JUNE 1979 The Analyst Vol. 104 No, 1239 Relevance of the Approximately Hyperbolic Relationship Between Fluorescence and Concentration to the Determination of Quantum Efficiencies Nigel Gains* Department of Biology, University of York, York, YO1 5DD and Alan P. Dawson School of Biological Sciences, University of East Anglia, Norwich, NR4 7TJ An absorptivity-related constant and a quantum efficiency-related constant can be derived, by a simple graphical procedure, from data obtained in a standard fluorimeter. The quantum efficiency of an unknown fluorophore can be determined by comparison of its quantum efficiency-related constant with that of a fluorophore of known quantum efficiency. The absorptivity can similarly be determined using the absorptivity-related constant. This method relies on the approximately hyperbolic relationship that exists between light absorbed and chromophore concentration.The approximation may be derived from the Beer - Lambert equation and the limits of its validity have been tested using theoretical and experimental data. Keywords : Beer - Lambert equation ; fluorescence ej5ciencies The measurement of quantum efficiencies is most accurate when determined with optically dilute solutions with absorbances between 0.01 and 0.05.1-4 The measurement of low absorbances in a spectrophotometer is prone to inaccuracies. Further, as the spectral purity of the exciting light in the fluorimeter may not be the same as that in the spectrophotometer the effective absorbance in each may be differents** In a recent papel-4 a method was described for determining the absorbance of a solution by measuring the fluorescent intensities at two points along the absorbance path, thus overcoming the necessity for a separate measurement of absorbance in a spectrophotometer. This method would require some modification for right-angle fluorimeters and cannot be used with front-faced fluorimeters.However, it is also possible to determine an absorptivity-related constant and a quantum efficiency-related constant for a fluorophore in a standard, unmodified fluorimeter. The procedure depends on the approximately hyperbolic relationship between fluorescence and fluorophore concentration. A graph of fluorescence against fluorescence x [fluorophorel-1 approximates to a straight line and can be extrapolated to give intercept values at both zero and infinite fluorophore concentration.The derivation of this approximately hyperbolic relationship and the limits of its validity are shown. The determination, and potential accuracy, of absorptivity and quantum efficiency using this method are discussed. The work described in this paper was carried out as a control for some biological experi- ments. Its potential relevance to the determination of quantum efficiencies was realised only after reading a recent paper.* * Present address : Eidgenossische Technische Hochschule, Laboratorium fur Biochemie, ETH-Zentrum, 481 CH-8092 Zurich, Switzerland.482 GAINS AND DAWSON : FLUORESCENCE - CONCENTRATION RELATIONSHIP Analyst, VoZ. 104 Theoretical Derivation of an Approximately Hyperbolic Relationship Between Light Absorbed (Io - I ) , or Fluorescence, and Fluorophore Concentration The Beer - Lambert equation may be rearranged to give where I , is the intensity of the incident light, I is the intensity of the unabsorbed light, E is the absorptivity, c is the concentration and d is the path length. This can be expanded to give ( 2 ~ 3 ~ c d ) ( 2 .3 ~ ~ 4 ~ 2! 3! 