6
Proc. Nat. Acad. Sci. USA Vol. 70, No. 3, pp. 870-875, March 1973 Fluctuation Spectroscopy: Determination of Chemical Reaction Kinetics from the Frequency Spectrum of Fluctuations (chemical kinetics/relaxation methods/electrolyte/noise) GEORGE FEHER AND MIKE WEISSMAN University of California, San Diego, Department of Physics, La Jolla, Calif. 92037 Communicated by Bruno H. Zimm, December 21, 1972 ABSTRACT The kinetic parameters ofa chemical reac- tion were obtained from analysis of the frequency spec- trum of the fluctuations (i.e., "noise") in the concentra- tions of the reactants. In "fluctuation spectroscopy," no external perturbation is applied and the system remains in macroscopic chemical equilibrium during the experiment. Results obtained by this method for the dissociation reac- tion of beryllium sulfate agree well with those obtained by relaxation methods in which the approach to equilib- rium is analyzed. Other noise sources not originating from a chemical reaction were observed and analyzed. The most prominent of these arose from the flow of an electrolyte through a capillary. The method of fluctuation spec- troscopy should be applicable to problems of physical, chemical, and biological interest. Chemical equilibria are maintained by a balance between the rates of competing reactions. At equilibrium the average concentrations of the individual reactants remain constant. For a determination of -the individual reaction rates, tech- niques that monitor the time dependence of the concentration of a reactant as the system comes to equilibrium have been developed (1, 2). A particularly powerful method pioneered by Eigen and coworkers (3, 4) is the chemical relaxation method, in which the system at equilibrium is subjected to a sudden perturbation, and the subsequent time dependence of the equilibration process is monitored. In the present work, we discuss and demonstrate experi- mentally a method of obtaining kinetic parameters without applying an external perturbation to the system. The method is based on the basic principle of statistical mechanics that states that the concentration of a reactant in a system at equilibrium fluctuates around its equilibrium value. The frequency spectrum of these fluctuations are related in a known way to the time dependence of the equilibration process (5). Thus, by spectral analysis ("fluctuation spec- troscopy") of the concentration fluctuations, one obtains, in principle, the same kinetic information as from relaxation methods. Any physical parameter that "tags" the reactants (e.g., optical absorption, circular dichroism, fluorescence, di- electric constant, magnetization, electrical charge, volume, etc.) will reflect the concentration fluctuations of the reactant and can, therefore, be used. Measurement of concentration fluctuations due to diffusion in a chemically inert system has become a standard experi- mental technique (6). The effect of a chemical reaction on the spectrum of scattered light has been investigated theoretically by several authors (7) and experimentally by Yeh and Keeler (8).* Recently, Magde et al. (9) have observed the fluctuations in the fluorescence emission from a chemically reactive system. In both of these experiments, the major part of the spectrum of the detected light is governed by diffusion. In this study we have determined the kinetic parameters of the dissocia- tion process in beryllium sulfate by studying the frequency spectrum of the conductivity fluctuations of the electrolyte. This system has been studied by relaxation methods for many years (3, 10, 11) and is well understood. It serves, therefore, as a good model system to test the method of fluctuation spectroscopy. A brief account of this work was given earlier (12). During these experiments, other noise sources not originat- ing from a chemical reaction were observed and analyzed. The amplitude and frequency spectrum of these fluctuations contain information about several parameters of the system (e.g., its dimensions, the number of "active" molecules and their rate of transport). THEORETICAL CONSIDERATIONS Fluctuations in a simple reacting system Consider first the simple reacting system described by ki2 NA 2 NB k2h [1] where kl2 and k21 are the kinetic parameters to be determined. At equilibrium there are, on the average, NA and NB mole- cules in states A and B, respectively. NA and NB fluctuate about this equilibrium as schematically illustrated in (Fig. 1 top). We wish to find the frequency spectrum of the fluctua- tions in one of the reactants, say [UNA(v) ]2. The integral of this quantity over all frequencies is the mean square fluctua- 2 tion, (6NA)2, given by (NA - NA)2 = NA2 - NA . To evalu- ate this quantity, let us define p as the probability of finding a molecule in state A, i.e., NA = pN and NB = (1-p)N. N NA can be viewed as E NA), where NAj = 1 if the jth mole- j=1 cule is in state A and zero if it is in state B. It follows, there- 2 fore, that NAj2 - NAj = (1)2p - [(l)p]2 = p(l - p).- For * The validity of these experimental results is being questioned by several groups. The possibility of an experimental artifact is at present being investigated by Yeh (Y. Yeh, private communi- cation). 870

