工学院大学研究報告第127号

令和2年2月ランベルト・ベールの法則、質量作用の法則、およびヘンリーの法則に原子分子論を適用する方法
高 見 知 秀＊1

Application of Atomic Molecular Theory to Beer–Lambert Law, Law ofMass Action, and Henry’s Law
Tomohide TAKAMI＊1

Abstract
本論文は化学を学ぶ初学者を教える先生方のために書かれたものであり、量子化学のような専門的な知識を必要としません。化学で使用されるいくつかの法則は、原子分子の概念が科学的に受け入れられる前に確立されており、それらは原子分子論、すなわちすべての物質は元素からなる、ということに基づいて再考される必要があります。本論文では、ランベルト・ベールの法則、質量作用の法則、およびヘンリーの法則に原子分子論を適用する方法について述べます。また、グーグル翻訳がある時代に日本語で書く意味はないため、本論文は英語で記述しました。
This paper is written for those who teach beginners learning chemistry in university and does not require
the subject of study such as quantum chemistry. Some laws used in chemistry were established before theconcepts of atom and molecule had been scientifically accepted, and they need to be reconsidered based on
atomic molecular theory; all materials are made from elements. Here, the author describes a way to develop
the Beer–Lambert law, the law of mass action, and Henry’s law from the perspective of atomic moleculartheory. It is also noted that this paper is written in English, not Japanese, because there is no point in writingin Japanese when there is a Google translation.
Keywords: Beer–Lambert law, the law of mass action, Henry’s law

molecules. Thus, thermodynamics has been re-1．Introduction

derived from the point of view of molecular theory

When a bongo player, Richard Feynman, wasevolving into statistical thermodynamics. However,asked to think of a sentence that would impart thesome laws in analytical and basic chemistry still

most important scientific knowledge, he answeredexist without reference to the molecular theory.

thatmatterwasdemonstrated
molecules

madeoftheoreticallywith

hisatoms.1
theexplanationEinstein

existenceofofBrownian

Here, the author demonstrates how the Beer–Lambert law, the law of mass action, and Henry’s
law can be based on molecular theory.

motion.2 Perrin performed the precise experiments
to prove the existence of molecules.3 Since the2．Beer–Lambert Law

proofs of the existence of atoms and molecules,
many improvements and discoveries have occurred.

It should be noted that this section was inspired

Whereas, some important laws and theories,

by the homepage4 and textbook5 written by Koichi

including thermodynamics, were established before

Ohno, Professor Emeritus of Tohoku University

theindicated last century. It is also noted that Akul

generalacceptanceoftheexistenceofMehta mentioned the similar approach on line in＊1

工学院大学教育推進機構基礎・教養科教授

2012.6― 1 ―工学院大学研究報告

The Beer–Lambert law was typically taught by𝑎 𝑁

hypothesis that the logarithm of the transmittancethroughasubstanceisproportional

to𝑎(1.3),(1 − ) ~ 𝑒 −𝑆𝑁

starting with a differential equation based on the𝑆and the approximation is explained later.

itsThen, the following expression can be obtained,

concentration and the optical path length. However,𝑎 𝑁𝐼𝑎(1.4).

ln 𝐼 = ln (1 − 𝑆 ) ∼ − 𝑆 𝑁

textbooks do not, in general, show how that0hypothesis is derived. Here, the Beer–Lambert law
is derived from molecular theory.

The concentration, c, of the sample liquid in the cell

Figure 1 displays a schematic illustration of thededuction of the Beer–Lambert law. First, wein our model for the deviation of the Beer–Lambert
law isconsider a single molecule (n = 1) with cross section

𝑐=a in a cell with volume V = SL (S: cross section of𝑁/𝑁𝐴𝑉

𝑁= 𝑆𝐿𝑁(1.5),
𝐴the cell, L: optical path length in the cell), as shownon the left of Fig. 1. The transmission probability T1

where NA is Avogadro constant (6.0221023 mol−1).

isWe define the molar attenuation coefficient ε as𝜀 ≡ log10 𝑒 × 𝑎𝑁A = 2.303𝑎𝑁𝐴
𝐼𝑎𝑇1 = 𝐼1 = 1 − 𝑆(1.1).

