has confirmed this conclusion. The reaction solution contained approximately 0.75F ammonia, 0.15F ammonium chloride, 0.15F sodium perchlorate, and 0.01F thioacetamide. Samples of the reaction mixture u ere taken at timed intervals and were analy~ed for sulfide. After the complete removal of sulfide as hydrogen sulfide by acidification of the sample (IO), the thioacetamide concentration iyas determined spectrophotometrically at 262 mp. The amount of sulfide formed a t any time was always smaller than the quantity of thioacetamide which had disappeared, the difference increasing with the time of reaction. For example, the amount of sulfide present after 60 minutes was about eight-tenths of the amount of thioacetamide which had reacted. M7e believe that the difference in the rates of disappearance of thioacetamide and formation of sulfide is due t o the oxidation of sulfide by oxygen during
the course of the kinetics measurements and that, in fact, the two rates are identical. Definite proof that the rate of formation of sulfide is equal to the rate of disappearance of thioacetamide has been obtained in a study of the precipitation of lead(I1) sulfide by thioacetamide from ammoniacal EDTA solutions. The lead(I1)-EDTA complex does not undergo a direct reaction with thioacetamide; however, this complex does react rapidly with the suEide ions generated by the amnionia-thioacetamide reaction. The rate of precipilation of lead(I1) sulfide follows quantitatively the rate of disappearance of thioacetamide predicted from the data reported in the present investigation. Lnder the conditions of these experiments, sulfide escapes air oxidation because of its fast reaction with the lead (11)-EDTA complex. Full details of this precipitation study will be described in a subsequent paper.
ST. JOHN, W. J.
McCARTHY, and J.
T
are many factors which influence the measured signal and noise in analytical techniques based upon the measurement of luminescence (fluorescence, phosphorescence, and delayed fluorescence) of molecules in the condensed phase. So far no detailed treatHERE
1828
@
ANALYTICAL CHEMISTRY
E. H., Talanta
12,
357 (1965).
(3) Klein, D. H., Swift, E. H., Ibid., p. 349. (4) Ibid., p. 363.
(5) Levens, A. S., “Graphics in Engineering and Science,” p, 386, Wiley, New Pork, 1954. (6) Meites, L., “Handbook of Analytical Chemistry,” p. 1-34, McGraw-Hill, New York, 1963. (7) Peters, D. G., Swift, E. H., Talanta 1, 30 (1958). (8) Rosenthal, D., Taylor, T. I., J. Am. Chem. SOC.79, 2684 (1957). (9) Swift, E. H., Anson, F. C., in “Advances in Analytical Chemistry and Instrumentation,” C. N. Reilley, ed., Vol. 1.,I. ). 293. Interscience. New York. 1960. (10) Swift, E. PI., Butler, E. A,, ANAL. CHEW28, 146 (1956). RECEIVEDfor review -4ugust 8, 1966. Accepted September 22, 1966.
.
D.
WINEFQRDNER
Department of Chemisfry, University o f Florida, Gainesville, Ha.
b A general approach is used to derive equations which are concerned with the influence of experimental and spectral parameters on the photodetector signal, signal-to-noise ratio, minimum detectable sample concentration, and an analytically useful monochromator slit width in luminescence spectrometry. The general equations presented in this manuscript relate all important parameters involved in luminescence measurements. By the use of these equations, the influence of variation of any parameter on the shape of the analytical curve, the minimum detectable sample concentration, and the precision of measurement can be predicted. The use of these equations facilitates optimization of each parameter. Estimates of phosphorimetric minimum detectable Concentrations of several organic molecules obtained using the derived equations compared quite well with measured values using commercial equipment.
(1) Butler, E. A., Peters, D. G., Swift, E. H., AXAL.CHEM.30, 1379 (1958). (2) Klein, D. H., Peters, D. G., Swift,
oise Theory In ectrometry
plications of olecular Lurnine P. A.