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Proc. Nat. Acad. Sci. USAVol. 70, No. 3, pp. 870-875, March 1973

Fluctuation Spectroscopy: Determination of Chemical Reaction Kineticsfrom the Frequency Spectrum of Fluctuations

(chemical kinetics/relaxation methods/electrolyte/noise)

GEORGE FEHER AND MIKE WEISSMAN

University of California, San Diego, Department of Physics, La Jolla, Calif. 92037

Communicated by Bruno H. Zimm, December 21, 1972

ABSTRACT The kinetic parameters of a chemical reac-tion were obtained from analysis of the frequency spec-trum of the fluctuations (i.e., "noise") in the concentra-tions of the reactants. In "fluctuation spectroscopy," noexternal perturbation is applied and the system remains inmacroscopic chemical equilibrium during the experiment.Results obtained by this method for the dissociation reac-tion of beryllium sulfate agree well with those obtainedby relaxation methods in which the approach to equilib-rium is analyzed. Other noise sources not originating froma chemical reaction were observed and analyzed. The mostprominent of these arose from the flow of an electrolytethrough a capillary. The method of fluctuation spec-troscopy should be applicable to problems of physical,chemical, and biological interest.

Chemical equilibria are maintained by a balance between therates of competing reactions. At equilibrium the averageconcentrations of the individual reactants remain constant.For a determination of -the individual reaction rates, tech-niques that monitor the time dependence of the concentrationof a reactant as the system comes to equilibrium have beendeveloped (1, 2). A particularly powerful method pioneeredby Eigen and coworkers (3, 4) is the chemical relaxationmethod, in which the system at equilibrium is subjected to asudden perturbation, and the subsequent time dependence ofthe equilibration process is monitored.

In the present work, we discuss and demonstrate experi-mentally a method of obtaining kinetic parameters withoutapplying an external perturbation to the system. The method isbased on the basic principle of statistical mechanics thatstates that the concentration of a reactant in a system atequilibrium fluctuates around its equilibrium value. Thefrequency spectrum of these fluctuations are related in aknown way to the time dependence of the equilibrationprocess (5). Thus, by spectral analysis ("fluctuation spec-troscopy") of the concentration fluctuations, one obtains,in principle, the same kinetic information as from relaxationmethods. Any physical parameter that "tags" the reactants(e.g., optical absorption, circular dichroism, fluorescence, di-electric constant, magnetization, electrical charge, volume,etc.) will reflect the concentration fluctuations of the reactantand can, therefore, be used.Measurement of concentration fluctuations due to diffusion

in a chemically inert system has become a standard experi-mental technique (6). The effect of a chemical reaction on thespectrum of scattered light has been investigated theoreticallyby several authors (7) and experimentally by Yeh and Keeler

(8).* Recently, Magde et al. (9) have observed the fluctuationsin the fluorescence emission from a chemically reactive system.In both of these experiments, the major part of the spectrumof the detected light is governed by diffusion. In this studywe have determined the kinetic parameters of the dissocia-tion process in beryllium sulfate by studying the frequencyspectrum of the conductivity fluctuations of the electrolyte.This system has been studied by relaxation methods for manyyears (3, 10, 11) and is well understood. It serves, therefore,as a good model system to test the method of fluctuationspectroscopy. A brief account of this work was given earlier(12).During these experiments, other noise sources not originat-

ing from a chemical reaction were observed and analyzed.The amplitude and frequency spectrum of these fluctuationscontain information about several parameters of the system(e.g., its dimensions, the number of "active" molecules andtheir rate of transport).

THEORETICAL CONSIDERATIONSFluctuations in a simple reacting system

Consider first the simple reacting system described byki2

NA 2 NBk2h

[1]

where kl2 and k21 are the kinetic parameters to be determined.At equilibrium there are, on the average, NA and NB mole-cules in states A and B, respectively. NA and NB fluctuateabout this equilibrium as schematically illustrated in (Fig.1 top). We wish to find the frequency spectrum of the fluctua-tions in one of the reactants, say [UNA(v) ]2. The integral ofthis quantity over all frequencies is the mean square fluctua-

2

tion, (6NA)2, given by (NA -NA)2 = NA2 - NA . To evalu-ate this quantity, let us define p as the probability of findinga molecule in state A, i.e., NA = pN and NB = (1-p)N.

