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LECTURE TEXT

First-order differential equations in chemistry

Gudrun Scholz •Fritz Scholz

Received: 7 August 2014 / Accepted: 13 September 2014 / Published online: 25 November 2014

ÓSpringer International Publishing 2014

Abstract Many processes and phenomena in chemistry,

and generally in sciences, can be described by first-order

differential equations. These equations are the most

important and most frequently used to describe natural

laws. Although the math is the same in all cases, the stu-

dent may not always easily realize the similarities because

the relevant equations appear in different topics and con-

tain different quantities and units. This text was written to

present a unified view on various examples; all of them can

be mathematically described by first-order differential

equations. The following examples are discussed: the

Bouguer–Lambert–Beer law in spectroscopy, time con-

stants of sensors, chemical reaction kinetics, radioactive

decay, relaxation in nuclear magnetic resonance, and the

RC constant of an electrode.

Keywords Differential equations Bouguer–Lambert–

Beer law Time constants Chemical kinetics 

Radioactive decay Nuclear magnetic resonance RC

constant

Introduction

‘‘Differential equations are extremely important in

the history of mathematics and science, because the

laws of nature are generally expressed in terms of

differential equations. Differential equations are the

means by which scientists describe and understand

the world’’ [1].

The mathematical description of various processes in

chemistry and physics is possible by describing them with

the help of differential equations which are based on simple

model assumptions and defining the boundary conditions

[2,3]. In many cases, first-order differential equations are

completely describing the variation dyof a function y(x)

and other quantities. If yis a quantity depending on x,a

model may be based on the following assumptions: The

differential decrease of the variable yis proportional to a

differential increase of the other variable, here x, i.e.

–dy*dx. This decrease –dyshould depend on the

function yitself: –dy*ydx, and together with a so far

unknown constant a, results in the equation

dy¼aydxð1Þ

Thus follows the ordinary linear homogeneous first–

order differential equation:

dxþay ¼0ð2Þ

The characteristics of an ordinary linear homogeneous

first–order differential equation are: (i) there is only one

independent variable, i.e. here x, rendering it an ordinary

differential equation, (ii) the depending variable, i.e. here y,

having the exponent 1, rendering it a linear differential

equation, and (iii) there are only terms containing the

Electronic supplementary material The online version of this

article (doi:10.1007/s40828-014-0001-x) contains supplementary

material, which is available to authorized users.

G. Scholz

Department of Chemistry, Humboldt-Universita

¨t zu Berlin,

Brook-Taylor-Str. 2, 12489 Berlin, Germany

F. Scholz (&)

Institute of Biochemistry, University of Greifswald,

Felix-Hausdorff-Str. 4, 17487 Greifswald, Germany

e-mail: fscholz@uni-greifswald.de

123

ChemTexts (2014) 1:1

DOI 10.1007/s40828-014-0001-x

variable yand its first derivative, rendering it a homoge-

neous first–order differential equation.

This equation can be solved when, e.g. the boundary

conditions are such that yvaries between y

and y, when x

varies between 0 and x. Following a separation of vari-

ables, the integration of Eq. 2gives:

y¼adxð3Þ

y¼aZx

dxð4Þ

ln y

y0¼ax ð5Þ

y¼y0eax ð6Þ

Equation 6describes the exponential decrease of yas a

function of x.

This formalism will now be applied to some special

cases which occur frequently in chemistry, and finally all

discussed cases will be compared in a table.

The Bouguer–Lambert–Beer Law

The intensity of electromagnetic radiation (e.g. visible

light), i.e. exactly the radiant ﬂux I(unit W, watt or J s

–1

joule per second), diminishes along the path length

xthrough a homogeneous absorbing medium (e.g. a col-

oured solution). Figure 1depicts a cuvette and the changes

in radiant ﬂux along the optical path length.

The differential decrease –dIof radiant ﬂux by passing

through the differential length increment dxis supposed to

be proportional to the actual value of Iat x

. To understand

this, we must consider the physical background of the

decrease of radiant ﬂux: If the radiation is understood as a

ﬂux of photons, the absorption of radiation is the loss of

photons due to their ‘‘capture’’ by absorbing particles

(molecules, atoms or ions) in the cuvette. Clearly, the

effectivity of capture must be proportional to the number of

particles per volume, i.e. their concentration cin mol L

–1

as the probability that a photon hits a particle will be

proportional to its concentration. However, not each hitting

leads to an absorption event (capture of a photon). To take

into account the probability that a collision of a photon

with a particle leads to its capture, one defines an effective

cross section of the particles. This cross section has the unit

of an area because one may understand it as an effective

target area for the photons in contrast to the geometric

target area which a particle exposes to the photon ﬂux.

Instead of using the effective cross section, one may define

a constant j(Greek letter kappa) which theoretically can

have values between 0 and 1, giving the fraction of suc-

cessful absorption events. jis a value specific for the

particles and specific for the photon energy E

photon

, and

thus the frequency m(Greek letter nu) of the radiation, with

m=E

photon

/h(his the Planck constant

6.62606957(29) 910

–34

Js), and the wavelength k(Greek

letter lambda) with k¼hclightEphoton (clight being the

velocity of light in the respective medium).