LITERATURE ClTED
32607
ment of the experimental factors and the extent t o which these factors affect the measured detector signal, the noise, and the shape of analytical curves, has appeared in the literature. Therefore, in a practical analysis, the investigator is usually forced to rely on a trial and error approach to obtain the optimum experimental conditions. Trial and error optimization of parameters is generally time consuming and subject to considerable error. In this manuscript, a general approach to the theory is giyen, and equations are derived for the photodetector signal, signal - t o - noise ratio, and limiting detectable sample concentration as a function of sample characteristics and experiniental parameters. The derived equations are used to predict the influence of experimental parameters on the measured signal, the shape of the analytical curve, the signal - to - noise ratio, and the limiting detectable sample concentration. The approach used in the derivation is similar to that used by Winefordner and Vickers (%%’) in their theory concerning atomic emission flame spectrometry. DERIVATIONS OF SIGNAL AND SIGNAL-TO-NOISE EXPRESSIONS
Part I. The Signal. Several assumptions must be made which will facilitate discussion of the theory b u t
will not seriously limit its use. It will be assumed that the entrance and exit slit widths and heights are equal for each monochromator. bfost monochromators have equal entrance and exit slits or can be made t o have equal slits. No significant gain in resolution or in signal-to-noise ratio results if the entrance and exit slit widths and heights differ. Also, it will be assumed that the spectral band width of the excitation monochromator, s, is appreciably less than the half-intensity width of the excitation spectral band and that the spectral band width of the emission monochromator, s’, is appreciably less than the half-intensity width of the emission spectral band. The influence of spectral band width on absorbance and intensity values has been considered by Broderson (S) and others (4, 18). For convenience, it will also be assumed that the luminescence emission band shape is given by a Gaussian function of the wavelength (6). Finally it will be assumed that the intensity of absorption and emission are approximately constant over the spectral band widths, s and s’, respectively, and that the measured emission band is a result of only a single electronic transition. The most common experimental arrangement for measurement of luminescence (fluorescence, phosphorescence, or delayed fluorescence), consists Qf the
excit'ation, is the concentration in moles literm1 of species M, and the summation is taken over all absorbing species except the sample, S. Thus, the radiant power in watts absorbed by the sample, Pabslin the region hl to bp is given by the product of Equations 1 t o 3, namely, EXCITATION MONOCHROMATOR
Pabj
(4
= I " k , f l f ~ ,watts
The total radiant power absorbed, P B i b 3 , can be converted t o total radiant power emitted (13) by multiplying Pabs by the luminescence power efficiency @S for the specific process in concern. Therefore, the maximum radiant power emitted, p,,, in watts, is given b3TO EMISS I ON
Pem
MONOC H R OMA T O R Figure 1. General arrangement for sample cell used in luminescence measurements
b = width of cell parallel to excitation beam d = length of cell parallel to emlsrion beam (bz -+ bl) = region over which emlssion is measured (dz ---f dl) = region over which excitatlon occurs
excitation source and monochromator optically aligned perpendicularly to the emission monochromator and photodetector. The cell is usually placed as shown in Figure 1. In this section, the current, i, a t the anode of the photodetector (usually a multiplier phototube) will be related to the various parameters involved in the experimental system. By means of an approach similar to that used by Winefordner and Vickers (22) in atomic emission flame spectrometry and by consideration of all factors of importance in molecular luminescence measurements ( 6 ) , a general equation for the photodetector anodic current, i, produced as a result of luminescence radiation can be derived. A compilation of all symbols and their units will be found in Appendix I. The radiant power of the exciting radiation reaching point bo (see Figure 11, P,, in watts is given ($2) by
Pi
=
I" W H s T ,
($>
=
I'k,, watts
(1)
where I o is the intensity of the source of excitation in watts cm.-2 mp-' stern-', W is the slit width in cm. of the excitation monochromator, H is the slit height in om. of the excitation nionochromator, s is the spectral band width of the excitation monochromator in mp, TIis the transmission factor (no units) of the excitation monochromator, and (A,/F,Z) is the effective aperature of the excitation monochromator-Le., the solid angle, in steradians, subtended by the mono-