N

NA can be viewed as E NA), where NAj = 1 if the jth mole-j=1

cule is in state A and zero if it is in state B. It follows, there-2

fore, that NAj2 - NAj = (1)2p - [(l)p]2 = p(l - p).- For

* The validity of these experimental results is being questionedby several groups. The possibility of an experimental artifact isat present being investigated by Yeh (Y. Yeh, private communi-cation).

870

Proc. Nat. Acad. Sci. USA 70 (1973)

2 N 2

N independent molecules NA2- NA = E (NA,2 - NAJ)=j-1

Np(1 - p). We thus obtain for the mean square fluctuation22_

(SNA)2= NA2- NA

= f [6NA(V)]2dv = NA(1 - NA/N). [2]

This equation shows that the mean square magnitude of thefluctuations is independent of the reaction rate and is largestfor p = 0.5, i.e., NA = NBEWe next calculate the frequency spectrum of the fluctua-

tions [6NA (V) ]2. A perturbation imposed on the system willdecay exponentially as exp(-t/r), where l/r is given by k12 +k2l (3, 4). As long as the system behaves randomly, theautocorrelation function of the concentration fluctuations3NA(t)SNA(t + t1) is a constant times exp(-tl/r) (5). Thepower spectrum of the fluctuation is the Fourier transform ofthe correlation function (5). For an exponential autocorrela-tion function, this will be a Lorentzian, i.e., [6NA (V) ]2 =C[1 + (2TvT)2]-1. The constant C is determined by integrationof this expression over all frequencies (see Eq. 2). The fre-quency spectrum of the fluctuations per unit frequency in-terval (bandwidth) is then given by

4-rNA[l - (NA/N)]I[6NA(V)=12-[- (2 A/))2 [3]

Thus, from the amplitude and frequency dependence of thefluctuations one obtains both the number of molecules, NA,and T. Fig. 1 (bottom) shows a plot of Eq. 3.

Noise spectrum from a Chemical reactionin an electrolytic solution

The association-dissociation reaction of a divalent electro-lyte proceeds as follows (3)

ki2 k23Me++ + S04-- ± Me++(H2O)SO4-- ± MeSO4. [4]

k2l k32

where Me++ is any divalent metal. We used beryllium sulfate.Let n,, n2, n3 denote the molar concentration of Be++, S04--ions, Be++(H2O)SO4--, and BeSO4, respectively. Then wehave

Ko = kI2/k21 = n2/nl2 = 3.1 M-1

Ki = k23/k32 = n3/n2= 0.7

K2= Ko(1 + K,) = (n3+ n2)/nl2 = 5.2 M-1 [5]

where the numerical values for beryllium sulfate were ob-tained by the pressure jump (10) and temperature jump (11)techniquest. The rate-limiting step of the reaction (Eq. 4)is the expulsion of H20 from the inner coordination sphere,i.e., k23, k32<< k12, k21. Under these assumptions, the slow re-laxation time, T. which we measure in our experiments is givenby

1/,r = k52[1 + K1(2K~nl)/(l + 2K~n1)] [6]

NA(W)

NAk1 NB fl:k +kA W2l NB 1k2+ k21

NA

-TIME tI

4TrN (l-N/N)[l+(21r)2]j'

, V,(2rT)

t

FREQUENCY V

FIG. 1. The theoretically predicted fluctuations (top) andfrequency (power) spectrum (bottom) from the simple chemical

ka2reaction NA 4 NB. From the amplitude and frequency de-

k2lpendence of the fluctuations, the number of molecules, NA, andthe rate constant, r, is determined.

where nli is obtained from the salt concentration, n, by way ofthe rate constant K2.We next calculate the mean square of the voltage fluctua-

tions when a constant current passes through the electrolyteconfined in a capillary of length 1 and radius r. These fluctua-tions are related to the number of ions N1 as follows:

(OV)2 = I2d0(OR)2 = V2dc OR= (ON1) N1 aR)2R2 ~ N21 R 6N1/ [7]

where R is the resistance of the electrolyte in the capillary.The quantity in the last parenthesis would equal one if theconductance were exactly proportional to the number of ions.For a 30 mM solution of beryllium sulfate, we determined ex-perimentally [(N1/R) (bR/bN1) ]2 to be about 0.8.The fluctuations in the number of BeSO4 molecules, N3, are

related to the monitoring fluctuations, ON1, by ON1 =

ON3(bN,/1N3) = -0N3/(1 + 2nK0). From Eqs. 2 and 5we obtain

(ON1)2 niK1K (1 + nlK, 1 \2N12 N \1 + nlKi /K 1 + 2niKoJ

[8]

where ni refers to molar concentrations and N, to the numberof particles. Combining Eqs. 5, 7, and 8, we obtain for thefrequency spectrum of the fluctuations per unit bandwidth

(,)6R 2 K1Ko[OV(V) ]2 = a Vde, )N,) (6 X 1020) (i7rr2l)

<(1 + nKo\11 \2 4-r []i + nK )1 + 2niKO/ 1 + (2TvT)2

where a is a correction factor that takes into account thefinite spreading resistance at the ends of the capillary. Itsvalue is approximately given by [1 + (xrr/21) ]'- (13).

It is useful to compare the reaction noise voltage with theinherent frequency-independent Johnson noise originating in

t The value for Ki given in ref. 11 seems to be wrong. They referto the work of Kohler and Wendt (10), who obtained an averagevalue of Ki = 0.7 ± 0.2.

Chemical Reaction Kinetics 871

872 Chemistry: Feher and Weissman

FIG. 2. The experimental arrangement used to measure thefrequency spectrum from an electrolyte.

a resistor. Its value per unit bandwidth is given by (5)

[6V(v)VJjohnson = 4kTR = 4kTp/(a7rr2)

l = 5.8 X 10-2 cm and r = 3.0 X 10-3 cm, Eq. 11 gives a

value of y = 1.1. Since Johnson noise can be measured withan accuracy of a few percent, the reaction noise can beanalyzed with 'y significantly smaller than unity.

MATERIALS AND METHODS

The cell

The electrolytic cell was produced by epoxying a capillary(see Fig. 2) between two Pyrex reservoirs that hold the elec-trolytic solution. The capillary was made by drawing out a

thick-walled Pyrex tube that was subsequently sliced with a

diamond wheel to the desired length. The dimensions of thecapillary were obtained by a direct microscopic measurementand by a measurement of the resistance with an electrolyte ofknown conductivity. Taking account of the spreading re-

sistance (13), the two methods agreed to within a few percent.A constant current source was provided by a battery con-

nected to the cell through a resistance that was large com-

pared to the cell resistance. The large current electrodes,

[10]

where p is the resistivity of the electrolyte, k the Boltzmansconstant, and T the absolute temperature. Since the Johnsonnoise and the reaction noise are uncorrelated, the observedmean square voltage is the sum of these two contributions.From Eqs. 9 and 10, we obtain the ratio of the reaction noiseto the Johnson noise at zero frequency:

[WV(0) 12Reaction rKjKa3Ko _Vdc

[5V(0) I2Johnson 6 X 102kTp KI

XP(- ) +n'Kj; L1 + 2nKj [11]

The two variables that are at our disposal in trying to maxi-mize the above expression are the dimensions of the capillaryand the concentration, i.e., the resistivity of the electrolyte.In order to have a well-defined uniform electric field, we want a

capillary whose length 1 > r. Since the reaction noise is pro-

portional to V2d, whereas the Johnson noise is independent ofVdc, it is clear that we want to apply the highest voltagecompatible with the power dissipation within the capillary.The electrolyte is cooled by the wall of the capillary whosearea is 27rrl, whereas the power dissipated in the electrolyteis given by V2dlrr2/ (pl). From a balance of these two termswe see that, for a maximum allowable temperature rise ofthe electrolyte, the expression V2dc/12p that appears in Eq.11 is inversely proportional to r. To maximize y in Eq. 11, wewant, therefore, to use a capillary with the smallest radius.The smallest usable radius is given by the appearance ofnoise sources that are not associated with a chemical reaction(see Other Noise Sources). Once the dimensions of the capillaryhave been chosen, the maximum power dissipation V2dc/R isdetermined, and Y increases with decreasing concentrationni (see Eq. 11). The lowest practically usable electrolyte con-

centration is determined by the shunting resistance and timeconstant effects of the electronic circuitry.For a 30 mM solution of beryllium sulfate at room tempera-

ture with capillary dimensions as used in our experiments of

4.(C)

C-

g9 2t

Orr,- 3.