0Next, we consider number N of molecules in thesame cell, as shown on the right of Fig. 1. The event

then, the logarithm of the transmittance islogshown in the left of Fig. 1 occurs N-times when themolecules

aresmallenoughtoneglect(1.6);

𝐼0𝐼
𝑎= − log10 𝑒 ln ( ) = log10 𝑒 × 𝑁
𝐼𝐼0𝑆
𝜀 𝑁=𝑁theA 𝑆= 𝜀𝑐𝐿(1.7).

correlation between the molecules. Thus, theThus, we have derived the reason why the logarithm

transmission

ofprobability

Tis

theN-times

multiplication of the individual probabilities;
𝑎 𝑁𝐼

transmittance

0n=NSI𝐼
𝑎= 1−𝐼0
𝑆I0atothe

can be obtained is shown. From a Maclaurin-series
wecanobtainthe

following

approximation:

lnI0

proportionalNow, how the approximation of the equation 1.3expansion,

n=1

is

concentration c and the optical path length L.
(1.2).

𝑇 = 𝐼 = (1 − 𝑆 )the𝐼𝑎 𝑁𝑎𝑎
= ln (1 − ) = 𝑁ln (1 − ) ∼ 𝑁 (− )𝐼0𝑆𝑆𝑆(1.9),

I

so that we can obtain the approximated equations

𝑎𝐼𝑎 𝑁= 1−~ 𝑒 − 𝑆𝑁𝐼0𝑆

1.3 and 1.4.
TheFigure 1. Illustration of the Beer–Lambert law using
molecular theory. Left: Single-molecule scattering

physicalmeaningofthelogarithmic

dependence in the Beer – Lambert equation iswhere the molecule is in a thin slab (n = 1). Right:

described. Note that a/S = acLNA/N (from equation

Multiple-molecule scattering where the molecules1.5) = b/N (b ≡ acLNA) and N is a very large

are in the pile of the slabs (n = N).

number. Then, the following equation related to
Napier’s constant e could be obtained,𝑏 𝑁lim (1 − 𝑁) = (1/𝑒)𝑏

In other words, the thin slabs with the scattering

Figure 2 illustrates the physical meaning ofmolecules piles up N-times. For small number a/S,
the following approximation can be applied,(1.10).

𝑁→∞

Napier’s constant e. A sheet of paper with area S is
cut into a number N of tiny pieces of paper, and the― 2 ―ランベルト・ベールの法則、質量作用の法則、およびヘンリーの法則に原子分子論を適用する方法

tiny pieces of paper are randomly deposited on a“activemass”,10“chemicaldifferent sheet of paper with the same area S as theconcentration, because the concept of the amountoriginal sheet. Then, the probability to cover theof a substance did not exist when the law of mass

deposited area with a piece of the paper isaction was proposed. For the student learning the(S−S/N)/S = 1−1/N. The behavior of each piecelaw of mass action, however, there can be someshould be considered an independent event, so that

confusion over fact that no mass variable, m,

the ratio of the area that is not covered with theappears in the law of mass action,′pieces of the paper is the N-th power of the𝐾=probability, i.e., (1−1/N)N. Therefore, the ratio of theforce”,11or′[A′ ]𝑎 [B′ ]𝑏 ⋯[A]𝑎 [B]𝑏 ⋯(2.1),

deposited domain to the exposed domain isfor the general reaction

approximately 1−e−1 : e−1 = e−1 : 1 when the number𝑎A + 𝑏B + … ⇄ 𝑎′A′ + 𝑏′B′ + …

N is large enough because
1 𝑁lim (1 − 𝑁) = 1/𝑒(2.2).
Therefore, when evaluating this expression for a(1.11).

particular reaction, students are liable to make the𝑁→∞

This is the meaning of Napier’s constant according

error of using the mass percentage (%) in place ofto the stochastic interpretation.

the concentration [A], which should be a molarpercentage (e.g., in units of mol L−1 for each of thecomponents of a solution-phase reaction).
Although the historical discovery of the law ofmass action should be respected, the words of thelaw of “mass” action should be reconsidered, e.g.,“the law of chemical equilibrium”.12 could be used
as an alternative.