chromator (A, is the area in cm.* of the collimating mirror or lens, and F , is the focal length in cm.). The parameter k , is the optical constant for the excitation monochromator and is given by the product TT'HST~(A,/F,~).The spectral band width, s, is given by s = Rd(W W,,,), where W,,, is the minimum resolving power slit width which will generally be negligible (22) compared to W when making molecular luminescence measurements. For simplicity, it will be assumed here that the entrance optics are such that the entrance slit of the excitation monochromator is fully and uniformly illuminated and that the radiation just fills the collimator. The fraction, fi, of the incident radiant power absorbed by the sample, S,in concern in the region bl to bz is given (18) by
+
f1
= exp(-2.3asblCs)
-
exp( -2.3asbzCs)
(2)
where as is the molar absorptivity coefficient of the sample in liters mole-' ciii.-l a t the wavelength, A, being used for excitation, and Cs is the concentration of the sample in moles liter-I, I n general, there mag be other absorbing but non-luminescing species present in the sample solution which reduce the radiant power absorbed by the sample by a factor ( d ) , j z . The factor f2, is given by .f~= exp(-2.3hz
a,wC4w)
(3)
M
where u,w is the molar absorptivity coefficient of species idif in liters mole-1 ern.-' at the wavelength, A, used for
=
Pabs@S
= Iofif2@s,watts
(5)
The radiant power emitted can be converted into maximum intensity emitted (vatts,'cm.2 of cell surface, ster. +I, mp) by dividing the radiant power emitted, Pem. by the sample surface area, A,; by 4n, the number of steradians in a sphere; and by the halfintensity emission band width in nip, A i ' of a Gaussian emission peak. The intensity of emission, I,, is then given bY
matts cm.-2 ster.-l nikt-l
(6)
A fraction of the emitted radiation may be reabsorbed by the sample (self-absorption) or by extraneous species. The fraction of the luminescenee radiation reaching the cell surface is accounted for by f3 (see appendix I1 for the derivation of f3). The power of radiation in matts passing through the exit slit of the emission monochromator and reaching the photodetector is given by I M k , ' , where the factor k,' is the optical constant of the emission monochromator and is defined by a product identical to that fork,, Equation 6 applies only if the sample is being excited continuously and if the luminescence is being continuously measured. In fluorimetric studies, this is generally the case. However, in phosphorimetry or in delayed fluoreecence studies, a dynamic shutter mechanism such as a rotating can (7, 9, 19, 20) or a Becquerel type disk (13) is usually used t o modulate out-ofphase the exciting radiation and the luminesceiice radiation in order to minimize the signal due to fluorescence and incident light scattering. Therefore the measured photodetector signal depends upon the time elapsed between initiation of excitation and measurement of the luminescence signal. 0'Haver and Wiiiefordner (11j have derived an expression to account for the loss of signal due to intermittent exVOL. 38, NO. 13, DECEMBER 1966
e
'1829
citation and observation. They defined a parameter, a‘, to account for this effcct, where 01’ is the ratio of the observed power emitted when uying mtciniittcnt excitation and observation to the power emitted if using continuous escitation and observation. Of course 01’ is unity when using continuous excitation arid observation. It mill be assumed that the time dependency of the decay (and growth) of luminescence processes can be described by a single exponential term (6,is). It has been shown recently (1, 5 , 16) that some luminescence decays are, in fact, not adequately described by a single exponential term; ho\\-evcr, this deviation is generally minor and can be neglected in most cases (5) as will be done here. The , luminescence is dedecay time, T ~ of fined ab the time for the intensity to decrease from the original value to l / e of that value. Therefore, if t i8 the time a t which the luminescence is measured afier coniplcte termination of the excitiiig radiation, then the eniilted intensity and the resultant photoanodic current, i , is proportional t o the exponcntial factor, eq(-t/TJ. Of course, if the exciting radiation is not terminated by a shutter mechanism and if stcady state luminescence is being mcamred, then the exponential term is unity. The photoanodic current, i, due to thc radiant power reaching the photocathodic suriace is found by multiplying the radiant power in watts reaching the photocathode by y , the sensitivity factor, which has units of ainpercs a t the photodetector anode per watt of radiant power incident on the photocathode. Therefore, the photoanodic current is given by
(7) Two limiting and experimentally useful caws of Equation 7 can be defined which rebult in considerable simplification of the cquation. The limiting cases describe nearly all experimental situations and can be used t o investigate the possibilities of optimizing esperimenial conditions. Liniiting Case I r e p esciits the luminescence of a single component in a dilute solution of a solvent which does not appreciably absorb at the wavelengths of interest. This case approximately describes nearly all quantitative luminescence measurements. Liniiting Case I1 represents the luminescence of an extremely concentrated solution of a single component and describes the so-called quantum counter effect. Limiting Case I. This case is the one uaually considered in text books and reference works although many more assumptions are made to 9 830 *