CWI

(5

H-:Cut

c-

c() 2Z5

T

I.0

.o

1.0

0

FREQUENCY M[Hz]

FIG. 3. The noise spectrum from a 30 mM solution of beryl-lium sulfate. 1024 one-second sweeps were averaged to obtain the

traces (see Fig. 2). (a) and (b) The Noise spectrum at two tem-

peratures originating from the chemical reaction (see Eq. 4)

superimposed on the Johnson noise. The dashed trace representsthe theoretical fit (Eq. 9). (c) Johnson noise obtained with no

voltage applied to the cell.

-32r=6.OXIO cm a

\ | RCELL 7.8x 10

\dc: 4.6x10 Amps

----THEORY

(vV2-30 JH

( JOHNSON

o I

b

a RCELL = 6.4X 10 a

H-Vg/z 52 Hz

-~~~~~~~~~~~~~~~~~~7T(V )JOHNSON

IO.'

.0 C

Idc = T = 230CRCELL 6.5x It0Q

1.0 F-n-a>-(IV )JOHNSON

o II I

0 100 20(

I

Proc. Nat. Acad. Sci. USA 70 (1973)

Proc. Nat. Acad. Sci. USA 70 (1973)

A (about 4 cm2), and smaller potential electrodes, B (about0.5 cm2), were made out of platinum. The current electrodeswere covered with platinum black (14). The cell was mountedin a heavy [0.62 cm (0.25 inch) wall thickness] aluminum boxwhose temperature could be varied and monitored between00 and 500C. The box was placed on a shockproofed table tominimize microphonic effects. The temperature rise of theelectrolyte in the capillary due to the current heating wasobtained from the measured resistance and the known tem-perature dependence of the resistivity of the electrolyte(about 2% per 0C). The cell resistance was measured with aconventional ac Wheatstone bridge. The electrolytic solutionwas made from reagent grade beryllium sulfate dissolved indouble-distilled water to the desired concentration. The solu-tion was passed through a 0.22-,gm Millipore filter for removalof lint and other particulate impurities. The solution wasdegassed before it was put into the cell. The cell was thor-oughly cleaned before use. Care was taken to fill the two armsof the cells to the same level. If this was not done, an addi-tional noise source was introduced (see Other Noise Sources).

The electronics

The voltage across the potential electrodes is made up of a dccomponent Vdc and the fluctuating component bVac, whichcontains the desired information. The latter is amplified andfed to a 200-channel spectrum analyzer and squarer. Thisinstrument analyzes the voltage fluctuations at 200 differentfrequencies, in our case in 1-Hz increments from 0 to 200 Hz.The bandwidth of the instrument is 1 Hz. It is followed by anintegrator that stores and averages successive experimentalruns in order to improve the signal-to-noise ratio of the"noise." Its output is connected to an x-y Recorder, whichplots [6V(v) ]2 as a function of frequency. A block diagram ofthe electronic circuitry is shown in Fig. 2.

EXPERIMENTAL RESULTS

The noise spectrum from a 30 mM solution of beryllium sul-fate was obtained as described in the previous section. Theexperimental results are shown in Fig. 3. Fig. 3c shows thenoise spectrum with no dc current flowing through the elec-trolyte. A similar spectrum was obtained from a wire-woundresistor having the same resistance as the electrolyte. Forthese two cases, one expects a frequency-independent (whitenoise) spectrum due to Johnson noise (Eq. 10). A slight devia-tion (about 10%) is observed at higher frequencies. Thisdeviation is attributed to the imperfect frequency responseof the spectrum analyzer and was taken into account inanalyzing the spectra.The reaction noise from the electrolyte at two different

temperatures is shown in Figs. 3a and b. The dashed linerepresents the theoretical fit (Eq. 9) superimposed on theJohnson noise (Eq. 10). No data were taken below a fre-quency of about 20 Hz. At these low frequencies additionalfluctuations, not connected with the chemical reaction, wereobserved (see next section). The amplitude of the Johnsonnoise was obtained with no adjustable parameters. Foranalysis of the reaction noise spectra, both the amplitude[V(0) ]2 and ahlf-width P,/2 were chosen to give the best fit.We estimate the statistical error of obtaining these twoquantitatives as 4-10%. In principle, V1/2 is the only adjust-able parameter, since [V(0)(]2 can be obtained from Eq. 9.However, in view of the uncertainties in the values of the