1 - (2.3~cd) + -- - - For small values of ~ c d the cube term and above may be ignored and the following approxi- mation can be made : Also, for small values of Ecd the approximation can be made that 1 - 2.3~cd I--- 2 2.3~cd 1 + 7 This gives 2I02.3ecd 2 + 2.3ccd I o - I = or by rearrangement 2 ( I , - I ) . . .. .. * * ( 2 ) I, - I = 21,) - -~ 2.3~cd The first of these approximations is valid t:o within +4% provided that a d is less than 0.17, and the second approximation to within -4% provided that Ecd is less than 0.2.Details of the errors produced when these approximations are combined are given in a later section. The hyperbolic function described by equation (2) approximately describes the relation- ship between light absorbed and concentration in fluorimeters that have the optical arrange- ments shown in Fig. l(a), (b) and (c). Equations have been derived that describe the absorption of light (Ito - 1') in the path length, d, (where d2 = d - dl), monitored in right- angle fluorimeters with the optical arrangement shown in Fig. l ( d ) . 5 They are formally similar to the following: Equation (3) may be approximated in a similar fashion to equation (1) to give ;!(Ifo - 2 .3 ~ ~ * * (4) This equation, like equation ( 2 ) , describes a hyperbolic relationship between light absorbed (Ito - I f ) and concentration. In fact, equation (2) is a special example of equation ( 4 )June, 1979 AND THE DETERMINATION OF QUANTUM EFFICIENCIES 483 with d equal to d,; it is therefore not considered separately below. (2d - d,) approximates to 2d then equation (4) approximates to If d , is small so that If, as it is ideally, fluorescence is directly proportional to the amount of light absorbed then fluorescence should bear an approximately hyperbolic relationship to concentration in all the fluorimeter designs shown in Fig. 1. Equation (4) may be rewritten as where F is the fluorescence intensity, Y is a geometrical factor andais the quantum efficiency.If light absorbed, or fluorescence, is plotted against light absorbed (or fluorescence) x [fluorophorel-l then, to the limit to which the approximations in equations ( 2 ) , (4) and (5) hold, a straight line should result. As c tends to infinity I t , - I', or F , tends to a maximum value Imax., or Fmax., such that As c tends to zero (Ito - I')/c, or F/c, tends to a limiting value, such that Dividing the second intercept [equation (7)] by the first [equation (6)] gives a constant, K c d , such that 2 . 3 ~ ( 2 d - d,) 2 .. .. .. (8) .. Ked = Determination of Quantum Efficiencies The intercept on the fluorescence axis in a graph of fluorescence against fluorescence x [fluorophorel-1 is a t infinite concentration and infinite absorbance and is therefore indepen- dent of them [equation (S)] and of the spectral purity of the exciting light.It is dependent on the geometrical factor, the intensity of the illuminating light, a function of the two optical path lengths, 2d2/(2d - d,), and the quantum efficiency. Therefore, the direct comparison of this intercept value with that of a standard fluorophore with a known quantum efficiency will give the quantum efficiency of the unknown fluorophore: The minimisation of the errors arising from the mathematical assumptions in this procedure are discussedbelow. The procedure may also be subject to at least one non-mathematical error arising from overlap of the excitation and emission spectra. This may, as in other procedures, be minimised by using dilute ~olutions.l-~ Determination of Absorptivity The absorptivity of an unknown fluorophore can be found from the ratio of its Ked value [equation (S)] to that of a standard fluorophore of known absorptivity using the following equation : Kc- - 'unknown Kedatandard 'Btandard This procedure is equivalent to deriving a value of (2d - d,) from equation (S), using a fluorophore of known quantum efficiency and absorptivity, and then substituting this value484 GAINS AND DAWSON : FLUORESCENCE - CONCENTRATION RELATIONSHIP Analyst, VoZ.104 into the same equation to derive the absorptivity, from the KEd value, of an unknown fluoro- phore. Alternatively, the absorptivity can be calculated directly from equation (8) i f , for a right-angle fluorimeter [as in Fig.l ( b ) and (41, the emitted light analysed by the photo- multiplier is assumed to be sampled evenly about the centre of the cuvette. In this instance the term (2d - d,) is equivalent to the path length of the fluorimeter cell (d, in Fig. 1). As Ked is found by extrapolation to zero absorbance, it may be assumed that the level of illumination is the same throughout the fluorimeter cell. The second assumption, that the photomultiplier optics are symmetrical about the centre of the cuvette, depends not only on the accuracy to which the fluorimeter is built but also that to which the cell is made. The path length, found using this assumption, can be compared with the effective path length measured as described above. The value of the absorptivity determined using these pro- cedures will depend on the spectral purity of the exciting light.Materials and Methods Fluorescence was measured in two fluorimeters with different optical arrangements. One was a front-faced fluorimeter [Fig. l(a)] constructed at the University of East Anglia. The light source was a 40-W quartz iodide lamp. An image of the lamp element was focused through a Wratten 18A filter on to the cuvette so that it occupied the whole of the front face. The maximum intensity of the exciting light was at 380 nm. The emitted light was analysed by a Bausch and Lomb monochromator (No. 33-86-02) at 480nm, with a band width of 10 nm. The optical path length was 15 mm. The other fluorimeter was a Perkin- Elmer MPF3 spectrofluorimeter with an optical arrangement similar to that in Fig.l ( d ) . The exciting light was at 380 nm, and the emitted light was analysed at 480 nm, each with a band width of 4 mm. A 10-mm square cuvette was used. In both fluorimeters the contents of the cuvette were stirred continuously and maintained at 30 "C. .:. ?..D . r l .. ... 0 Light source a Photomultiplier (dl Q .. .. (c) Q .. . . .. .. .. . . Fig. 1. Four possible arrange.ments for a fluorimeter : d, = length of solution preceding the part monitored by the photomultiplier ; d , = monitored path length; d = total optical path length; and d , path length of fluorimeter cell. Absorbance was measured with a Zeiss spectrophotometer at 380nm using a cuvette with a 10-mm path length. The data were fitted by means of a least-squares fit to equations (9), (lo), (11) or (12).Each squared value, that is the square of the differences between the computer-generated values and the experimental value, was weighted by multiplying it by the reciprocal of the square of the experimental value from which it was derived. This is a fit based on the squares of the relative differences, as opposed to the squares of the absolute differences. This form of weighting is most suited to data that either have an insignificant error or have a similar relative error. It is also necessary to use this or a similar weighting if it is desired to fit a straight line through all the points of a chosen part of a curve rather than mainly through the higher values (Table I), or, as in Fig. 2 and Table 11, to fit a hyperbolic curve through data generated from the Beer - Lambert equation.June, 1979 AND THE DETERMINATION OF QUANTUM EFFICIENCIES 485 Magnesium 8-anilinonaphthalene-l-sulphonate (ANS) was obtained from Eastman Kodak Co., Rochester, N.Y., U.S.A. Triton X-100 and Tris were obtained from BDH Chemicals, Poole, Dorset.All other reagents were of analytical-reagent grade. Fluorescence and light absorbance are measured in arbitrary units. These arbitrary units are constant for the spectrophotometer and for both of the fluorimeters, but are different for each of them. TABLE I ERRORS INVOLVED IN THE ASSUMPTION T ~ A T THE RELATIONSHIP BETWEEN LIGHT ABSORBED OR FLUORESCENCE AND CONCENTRATION IS LINEAR For front-faced optics [or for right-angle optics as in Fig. 1 (b) and (c) where d = d,] values of a d were generated by substituting all the integral values of I , - I, up to the value shown in the left-hand column, into the Beer - Lambert equation using a value of 100 for I,.These values of acd were fitted, using a least- squares fit, to equation (9) where z should approximate to 2.303. For right-angle optics [as in Fig. l(d) where d > d2] the