ANALYTICAL CHEMISTRY
simplify the expression even further. If there are no absorbing species present in the measured solution other than the luminescent species then l f
.,-
to unity. Also, i t is assumed that excitation occurs over the entire cell (b2 = b and bl = 0) and emission is observed over the entire cell length (& = d, dl = 0). If, in addition to these assumptions, the saniple is dilute, then f~ = [2.3asbCs], and fa = 1.0. Finally, it will be assumed that all measurements are taken with identical excitation and emission monochromators having identical optical constants, Le., k , = k’a, and that all emission measurements are taken at the emission peak. Under these conditions, which are approximately obeyed for most quantitative measurements, the photoanodic signal, i, is given by
In this case, i is proportional to the concentration of luminescing species, CS, as is usually predicted by simpler approaches (6, IS. 17) and is esperimentally observed for dilute solutions. Limiting Case IT. If in addition to the conditions stated for Case I, the sample is extremely concentrated, such as in quantum counters, then fi = 1.0, and if reabsorption of luminescence by sample molecules is negligible, then f 3 = 1.0. Under these conditions, the photoanodic current, i, will be given by
In this case, i is independent of the sample concentration as long as the sample is sufficiently concentrated to produce the qkantum counter condition. Also, i will be proportional to the power yield ?Pa; this is the basis of the quantum counter and its use for measurement of energy or quantum yields. Analytical Curves. The shape of the analytical curve of i US. sample concentration, CS, can be predicted from the above two equations. ilt low concentrations the slope of a log i vs. log Cs plot should be unity, and a t high concentrations, it should be zero. At very high values of CS, @S will often decrease causing a decrease in i with Cs at large CS. For any given experimental system, the absolute value of i on the upper plateau of slope equal to zero should be dependent entirely on the value of as. Part 11. The Noise. The total Z t - m e a n square (rms) noise current, Air in amperes, at the photoanode is given (99,2s) by
where is the rms phototube noise due to the shot effect, aisis the rms effective luminescence flicker noise, and is the rim luminescence convection noise. Both the exciting and luminescence radiation are caused to flicker by convection (and bubbling) of the therniostating medium-e.g., liquid nitrogen in some luminescence studies. The effective luminescence flicker is composed of contributions from aciual source fluctuations due to inherent instability in the source of excitation and from fluctuations in the source intensity a t any point due to convection and bubbling of the thermostating medium. Convection flicker, on the other hand, is due entirely to fluctuations in the luminescence resulting from convection and bubbling of the thermostating medium. The noise terms in Equation 10 add quadratically because they are independent noises (22). For most experimental systems used for analytical measurements, the electrometer noise will be negligible compared to the other noises in Equation 10. The rms Johnson noise due to thermal agitation of electrons in the load resistor is also negligible in most practical studies compared to other noises in Equation 10 and so is not included in the above equation. Both electrometer noise and Johnson noise, if significant, can be considered in the manner used by Winefordner and Vickers (2.2, 23) and Winefordner and Vcillon @ I ) , respectively. The rms shot noise, G,is given by
where Af is the frequency response bandwidth of the electrometer-readout system, id is ths photoanodic dark current, in amperes, iB is the photoanodic current, in amperes due to background luminescence, i is the photoanodic current, in amperes, due to the sample luminescence, and k D is the detector constant which is given by the product 2e,BJI, where e, is the charge of an electron in coulombs, and BM is the overall gain of the phototubei.e., gain from the cathode and each successive dynode and is given by G c
+ Gz-’ + G S - ~+ . . . .+ G1 + Go,
where G is the gain per dynode and x is the number of dynodes in the photodetector. The total rms luminescence flicker noise, in amperes, ais, due to the luminescence background and sample is given by an expression similar to the one developed for atomic absorption flame spectrometry by Winefordner and Vickers (as),namely,
where 6 is the effective source flicker factor in units of sec.l’z defined as the