From the preceding discussion follows that the differ-

ential equation

dIðxÞ¼IðxÞjcdxð7Þ

adequately describes the decrease of radiation ﬂux. Since j

is specific for the energy of absorbed photons, this equation

Fig. 1 Electromagnetic radiation is trespassing a cuvette filled with a

homogeneous absorbing medium. I0is the radiant ﬂux before entering

the cuvette, Iis the radiant ﬂux leaving the cuvette, dIis the

differential decrease of radiant ﬂux by passing through the differential

length increment dxat xi

Fig. 2 The decay of radiation ﬂux when passing through the

absorbing medium

1Page 2 of 12 ChemTexts (2014) 1:1

123

relates to monochromatic radiation (radiation with one

constant frequency, i.e. photon energy). The meaning of

Eq. 7can be understood with the help of Fig. 2:Ifxemarks

the overall length which the electromagnetic radiation

passes through the absorbing medium, and the intensity

(radiation ﬂux) of the incident light is I0(at x¼0), then at

a path length xithe intensity of light will have dropped to Ii

and the slope of IðxÞ¼fðxÞ, i.e. dIðxÞ

dxwill be proportional

to Iiand jand c.

Integration of Eq. 7and some rearrangements have to be

performed as follows:

dIðxÞ

IðxÞ¼jcdxð8Þ

ZI

dIðxÞ

IðxÞ¼jcZxe

dxð9Þ

ln I

I0¼jcxeð10Þ

ln I0

I¼jcxeð11Þ

log I0

I¼1

ln 10 jcxe0:4343jcxeð12Þ

The ratio log I0

Iis called absorbance A, and the product

0:4343jis called the molar absorption coefficient e(Greek

letter epsilon) or molar absorptivity. The path length of the

radiation xeis usually given the symbol l. The Bouguer–

Lambert–Beer Law is thus normally written as:

A¼ecl ð13Þ

Outside of this purely mathematical analysis, it needs

to be mentioned that Eq. 13 has a restricted range of

validity: it is a good description of real systems only

at low concentrations. At higher concentrations

(sometimes already above 10

–5

mol L

–1

) intermo-

lecular interactions of the absorbing particles, and

chemical equilibria can lead to deviations (apparent

variations of the molar absorption coefficient). Fur-

ther, another contribution to the absorption coeffi-

cient depends on the refractive index nof the

solution. Because the refractive index may signifi-

cantly vary with the concentration of the dissolved

analyte, it is not e, which is constant, but the molar

refraction and the term en=ðn2þ2Þ2should be used

instead of e[4].

The time constant of a sensor

Sensors measure a physical or chemical quantity and

transduce it to an output signal which is read, monitored

or stored. Possible physical quantities are temperature,

pressure, radiative ﬂux, magnetic field strength, etc.

Chemical quantities are mainly concentrations and activ-

ities of molecules, atoms and ions. The recorded signals

are usually voltages or currents. The most typical feature

of a signal is that the results are one dimensional, e.g. the

output signal is a single quantity, i.e. one measures only

that signal and not a dependence of that signal on another

given quantity. Most devices for chemical analysis pro-

duce two-dimensional read-outs, e.g. optical spectra in

which the absorbance is displayed as a function of

wavelength (E¼fðkÞ), voltammograms in which currents

are displayed as function of electrode potential or X-ray

diffractograms, in which the intensity of diffracted rays is

displayed as function of diffraction angle, etc. In modern

instrumentation, one has even expanded the

Fig. 3 A comparison of the three common dimensionalities of analytical devices

ChemTexts (2014) 1:1 Page 3 of 12 1

123

dimensionality to three, when, as an example, optical

spectra (E¼fðkÞ) (or mass spectra, i.e. ion intensities

versus the mass-to-charge ratio of ions) are displayed as a

function of elution time of a chromatogram. Figure 3

gives a comparison of the common dimensionalities of

analytical measurements.

Since any measurement needs time, there is nothing like

an instantaneous establishment of a signal. This is easy to

see when using a sensor, e.g. a pH electrode: There is

always a certain time period in which the reading changes

until we finally have the impression that a constant end

value is reached. The same is true also for two- or three-

dimensional measurements, but we cannot easily detect it

because the variation of the measured signal (e.g. the

absorbance) anyway changes as a function of the varied

quantities (e.g. the wavelength) and thus with time. Nor-

mally, the wavelength is changed with the so-called scan

rate dk=dt(rate of recording the spectrum), and generally

(see Fig. 4), the quantity xis varied with a scan rate dx=dt

concentration step. The introduction of the sensor into a

solution can be regarded as a concentration step. Figure 5

depicts two different kinds of response of a sensor on a

concentration step.

Figure 5depicts two basic types of time responses of

sensors. The different sensor behaviours shown in B and C

can be modelled with the help of different differential

equations. Whereas the response curve shown in B can be

modelled with a first-order differential equation; the curve

shown in C needs higher-order differential equations [5]. At

this point, it is necessary to note that it is impossible to realize

a concentration step with infinite rate of concentration rise, as

shown in Fig. 5a. This means, when the temporal response

properties of a sensor are studied, this concentration rise has

to be much quicker than the response of the sensor. Further,

also the response shown in Fig. 5b is to some extend an

idealization, and in reality there may be always a sluggish

response at the start, but it may be on such short time scale

that it escapes our recognition. The response curve shown in

Fig. 4 Possible distortion of a

spectrum when the rate of

changing xwith time above a

limiting value ðdx=dtÞlimit

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