generated values of Ecd were substituted with their corresponding values of ( I , - I ) and with a value of acd2 equal to acd/100 into equation (3). These values of acd were fitted to the equation (I’,, - 1’) = I,zacd2. The maximum errors between the values of I , - I substituted into the Beer- Lambert equation and the corresponding values derived from the equation to which they were fitted are given.A . It should be noted that acd and not acd3 has been used in these calculations. For front-faced optics, and for right-angle optics where d2/d = 0.01- Light absorbed, 0 3 4 5 6 7 8 9 10 y) acd 0.0000 0.0132 0.017 7 0.022 3 0.026 9 0.031 5 0.036 2 0.041 0 0.0458 Front-faced optics r A \ Maximum difference, z % 2.303 2.279 +0.51, -0.51 2.274 +0.77, -0.76 2.268 +1.03, -1.02 2.262 +1.30, -1.28 2.256 +1.57, -1.54 2.250 +1.84, -1.80 2.244 +2.11, -2.06 2.237 +2.39, -2.33 B. For right-angle opics and for various values of d2/d- d2ld 0.001 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Light absorbed, 6 6 5 5 5 6 6 6 7 7 8 9 % €Cd 0.022 3 0.022 3 0.022 3 0.022 3 0.022 3 0.026 9 0.026 9 0.0269 0.03 1 5 0.031 5 0.036 2 0.041 0 Right-angle optics f L I Maximum difference, z % 2.303 2.256 +1.03, -1.00 2.245 +1.56, -1.51 2.233 +2.09, -2.01 2.221 +2.64, -2.52 2.209 43.12, -3.03 2.197 +3.76, -3.54 2.185 +4.34, -4.05 2.173 +4.93, -4.56 Maximum difference, % +2.09, -2.01 +2.09, -2.01 f2.00, -1.92 f1.89, -1.82 +1.78, -1.72 +2.11, -2.03 f1.97, -1.91 +1.84, -1.78 +2.06, -2.00 f1.90, -1.84 f2.03, -1.98 f2.13, -2.06 Results Computer-generated Model Showing the Approximately Hyperbolic Relationship Between Light Absorbed (10 - I ) and ecd For a front-faced optical arrangement Theoretical values of a d were obtained by substituting integral values of I into the Beer - Lambert equation using a value of 100 for I,.Values of (I, - I ) , up to lo%, with their corresponding values of ecd were fitted to the equation1s2 (I, - I ) = I,+cd .... .. * (9) where z is an operational constant. Values of (I, - I ) , up to 70%, with their corresponding value of Ecd were fitted to the equation486 GAINS AND DAWSON : FLUORESCENCE - CONCENTRATION RELATIONSHIP Analyst, VoZ. 104 (K.,:? € C d ) * * .. . . (10) where KEcd is the absorbance that reduces I,,,. to half. and the hyperbolic assumptions are shown in Tables I and 11. The errors involved in the linear CONSTANTS FOUND FROM AND ERRORS INVOLVED IN THE ASSUMPTION THAT THE RELATIONSHIP BETWEEN LIGHT ABSORBIED AND CONCENTRATION IS HYPERBOLIC Values of ecd were generated as described in Table I. The generated data were fitted, using a least-squares fit, to equation (10). Also given are the maximum errors between the values of I, - I substituted into the Beer - Lambert equation and the corresponding values derived from the least-squares fit to equation (lo), and a corrected value, I’max., for I,,,.(where I’max. = I,,,. x 0.8686 x KECd-l). It should be noted that ecd and not ecd, has been used in these calculations. A . For front-faced optics- The resulting values of I,,,. and KEcd are given. Light absorbed, 0 3 4 5 10 20 30 40 50 60 70 % ECd 0.0000 0.013 2 0.0177 0.022 3 0.045 8 0.096 9 0.1549 0.221 9 0.301 0 0.397 9 0.522 9 I,,,. 200.0 198.7 f 0.2 198.3 f 0.2 198.0 f 0.2 196.3 f 0.3 192.7 & 0.4 188.9 f 0.5 185.0 & 0.6 180.7 & 0.7 175.9 f 0.8 170.8 f 0.9 Kecd Maximum difference, % 0.868 6 0.8627 f 0.0009 0.861 3 f 0.0009 0.8598 f 0.0010 Values less than 0.01 0.85f!2 f 0.0014 -0.01, +0.01, -0.01 0.8362 f 0.0019 -0.05, +0.03, -0.06 0.8192 f 0.0024 -0.14, $0.08, -0.15 0.8007 f 0.0030 -0.28, +0.16, -0.32 0.7804 & 0.0035 -0.49, +0.27, -0.58 0.7576 f 0.0041 -0.77, +0.47, -0.96 0.7321 f 0.0048 -1.20, +0.69, -1.62 I’max.200.0 200.1 200.0 200.0 200.0 200.2 200.3 200.7 201.1 201.7 202.6 B. For right-angle optics, when d,/d = 0.01- Light absorbed, % rcd I m a x . K E e d Maximum difference, yo I’max. 0 0.000Q 100.0 0.4343 200.0 10 0.0458 94.9 0.4 0.411.9 5 0.0021 -0.07, $0.04, -0.07 200.1 20 0.0969 89.8 f 0.6 