as in fluorescence or in fast decaying phosphors, then Afs and A ~ Bwill be equal to Af. The parameters T L and rBneed be included in Equations 14 and 15, respectively, only if ~ T L > > Af-’ and 4TB>> Af-l, which, with normal electrometer-readout systems, is the case only if the luminescence species has a relatively long lifetime. appendix I11 includes a summary of experimental methods used for evaluation of the noise terms given in this section and measured values of the noises obtained for the studies carried out for the manuscript. The total rms noise current, G,obtained by substituting appropriate expressions given above into Equation 10, is given by
-
I
-3
0
L o g S t t Width‘ ( c e n t i m e t e r s )
+I
Air = [ A f k ~ ( i , f
*2
4
=
+
(13) The 6 and E add quadratically because they are independent fluctuations. The parameter Afs in Equation 12 is the effective frequency response bandwidth for source flicker and is given approximately (16) by 4 8 2
+
(14) where all terms have been defined above. The paranieter A ~ Bis given approximately (15) by [ ( ~ T B $) ~
Aff-2]-”2
(15)
where T B is the luminescence lifetime of the background in seconds and all other terms have been defined above. The total rms convection (and bubbling) noise, &, in amperes due to the fluctuation in the luminescence background and sample is given by an expression similar to Equation 12
-
A& =
E
(17)
iB]2]1’2
The current is is given by an expression similar to Equation 8 except that all terms pertaining to the sample, 8, are replaced by appropriate terms for the background, B. I n an actual experimental situation the actual identities of the background species are seldom known, and it is difficult to evaluate i ~ . However, i~ is easily obtained by direct measurement. Part 111. The Signal-to-Noise Ratio and Optimization of Experimental Parameters. The signal-to-noise ratio is approximately given by the ratio of Equations 8 to 17,
2.310k,2a8bCs@8ycrt 4nA,AXt d ~ . f k , ( i + ~ iB + i) +
tz[das i + d4fB iB]z+ € z ~ f ( i +
$2
(184
or
€2
Afs = [ ( ~ T s ) ’ Afs-2]-1’z
AfB =
~’A.f[i f
where all terms have been previously defined. The factor E in Equations 13 and 16 are the same value if and only if the distance traversed by the incident radiation through the thermostating medium is the same as the distance traversed by the luminescence radiation through the thermostating medium. This, of course, is usually the case; if i t is not the case, then E in Equation 13 applies only to the exciting radiation, and the E in Equation 16 applies only to the luminescence radiation.
2 = Air
+ i) f
t2[dafs i f dafsiB]’ f
Figure 2. Calculated plots of signal-to-noise ratio vs. monochromator slit width for several sample concentrations of a hypothetical molecule
ratio of the rms fluctuation in the effective source intensity to the effectire source intensity. The parameter 5: is composed of two factors, 6 and E, which account for the actual source flicker and the flicker introduced by convection of the thermostating medium. The paranieter 6, the source flicker factor is defined as the ratio of the rms fluctuation in the actual source intensity to the actual source intensity and has units of sec.1/2, The term E , the convection flicker factor, is defined as the ratio of the rms fluctuation in the source intensity at any point due to convection and bubbling of the thermostating medium to the source intensity a t the same point and has units of sec.l’z. The term 4 is given by
i~
daf(i -+ i
~ )
(16)
It should be pointed out that Equations 12 and 15 account for fluctuations due to the thermostating medium; these expressions, however, must be taken separately because Afs is determined by the lifetime of the luminescent sample molecule; a luminescent molecule is an effective integrator for rapid fluctuations in exciting radiation. The bandwidth, Af, on the other hand, is independent of the luminescent lifetime of the sample. If the convection noise ,j small, e.g., in most fluorescence measurements a t room temperature, then E is essentially aero and Equation 13 becomes 4 = 6. If A j s and A ~ Bare approximately equal, then Xs will be given by =E dafBi, a t the limiting detectable sample concentration. If the luminescence lifetime is short, e.g.,
where Ks and KB are defined in Appendix I. The above expression is analytically useful for predicting the variation in i/zas a function of any experimental parameter, e.g., TV. By plotting the signal-to-noise ratio us. the value of any experimental parameter, it is possible to determine if there is an optimum value of that parameter. If no optimum value exists for a particular parameter, it is possible by such a plot to obtain the best possible experimental conditions for the experimental system in use and to give the largest signal-tonoise ratio which will result in obtaining the highest sensitivity for a particular analysis and the greatest precision of measurement of any concentration. The most easily varied experimental parameter is the monochromator slit VOL 38, NO. 13, DECEMBER 1966
e
1831
Comparison of Calculated and Observed Minimum Detectable Concentrations for Four Compounds by Phosphorimetry’
Table I.