100

80

60

40~

-E

I.-I:;

LI-

--I

3.2 3.4

--103/T [°k]3.6

FIG. 4. Half-width of the noise power spectrum from beryl-lium sulfate against the reciprocal temperature. From the slopeof the solid line, an activation energy of 8.2 ±4 0.4 kcal/mol isdeduced (Eq. 12). This is in good agreement with the value of8.5 ±4 0.4 kcal/mol obtained by the relaxation method (10, 11).

rate constants, the amplitude was also left as an adjustableparameter to be checked against the theoretically predictedvalue. This was done by calculating the ratio [6V(0) I2Reaction/[6V(0) ]2Johnson from Eq. 11. For 240C and 13WC, the predictedvalues are 1.1 i 0.3 and 2.5 ± 0.6, respectively, where theerror is due to the quoted uncertainties in the values of theequilibrium constants (10, 11). The experimentally observedratios (see Figs. 3a and b) are 1.5 i 0.2 and 3.2 i 0.3, inagreement with theory. This result leaves little doubt thatwe are indeed observing the reaction noise from the elec-trolyte. From the concentration dependence of the fluctua-tions, improved values of the equilibrium constants Ki andK, could be obtained.

Let us now compare the kinetic parameters obtained in thiswork by fluctuation spectroscopy with those obtained byrelaxation methods (10, 11). Our results are presented on anArrhenius plot in Fig. 4. The effective activation energy EA

100

->FREQUENCY [Hz]

200

FIG. 5. Noise spectrum from a 10 mM solution of zinc sulfateflowing through a capillary with dimensions as indicated. Thedashed line is a fit to a Lorentzian. From the integrated area underthe curve, the number of ions N, can be obtained (see Eq. 14).The characteristic time r corresponds to the emptying time ofthe capillary. The spikes (e.g., at 120 Hz) are due to micro-phonics. Care was taken to eliminate this noise source in theexperiment shown in Fig. 3.

o I

(40C) (30%C) (20'C) (10)

1, 1 II I\

- OV)JOHNSON AC _ _ __ _

Chemical Reaction Kinetics 873

-F-

r--lIr

2CICI.--iC>

N-:7;.

tal

8,3:M11

.:ICIO

cr-LLJCL

CliLiCD_:II'-.9LLJV)C>

t

874 Chemistry: Feher and Weissman

is obtained from the slope, i.e.,

EA = R [d (lnvi12)/d(1nl/T)] = 8.2 i 0.4 kcal/mol. [12]

This value compares very well with the activation energy of8.5 i 0.4 keal/mol obtained from relaxation measurements(11). The relaxation rate 1/r obtained at 250C is 350 i 40sec' (corresponding to P1/2 = 55 Hz; see Fig. 4). This is againin good agreement with the value of 375 sec1 obtained bythe pressure jump technique (10).

OTHER NOISE SOURCES

Besides the reaction and Johnson noise, there are severalother noise sources. These played a negligible role in the ex-periments described in the previous section, but under dif-ferent experimental conditions (e.g., different cell dimen-sions) they may be of importance and contain additional in-formation. We shall limit ourselves to a discussion of inherentnoise sources and leave out those that arise from imperfecttechniques (e.g., microphonics, electrode noise, fluctuationdue to gas bubbles or particulate impurities).