0.3801 f 0.0029 -0.34, +0.21, -0.37 201.0 30 0.1549 84.5 f 0.7 0.3628 f 0.0038 -0.88, +0.53, -1.00 202.3 40 0.2119 78.9 & 0.8 0.3358 & 0.0046 -1.77, +1.08, -2.16 204.1 C . For right-angle optics and for various values of d2/d- d2ld 0.001 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Light absorbed, 39 39 40 42 44 46 48 51 55 59 65 74 % cicd 0.215 0.215 0.222 0.237 0.252 0.268 0.284 0.310 0.347 0.387 0.456 0.585 Maximum difference, yo -1.66, +l.Ol, -2.02 -1.66, +1.01, -2.02 -1.61, +0.98, -1.97 -1.64, +1.00, -2.02 -1.64, +1.00, -2.04 -1.61, 1-0.98, -2.02 -1.55, $0.95, -1.97 -1.54, +0.94, -1.99 -1.55, $0.95, -2.05 -1.48, +0.91, -2.00 -1.45, +0.90, -2.01 -1.40, +0.86, -2.00 The computer-generated data were plotted in the form of light absorbed (Io - I ) against light absorbed x ( ~ c d ) - l (see Fig.2, upper curve). The line drawn is a computer fit of the data, for values of lo - I up to 70%, using equation (10). Smaller maximum values of lo - I were also used. From the computer analysis of these data, shown in Table 11, it can be seen that as the maximum values of I0 - I are decreased, the value of I,,,.more nearly approximates to the value of 210 [see equation ( 2 ) ] and Krcd more nearly approxi-June, 1979 AND THE DETERMINATION OF QUANTUM EFFICIENCIES 487 mates to 0.8686. The relative differences between the values of I , - I derived from the values of I substituted into the Beer - Lambert equation and those values of I , - I derived by substituting the corresponding values of Ecd into the hyperbolic function, derived from the least-squares fit, do not exceed -&2y0 (see Table 11). 201 \ 0 100 200 Light absorbed x (absorbance)-’/ arbitrary units I-’ pmol cm Fig. 2. Plot of light absorbed (I’, - 1’) against light absorbed x (absorbance) -l. The data were generated as described in Table 11.The upper curve (both and 0) is for a value of d,/d = 1.0, as in front- faced and some right-angle fluorimeters [see Fig. l(b) and (G)]. The line drawn through this curve is a computer fit to equation (10) using values of 1’, - I’ of up to 70% (e), assuming a hyperbolic function between I’o - I’ and absorbance. The lower curves in descending order are for values of d,/d equal to 0.9, 0.8, 0.6, 0.4 and 0.01. For a right-angle optical arrangement Theoretical values of Ecd were obtained by substituting integral values of I into the Beer - Lambert equation using a value of 100 for I,. Values of ccdl and a d Z were obtained by multiplying a d by dl/d and d,/d, respectively, where d = d, + d , [see Fig. 1 (a)]. These values of and ccd, were then substituted into equation (3) to obtain values of (16 - 1’).For comparative purposes these values were multiplied by d/d, before fitting them, with their corresponding values of ccd, to equation (10). The computer-generated data are plotted in Fig. 2 (lower curves) in the form of light absorbed (I’, - I’)d/d, against (Ito - I’)d/d, x (ccd)-l. A computer analysis of some of these data is given in Table 11. It shows the upper limits of (16 - I’)d/d, and ccd, to which the hyperbolic approximation holds, to within &2%, for various values of d,/d from 1.0 to 0.01. 6 Relationship Between Light Absorbed or Fluorescence and the Concentration of Dissolved Magnesium 8-Anilinonaphthalene- 1 - sulphonate The absorption of light by ANS in 0.005 M Tris - HCl (pH 7.6) and in 5% (m/V) Triton X-100 obeys the Beer- Lambert relationship, in that graphs of absorbance against con- centration give straight lines, Fig.3 (a). If the same data are plotted in the form of light488 GAINS AND DAWSON : FLUORESCENCE - CONCENTRATION RELATIONSHIP Analyst, VoZ. 104 absorbed (I, - I ) against (I, - I ) x [ANSIw1 the graphs appear to be linear within the experimental error and up to a value of 70% far I , - I [see Fig. (3)]. The data in Fig. 3 (b) were fitted to the following equation : The relative difference between any of the experimental and the computer-generated data was less than 2.2% (see legend to Fig. 3). Values