4．Henry's Law
Figure 2. Explanation of Napier’s constant e. The
ratio of the deposited area (blue) to the exposed area(yellow) is e−1 : 1, which is the physical meaning ofNapier’s constant.

Henry’s law was proposed in 1802, and WilliamHenry published an article on the solubility of gases
in water in 1803.13 Naturally, this law pre-dates thegeneral acceptance of the concept of molecules. The
most commonly used definition of Henry’s law isLykos demonstrated a similar route to the Beer–Lambert law following the process of deriving theNapier’s constant e = 2.71828... and the naturallogarithm;7 however, a thin "slab" was only used in
the model of the explanation. Using moleculartheory might allow students to understand theBeer–Lambert law more easily, based on theadsorption of a single molecule.

that the solubility of a gas is proportional to thepartial pressure of the corresponding gas.14,15
However, some teachers and textbooks present
the law by stating that the dissolved volume of a gas
is constant with increasing pressure in the case of
an ideal gas.16 This expression can generate
confusion for students.
Henry’s law can be expressed by the following
equation:

Sg = n/VS = kpP(3.1),

where Sg is the solubility of the dissolved gas par

3．Law of Mass Action
Guldberg and Waage proposed the law of massaction in 1864.8,9 According to the original paper,
the word “mass” in the law of mass action means

the volume of the solvent, n is the amount of
substance of the dissolved gas, VS is the volume ofthe solvent, kp is the Henry’s constant, and P is thepressure applied to the gas. The volume of dissolved― 3 ―工学院大学研究報告

gas Vg in the case of an ideal gas follows thereconsidered based on the concept of the amount ofequation

substance per amount of solvent; “the mole fraction

PVg = nRT(3.2),of the dissolved gas is proportional to the partial

where R is the gas constant and T is thepressure of the gas” as in most of the moderntemperature. This leads to the expression

university textbooks.

Vg/VS = kpRT(3.3),

6．Acknowledgments

and hence, when the temperature T is constant, thedissolved volume of the gas par the volume of theTheauthorthanksProf.

KoichiOhnoforsolvent Vg/VS is constant. Though the amountproviding the idea for the derivation of the Beer–

substance n dissolved in the solvent increase with

Lambert law from molecular theory. He would likethe applied pressure P, according to the equation

to thank Editage (www.editage.jp) for English

3.1, the volume of the dissolved gas Vg is constantlanguage editing. He is also grateful of Prof. K. W.

because the Boyle's law says that Vg is inverse

Hipps of Washington State University and theproportional to the applied pressure P. A textbook

reviewers of the e-Journal of Surface Science anddescribed that the volume of the dissolved gases in

Nanotechnology

the solvents were often converted to the same

manuscript and the corrections.

forcondition, e.g., standard state in case of thethevolumeofthedissolvedgas

wasreadingoftheReferencescomparison of the solubility using the volumes, andthat

careful

1. Feynman, R. The Feynman Lectures on Physics Vol. I,

proportional to the pressure when the gas was

Addison Wesley, Massachusetts, 1965, pp. 1–5.

dissolved at the constant pressure (as translated by

2. Einstein, A. Über die von der molecularkinetischen Theoriethe author),15 which makes more confusion to thederstudents.

Flüssigkeiten suspendierten Teilchen. Annalen der Physik.

Considering from the concept of the amount of

Wärme

geforderte

Bewegungvoninruhenden

1905, 322, 549–560.

substance n, this approach does not cause any

3. Perrin, J. Mouvement Brownian et réalité moléculaire. Ann.

confusion. Therefore, considering molecular theory,Chim. Phys. 1909, 18, 5–114.

Henry’s law has been defined as in most of the

4. Ohno, K. Welcome to the Page of Koichi OHNO(inmodern university textbooks17 that “the moleJapanese).

fraction of the dissolved gas” is proportional to its(accessed Jan. 2019).

pressure.