Minimum detectable concentration (moles/l.) at various slit widthsc Optimum conditions Cmin W,
Cmin
Cmin
Cmin
Cmin
Cmind
Cmin
Compoundb (W = 0.5) (W = 0 . 3 ) (W = 0 . 2 ) (Tt’ = 0 . 1 ) (TV = 0.15) (obsd.) 1.1 X 0.45 1 . 0 X 2.8 X 1.3 X 9.4 X 2.6 X 3X Retene 2.7 X 0.81 8 . 6 X 10-lo 3 . 5 X lo-’ 4 7 X 2.3 X 7.0 X 3X L-Tyrosine 4 X 10-9 6.6 X 2.7 X 4.0 X 10-8 1 . 2 X 10-8 1.7 X 0.51 1.T X Benzaldehyde 2 X 10-8 1 . 3 X loFQ 8 . 7 X 1 . 4 X 10-7 2 . 7 X 10-8 1.2 X 0.79 3.6 X Benzyl alcohol a All calculations were performed for the Aminco-Bowman Spectrophotofluorometer with phosphorescence attachment. All compounds were dissolved in ethanol. 0 A11 slit widths were in units of centimeters. d These calculated values of C,in should be compared with observed Cminvalues because W = 0.15 cm. is approximately the average value of the monochromator slit width for the Aminco monochromators (see text).
width, TT7, and hence it has been retained explicitly in Equation 18. In Figure 2, plots are given of signal-tonoise ratio us. monochromator slit width for various sample concentrations, Cs. See Appendix 111and IV for experimental procedures used to obtain parameters needed to calculate all constant terms in Equation 18 and for references t o computer programs necessary for rapid calculations. As can be noted from Equation 18b and the plots in Figure 2 , the signal-to-noise ratio varies approximately with TV4 a t very small values of W-i,e,, total noise is determined primarily by the shot noise which is due almost entirely to dark current, id, and so for small values of W, Equation 18b reduces to
Similarly for large values of tion 18 reduces to
W,Equa-
and so the signal-to-noise ratio a t large W’s is independent of slit width, TV (see Figure 2). Part IV. A Useful Monochromator Slit Width. It is evident from Equations 19 and 20 or from Equation 18 (also see Figure 2) t h a t there is no strictly optiniuin slit width where the signal-to-noise ratio reaches a maximum value and decreases a t both larger and smaller values of TI7. However, there is a n analytically useful slit width, TV,, for measurement of luminescence; this is the slit width which gives the greatest spectral resolution with near maximum signal-to-noise ratio. The slit width, TV,, can be calculated by equating Equation 19 to Equation 20 and solving for the slit width-Le., this is the value of TV which results a t the intersection point when the low TY curve is extrapolated to meet the high TV curve in Figure 2. The value of W , a t any sample concentration Cs is therefore given by 1832
m
ANALYTICAL CHEMISTRY
At the minimum detectable sample concentration, Cmin,i.e., the concentration GS corresponding to a value of i/G= 2, Equation 21 reduces to
the limit of detection, Cmin,the noise terms (in Equation 18) associated with the sample are necessarily negligible, and so K B >> K s and KB d a f B >> Ks It is assumed that AfB and Afs are comparable in magnitude or that Afs is smaller than AfB-these are generally good assumptions in practical situations. Therefore Cmin for any value of monochromator slit width, TV, is given by
428.
can be noted in Figure 2, the slit decreases in magnitude as width, ,TI’ Cs increases. It is quite possible that the slit width, W,,may be impossible to
attain in a practical experimental situation, which is the case in the experimentation performed in this paper. Because of the mechanical limitations of the slit mechanism and because the spectral band width corresponding to ?Vu may be appreciable compared to
the half-intensity width of the excitation or emission spectral bands, it may be impossible in practice to use a slit width as large as W,. Therefore, in a practical experimental situation the largest available slit width may have to be used but that slit width may be appreciably less than W,. It is interesting to note that a t the slit width, W,, the total current passing through the phototube is constant and consequently all noises are constants for any value of Cs. This is only true if Afs = A f B , Le., if 7s and T B