of I,,,, were found to be 169.8 in 5% (m/V) Triton X-100 and 175.7 in 0.005 M Tris - HC1 (pH 7.6). These values of I,,,.are lower than the value of 200 for ZI, [see equation (2:)]. However, it can be seen from Fig. 2 that the magnitude of I,,,, will depend on the maximum value used of the light absorbed. These experimental values of I,,,. compare favourably with the theoretical values shown in Table 11. From Fig. 3 (b) and from the analysis shown in the legend to Fig. 3 it can be seen that in practice the amount of light absorbed. (I, - I ) bears an approximately hyperbolic relationship to the concentration, provided thilt I,,,. is not fixed at a value of 21,. 180 I 0 20 40 60 80 100 [ANSI /pM 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Light absorbed x [ANSI-‘/arbitrary units x pM-’ Fig. 3. Relationship between light absorbed and concentration of magnesium S-anilinonaphthalene- l-sulphonate (ANS) .The same data are plotted in .two ways : (a) absorbance against concentration and ( b ) light absorbed (I, - I) against light absorbed x (concentration)-l. For both (a) and (b) ANS was dissolved in 0.006 M Tris - HC1 (pH 7.6) (O), or in 6% ( m / Q Triton X-100 (0). The data in (b) were fitted to equation (11). The following were found in Tris - HC1, with results obtained in Triton X-100 given in parentheses: the maximum difference between the experimental and generated data are -2.2%, +0.9%, -1.1% (-1.8%, +0.8%, -1.3%), I,,,. is 175.7 -+ 2.8 (169.8 f 2.0), K , is 188.1 & 3.7 p~ (126.5 & 2.0 p ~ ) , KEcd is 0.7543 (0.7295) ; the maximum value of I,, - I used was 60% (70%!. (The data are averages of ten separate titrations.) Similarly, a plot of fluorescence against fluorescence x [ANSI-l for the fluorophore dissolved in 5% (m/V) Triton X-100 is, over a limited concentration range, apparently a straight line.The front-faced fluorimeter was used for the former and the Perkin-Elmer MPF3 spectrofluorimeter for the latter. All the data in Fig. 4 (a) and some of the data in Fig. 4 (b) (closed circles) have been fitted to the following equation : This is shown in Fig. 4 (a) and (b).June, 1979 AND THE DETERMINATION OF QUANTUM EFFICIENCIES 489 F = Fm,,. ("> . . .. .. . . (12) K , + c Values of Fm,,, and K , are given in the legend to Fig. 4. 160- (6) 60- 0 - 0 - 0 40 20 - I I I I I I 1 1 1 1 I ~ 0 0.2 0.4 0.6 0.8 1.0 1.2 Fluorescence x [ANSI-'/arbitrary units x pM-' Fig. 4. Plots of fluorescence against fluorescence x [ANSI-l: (a) using a front-faced fluorimeter, (b) using a right-angle fluorimeter (both 0 and 0).For both, the fluorophore was dissolved in 5% m/ V Triton X-100. All the data in (a) and some of the data in (b) (a) were fitted by means of a least-squares fit t o equation (12). The lines drawn represent these fits. In both, the errors between the experimental and generated data were within f1.4y0, for (a) a 15-mm cell was used and K, is 78.2 f 1.7 PM and for (b) a 10-mm cell was used and K, is 119.1 1.6 p ~ . (The data are averages of three separate titrations.) Discussion For front-faced fluorimeters the assumption that fluorescence is directly proportional to fluorophore concentration is valid to within 52% provided that 8% or less of the exciting light is absorbed (Table I A).Similarly, the assumption that fluorescence bears a hyperbolic relationship to fluorophore concentration is valid to within -+2% provided that 70% or less of the exciting light is absorbed (Table I1 A). The errors involved in these assumptions should also be the same for the right-angle optical arrangements shown in Fig. 1 (b) and (c). For most commercial right-angle fluorimeters the optical arrangement is similar to that in Fig. 1 (d). In this instance the upper limit to which the linear and hyperbolic assumptions are valid to within &2y0 depends on the ratio d,:d [see Fig. 1 (d) and Table I B]. When this ratio is 0.1 or less the linear assumption is valid provided that 4y0, or less, of the light is absorbed and the hyperbolic