s law from a simple statement about the effect of pressure on the solubility of a gas to a description of the dependence on concentration of the vapor pressure of a solute in dilute solution and to the determination of the standard state of a solute in both liquid and solid solutions. We have seen that it is possible to demonstrate that Henry’s law is followed for the solubility of CO2 in acetone at pressures below 10 atmospheres, but difficult to demonstrate for binary liquid and solid solutions. In the latter cases, Henry’s law, like the ideal gas law, is only a limiting law, adequate at low mole fractions, but useful for practical purposes when high precision is not required. Why Henry’s law is not followed at very low concentrations is still a subject for theoretical investigation.

A discussion of Henry’s law in regular solutions is available in this issue of JCE Online.

aPb / XPb

1.05

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XPb / 10

Figure 7. A plot to test the applicability of Henry’s law to a solution of lead in mercury. Data from ref 26: experimental points, ---- weighted linear fit, y = (1.00 ± 0.02) − (40.0 ± 0.6)x.

0.03

Line extrapolated to X2 = 1

a2

0.02

0.01

0.00

0.01

0.02

0.03

X2 Figure 8. The extrapolation of the Henry’s law line for lead in mercury to reach the standard state. Data from ref 26: experimental points, ---- Henry’s law line.

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Supplemental Material

Literature Cited 1. Henry, W. Phil. Trans. Royal Soc. 1803, 93, 29–42. 2. Giacobbe, F. W. Fluid Phase Equilibria 1992, 72, 277–297. Lekvam, K.; Bishnoi, P. R. Fluid Phase Equilibria 1997, 131, 297–309. Krause, D., Jr.; Benson, B. B. J. Sol. Chem. 1989, 18, 823–873. Rettich, T. R.; Battino, R.; Wilhelm, E. J. Chem. Thermodyn. 2000, 32, 1145–1156. 3. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill Book Co.: New York, 1969; pp 200– 202. 4. Ramachandran, B. R.; Allen, J. A.; Halpern, A. M. Anal. Chem. 1996, 68, 281–268. Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences, 2nd ed.; McGraw-Hill: New York, 1994; pp 58–62. 5. Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. In Trans. Conn. Acad. 1878, Series 3, III, pp 108– 248, 343–524. Reprinted in The Collected Works of J. Willard Gibbs, Yale University Press: New Haven, CT, 1948; Vol. I, pp 135–138. 6. Planck, M. Treatise on Thermodynamics; Translated from the 7th German Edition; Dover Publications: New York, 1922; pp 247–249. 7. Ostwald, W. Solutions; Muir, M. M. P., Translator; Longmans, Green and Co.: London, 1891. 8. Bunsen, R. W. Annalen 1855, 93, 1. 9. Dalton, J. Mem. Manchester Philosophical Society 1808, 1. 10. Ostwald, W. Outlines of General Chemistry; Taylor, W. W., Translator; Macmillan and Co.: London, 1912; pp 320–321. 11. Lewis, G. N.; Randall, M. Thermodynamics; McGraw-Hill Book Co.: New York, 1923; pp 232–233. 12. Röck, H.; Schröder, W. Z. Phys. Chem. 1957, N.F. 11, 41– 55. 13. Scatchard, G.; Raymond, C. L. J. Am. Chem. Soc. 1938, 60, 1278–1287. 14. Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes, 3rd ed.; Reinhold Publishing Company: New York, 1950; p 18. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand Reinhold Company: New York, 1970; p 16. Klotz, I. M.; Rosenberg, R. M. Chemical Thermodynamics; Wiley-Interscience: New York, 2000; pp 344–347. 15. Veleckis, E.; Cafasso, F. A.; Feder, H. M. J. Chem. Eng. Data 1976, 21, 75–76. 16. Hwang, H.; Dasgupta, P. K. Environ. Sci. Technol. 1985, 19, 255–258. Staudinger, J.; Roberts, P. V. Chemosphere 2001, 44, 561–576. 17. Koga, Y. J. Phys. Chem. 1995, 99, 6231–6233. 18. Westh, P.; Haynes, C. A.; Koga, Y. J. Phys. Chem. B 1998, 102, 4982–4987. 19. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1999; p 217. 20. Cann, N. M.; Patey, G. N. J. Chem. Phys. 1997, 106, 8165.


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Research: Science and Education 21. Ramachandran, B. R.; Allen, J. M.; Halpern, A. M. J. Chem. Educ. 1996, 73, 1058–1061. Ramachandran, B. R.; Allen, J. M.; Halpern, A. M. Anal. Chem. 1996, 68, 281–286. Bamford, H. A.; Baker, J. E.; Poster, D. L. Reviews of Methods and Measurements of Selected Hydrophobic Organic Contaminant Aqueous Solubilities, Vapor Pressures, and Air-Water Partition Coefficients. NIST Special Publication 928, 1998. 22. Henry's Law Constants Web Page. http://www.mpchmainz.mpg.de/~sander/res/henry.html (accessed Aug 2004). 23. Lewis, G. N.; Randall, M. Thermodynamics; McGraw-Hill Book Co.: New York, 1923; p 258.

24. Klotz, I. M.; Rosenberg, R. M. Chemical Thermodynamics, 6th ed.; Wiley-Interscience: New York, 2000; pp 385–387. 25. Klotz, I. M.; Rosenberg, R. M. Chemical Thermodynamics, 6th ed.; Wiley-Interscience: New York, 2000; pp 393–396. 26. Hatfield, M. R.; Zapponi, P. P. Trans. Electrochem. Soc. 1939, 75, 473. 27. Klotz, I. M. Chemical Thermodynamics, 6th ed.; WileyInterscience: New York, 2000; pp 376–377. 28. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thertmodynamics of Fluid-Phase Equilibria; PrenticeHall: Englewood Cliffs, NJ, 1999; pp 216–217.

Appendix That H Em2 is a constant when Henry’s law is followed can be seen from the value of the enthalpy of mixing when Henry’s law is followed. The enthalpy of mixing is,

(

∆ mixing H = n1 H m° 1 − H m• 1

• + n 2 H m° 2 − H m 2

= n2

∆ mixing H = H solution − Hcomponents = n1H m 1 + n2 H m 2

(

− n1H m• 1 + n2 H m• 2

)

(16)

where Hm1 and Hm2 are the partial molar enthalpies in solution and H •m1 and H •m2 are the molar enthalpies of the pure components. If Henry’s law is followed by the solute, then Raoult’s law is followed by the solvent. Then, the partial molar enthalpies of both components are equal to the partial molar enthalpies in the respective standard states (27), so that

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(

)

(

H m° 2 −

H m• 2

)

)

(17)

since the Raoult’s law standard state for the solvent is the pure component. The excess enthalpy is given by (28),

H E = ∆ mixing H − ∆ mixing H Ideal = ∆ mixing H = n1 H mE 1 + n2 H mE 2

(18)

and H m1 = H •m1 for a solvent that follows Raoult’s law, so that H mE 2 = H m° 2 − H m• 2 which is a constant.


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(19)

Henry's Law: A Retrospective - Journal of Chemical Education (ACS

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