5. Ohno, K. Quantum Physical Chemistry, 1st ed.; (inhttp://iqce.jp/ohno/KOICHI_OHNO.HTMJapanese) University of Tokyo Press: Tokyo, Japan, 1989; pp

5．Conclusion

296 (Question 5-55), 390–391 (Answer 5-55).

A number of chemical laws were discovered or

6. A. Mehta, Ultraviolet-Visible (UV-Vis) Spectroscopy –proposed before the concepts of the atom, theDerivation of Beer-Lambert Law, on line since April 22, 2012,amount of substance, and molecular theory had(https://pharmaxchange.info/2012/04/ultraviolet-visible-

been established, and some of these laws should be

uv-vis-spectroscopy-%E2%80%93-derivation-of-beer-

re-expressed for the benefit of students learninglambert-law/).

chemistry. The Beer–Lambert law equation can be

7. Lykos, P. The Beer–Lambert Law Revised: A Development

derived from molecular theory and the hypothesis

without Calculus J. Chem. Educ. 1992, 69, 730–732.

thatis

8. Waage, P.; Guldberg, C. M. Forhandlinger: Videnskabs-

proportional to the concentration and optical path

Selskabet i Christiania 1864, 35–45; online available at

length can be explained via this approach. The

https://www.biodiversitylibrary.org/page/44007380#page/3

name of the law of mass action should be07/mode/1up (accessed Jan. 2019).

reconsidered based on the concept of the molar

9. Waage, P.; Guldberg, C. M. (Abrash, H. I. translated)amount of substance. Henry’s law, also, should be

Studies concerning affinity J. Chem. Educ. 1986, 63, 1044–

thelogarithmofthetransmittance― 4 ―ランベルト・ベールの法則、質量作用の法則、およびヘンリーの法則に原子分子論を適用する方法
1047.
10. Mysels, K. J. Textbook errors: VII. The laws of reaction
rates and equilibrium J. Chem. Educ. 1956, 33, 178–179.
11. Lund, E. W. Guldberg and Waage and the law of massaction J. Chem. Educ. 1965, 42, 548–550.
12. Adams, C. S. Teaching problems in presenting the law ofchemical equilibrium to students in analytical chemistry J.
Chem. Educ. 1940, 17, 22–29.
13. Henry, W. Experiments on the quantity of gases absorbed
by water at different temperatures and under differentpressures Phil. Trans. Royal Soc. 1803, 93, 29–43; 274–276.
14. Carroll, J. J. Henry’s law: a historical view J. Chem.
Educ. 1993, 70, 91–92.
15. Rosenberg, R. M., Peticolas, W. L. Henry’s Law: aretrospective J. Chem. Educ. 2004, 81, 1647–1652.
16. Yamanouchi, K. et al., "High School Chemistry, Revised"(Japanese),Feb.

10,2018 publishedversion;

Daiichi

Gakushusha: Tokyo, Japan, 2018; pp 56–57.
17. e.g., Atkins, P., de Paula, J., Keeler, J., "PhysicalChemistry ", 11th Ed.; Oxford University Press: Oxford, U.K.,
2018; pp 152–153.

Figure Captions
Fig. 1. (Color online) Illustration of the Beer–
Lambert law using molecular theory. Left: Singlemolecule scattering where the molecule is in a thinslab (n = 1). Right: Multiple-molecule scattering
where the molecules are in the pile of the slabs (n =N).
Fig. 2. (Color online) Explanation of Napier’sconstant e. The ratio of the deposited area (blue) tothe exposed area (yellow) is e−1 : 1, which is thephysical meaning of Napier’s constant.

― 5 ―工学院大学研究報告

n=1

n=N

SI0I𝐼
𝑎
= 1−𝐼0𝑆I0aI𝑎𝐼𝑎 𝑁− 𝑁
= 1−~𝑒 𝑆𝐼0
𝑆

Fig.1. (Color Online)

𝑆𝑁:Random deposition

SFig. 2. (Color Online)― 6 ―= 1−e−1 : e−1= e −1 : 1