assumption is valid provided that 39% or less of the exciting light is absorbed.As the ratio d,:d approaches unity the upper limits tend towards 8% and 74y0, respectively (Tables I B and I1 C). One possible criticism of the data in Figs. 3 and 4 is that the maximum fluorescence value may be caused by the saturation of ANS binding sites on the Triton X-100 micelles. How- ever, even at the highest fluorophore concentrations used there is an approximately &fold excess of Triton X-100 micelles over ANS ions. Although the results were similar the data could not be used to determine values of F,,,. and K , as the Beer - Lambert relationship was not followed. A slight upward curvature was found in graphs of absorbance against ANS concentration. Ethanol was also used as a solvent.490 GAINS AND DAWSON The accuracy to which the quantum efficiency and absorptivity can be determined depends on the error of the intercept values derived from the graph of fluorescence against fluorescence x [flu~rophorel-~.These values are dependent on the concentration range used (Table 11). For front-faced fluorimeters the error between I,,,. and the model value of 210 and between K,, and its maximum value of 0.8686 is about 2y0.per 0.05 absorbance. For right-angle fluorimeters the errors are about 5% at small slit widths (where d > d,) and will decrease towards 2% as d, approaches its maximum value of d. The theoretical errors in deriving I,,,. and K,, have been calculated using equal increments of (I’o - I’); the use of equal increments of Ed will slightly alter these errors.However, for both I,,,. and K,, the theoretical errors will be self-cancelling if the titrations of fluorescence against concentration of the unknown and standard fluorophores are made at the same, or reasonably similar, absorbance values. This can be seen from the values of I’max. (where I‘max. = Imax, x 0.8686 x shown in Table 11. If the fluorescence titration data for the standard and unknown fluorophores are such that the absorbance values of each bear a fixed relationship to one another then the value of I,,,. x K,,--l will be insensitive to the different absorbance ranges used, and the quantum efficiency may be calculated from the following equation : The errors in I,,,. and K,, are in the same direction and are very nearly equal. However, in this instance the absorptivity must be measured in a spectrophotometer or by the method described by Britten et aL4 Similarly, if the experimental data are such that the error on Imax. is unacceptably large, but that on the intercept of the fluorescence x [fluoro- phorel-l axis is acceptable, then the quantum efficiency may be calculated from equation (7) if the absorptivities are known. The data in this paper suggest that it is tlneoretically possible, by using the procedures discussed above, to determine the quantum efficiency to an accuracy of 2% or less. Whether or not these procedures are of any practical value can only be independently assessed by those who, unlike us, regularly use the already established methods for measuring quantum efficiencies. We thank Dr. M. J. Selwyn for helping us with the approximations that transform equation (1) into equation (2), P. D. Bolton for compiling the computer program and the University of York Computer Advisory Service for helping us to modify it to our needs. N.G. gratefully acknowledges the MRC for a Research studentship, the SRC for a Post- doctoral Fellowship, and the Centre National! pour la Recherche Scientifique for financial support whilst writing the paper. References 1 . 2. 3. 4. 5. Parker, C . A., and Rees, W. T., Analyst, 1960, 85, 587. Parker, C . A., “Photoluminescence of Solutions,” Elsevier, Amsterdam, 1968. Demas, J. N., and Crosby, G. A., J . Phys. Chem., 1971, 75, 991. Britten, A., Archer-Hall, J.. and Lockwood, G., Analyst, 1978, 103, 928. Brand, L., and Whitholt, W., Meth. Enzym., 1967, 11, 776. Received November 29th, 1978 Accepted January 16th, 1979