Noise due to energy fluctuations

The energy of a system fluctuates around its average energyby an amount (bE)2 = kT2c (5), where c is the heat capacityTof the system. Since E = cT, these energy fluctuations corre-spond to a temperature fluctuation (bT)2 = T2k/c and bythe temperature dependence of the resistivity of the elec-trolyte to resistance and voltage fluctuation (bV)2 given by

(bV)2/V2dc = (bR)2/R2

_(bT)2 = (T2k/c) RT [131

where 6R/(R-T) for 10 mM beryllium sulfate is about -2%per degree. For the experimental conditions depicted in Fig.3b, Eq. 13 gives (6V)2 6 X 10-'4 V2. This is about one orderof magnitude smaller than (bV)2 of the reaction noise. Thecharacteristic time r with which these temperature fluctua-tions decay can be estimated as follows: The rate of heatingof the electrolyte with a power in.put P equals P/c. Understeady-state conditions, the heating and cooling rates areequal, and a temperature rise AT is observed. The char-acteristic cooling (i.e., decaying) time is then given by r =ATc/P. For our experimental conditions (Fig. 3b) r - 4 msec.The characteristic time of the reaction kinetics was approxi-mately the same (3.1 msec), so that the value [bV(v) ]2 due tothe temperature fluctuations is expected to be about anorder of magnitude smaller than that obtained from the reac-tion noise. Thus, at room temperature it can contribute anerror of about 10% to the amplitude of the reaction noise.By replacing the beryllium sulfate solution with zinc sulfate,in which the reaction noise is not expected to show up(TReaction 10-8 sec), we found under the experimental condi-tions of Fig. 3b an approximate 5% increase in Johnson noisearound 30 Hz. Although this result is consistent with the

t Whether one uses c, or c, will depend on the details of the ex-perimental arrangement. Since, for our system, the differencebetween c, and c, is of the order of 1%, we need not worry aboutthis point. Similarly, the contribution of the chemical reaction toc is completely negligible.

above mechanism, further experiments are required to estab-lish it with certainty.

Noise due to fluctuations in the total number of molecules

If the molecules in the capillary can communicate with anoutside reservoir, their number N will fluctuate around anaverage value given by (bN)2 = N (5). The resulting fluctua-tions in the number of ions N1 will then be given by

N12 - (1/N1)(1 + niKm)[1/(l + 2niKz)]2 [14]

where the last bracket takes into account the change of N1with N. A comparison of Eqs. 14 and 8 shows that for ourexperimental conditions (Fig. 3b) the mean square value ofthese fluctuations is about one order of magnitude larger thanthat arising from the chemical reaction. However, becauseof the slow rate of these fluctuations, the power spectrum isshifted to very low frequencies and therefore did not con-tribute in Fig. 3 to an observable noise above 20 Hz. Thecharacteristic time of these fluctuations depends on themechanism of replacing the molecules inside the capillary.We shall discuss two of these mechanisms: diffusion andvolume flow.

Diffusion. The molecules diffuse out of the capillary with acharacteristic time r 12/(12D) where 12/12 is the meansquare distance from the ends of the capillary and D is thediffusion coefficient. For the capillary dimensions used inour experiments and with D __ 10-5 cm2/sec (15), the char-acteristic time r ' 30 sec. corresponding to a frequency ofabout 0.005 Hz. This is clearly outside the range of observ-ability in our experiments and, therefore, does not interferewith analysis of the kinetic data. Again, it should be notedthat for smaller structures (capillaries) the diffusion mecha-nism may constitute a significant noise source. In fact, in theoptical experiments the relevant diffusion length l is either thereciprocal scattering wave vector (8) or the beam diameter ofthe laser§ (9). In both of these cases it enters critically intothe interpretation of the experimental results.

Flow Through the Capillary. A simple way to replace thefluid in the capillary is to produce a pressure difference Apbetween its two ends that will force the liquid through it.For laminar flow, the time r it takes to empty a capillary oflength l and radius r is given by r = 812n1/(r2Ap), where -q isthe viscosity of the fluid (16). In order to observe this effectin a convenient frequency range, we used a cell with dimen-sions I = 1.8 X 10-2 cm, r = 2.5 X 10-s cm. The pressuredifferential was created by raising the level of the electrolytein one arm of the cell. In order to eliminate the reaction noisea 10 mM solution of ZnSO4 was used. The experimentallyobserved noise spectrum from a flowing electrolyte is shownin Fig. 5. The experimental trace could be fitted well with aLorentzian [1 + (2rvtr)2]-1 (see dashed line in Fig. 5), al-though there is no a priori reason to believe that this is theexpected functional dependence. The characteristic time rwas 4 msec and the ratio of the "flow noise" at zero frequency[bV(0) ]2Flow to Johnson noise was found to be 6.5 (see Fig 5).The calculated time it takes to empty the capillary with a

§ This could, in principle, be avoided by illumination of theliquid in a capillary whose diameter is smaller than the opticalbeam diameter (e.g., by the use of light pipes).

Proc. Nat. Acad. Sci. USA 70 (1978)

Proc. Nat. Acad. Sci. USA 70 (1973)

pressure head used in the experiment (7.4 mm of H20) is 5msec and the calculated ratio of "flow" to Johnson noise(using r = 4 msec) is 6.8, in excellent agreement with ex-periment. It should be noted that by these fluctuation methodsone determines directly the number of ions N, (i.e., the dis-sociation constant), which is not easily determined by othermethods.

DISCUSSION AND CONCLUSIONWe have demonstrated that the frequency spectrum of thefluctuations (i.e., noise) from a chemical reaction can be usedto obtain the kinetic parameters of the reaction. We haveconsidered here an electrolyte as a convenient model system,but the method should have general applicability to problemsof physical, chemical, and biological interest. For example,the melting of DNA or other poly nucleotides is accompaniedby a resistance clange (17), which makes the techniquedescribed here directly applicable. For enzyme reactions,monitoring of the fluctuations of an optical parameter mightbe advantageous. For the optical case, an expression analogousto Eq. 11 is easily derived in which the Johnson noise is re-

placed by the shot noise of the phototube. By use of an opticalinterferometer this unwanted shot noise may be greatly re-

duced.A general theorem (fluctuation-dissipation) states that

the kinetic parameters obtained from the frequency spectrumof the fluctuations are the same as those obtained fromchemical relaxation measurements (5). Since the latter hasbeen developed to a high degree of sophistication during thepast two decades, one may wish to compare the two methodsand to inquire whether fluctuation spectroscopy has any ad-vantages to offer (besides the personal satisfaction of usingrather than fighting noise). One basic difference is thatfluctuations are measured on a system that is in macroscopicchemical equilibrium, whereas in the relaxation method, thesystem has to be perturbed. Such pertubation may, in certainsystems, produce an appreciable error. Another possibledifficulty of the relaxation method may be the inability tofind the proper physical parameter to perturb the systemwith. Consider, for example, a racemic mixture (equal number)of molecules with right- and left-handed symmetries. Apressure or temperature jump cannot be used (since themixture remains racemic) to investigate the kinetics of trans-formation between the two states. However, the frequencyspectrum of the fluctuations of the absorption of circularlypolarized light could provide this information.Another basic difference between the two methods is:

Fluctuations arise essentially from the "granular" nature ofmatter. Consequently, the number of reacting molecules entersdirectly in the amplitude of the fluctuations. One can, forinstance, distinguish by the fluctuation method whether one

has a system with n molecules each with an extinction co-

efficient e or a system with nx molecules having an extinctioncoefficient e/x. The relaxation method cannot distinguishbetween the two systems. These advantages of the fluctuationmethod have to be weighed against the more stringent experi-mental requirement imposed by the handling of small (noise)signals.During our investigation of the "chemical noise," other

sources of noise were encountered. The most prominent ofthese arose from a flow of the electrolyte through the capillary.These fluctuations are related directly to the number ofcharge carriers. One can use, therefore, the "flow noise" to

determine the molecular weight of macromolecules, in com-plete analogy with the classical light-scattering method ofdetermining molecular weights. Although most of our experi-ment were designed to make these noise sources small incomparison to the chemical reaction noise, they can be ofimportance under different experimental conditions. Forexample, these fluctuations should be considered in analyzingthe noise from microelectrodes (18) and small orifices (19).These voltage fluctuations also could be used to investigatethe transport of molecules through small channels, e.g., inbiological membranes. Voltage fluctuations from membranesand nerves have already been observed by several investi-gators (20).We thank R. A. Isaacson for his expert technical assistance, in

particular with the assembly of the cell, and Drs. H. B. Shore, D. R.Fredkin, B. H. Zimm, and W. A. Hagins for several helpful andstimulating discussions. M.W. was the recipient of a NSF pre-doctoral fellowship. This work was supported by USPHS GrantGM-13191.

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New York) 2nd ed. pp. 205-214.16. Vennard, J. K. (1963) in Elementary Fluid Mechanics (John

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