It is important to evaluate and choose appropriate setups of concentration and path length for absorbance measurements because they will determine the dynamics of absorbance measurements. A path length of 1 cm is widely used in most of the studies for UV-Vis absorbance measurement, and the concentration range should be selected in a linear range, where the BLB law can be applied. To reexamine the validity of the BLB law and compare it with our proposed model, we chose the setups of concentration and path length within the absorbance range, that the BLB model was proven to be valid with a constant path of 1 cm. We designed systematic setups with various combinations of concentration and path length (1, 0.5, 0.2, and 0.1 cm), as shown in Tables 1, 2. For the setups of potassium dichromate (Table 1), the absorbances of the subsamples No. 1 and 5 are known to be linear with the same path length of 1 cm under various wavelengths (Burke & Mavrodineanu, 1977). Subsamples No. 1 to 4 were designed to have the same product ( c ∙ l ) of concentration ( c ) and path length ( l ), and the concentrations of the subsamples No. 2 to 4 were calculated based on the value of c ∙ l of the subsample No. 1. The subsamples No. 5 to 8 were also designed with the same principle of No. 1 to 4 with smaller and proportional concentrations. According to the BLB law, samples with the same values of c ∙ l will have the same absorbance. With the abovementioned design, it is easy to examine if the BLB is valid, and if not, the deviation will be shown clearly among the results. The subsamples No. 9 to 11 were added additionally to generate other combinations in the known linear range. The setups of microalgae (Table 2) were designed based on the same logic as for the potassium dichromate (Table 1). The absorbances of subsamples No. 1 and 5 are known to be in a linear range under the wavelength of 750 nm for P. tricornutum and C. vulgaris based on our laboratory database, which is usually used for the estimation of their biomass concentrations. As for the setups of the pigments (Table 2), the concentrations of No. 1 and 5 were determined based on a pre-test of their absorbance spectrum.

The medium used for potassium dichromate was 0.001 N perchloric acid, which was diluted from a stock solution of perchloric acid (density = 1.53 kg L^-1, weight percentage = 60%, Producer: Merck, Germany) with double-distilled water in a volumetric flask. The potassium dichromate powder was weighed and dissolved in 0.001 N perchloric acid in volumetric flasks to prepare 1,000 mg L^-1 and 100 mg L^-1 stock solutions. These stock solutions were further diluted to prepare eleven subsamples with the designed concentrations shown in Table 1. Pipettes and falcon tubes were used for these further dilutions of stock solutions. Three sets of aforementioned stock solutions of potassium dichromate were prepared separately to repeat the absorbance measurements of the subsamples.

Fresh algae cultures taken from the PBRs were first dewatered by centrifugation at 4000 RPM for 10 min and then re-suspended in 0.9% NaCl solution. This process of dewatering and re-suspension was repeated three times to keep the consistency of the sample medium for absorbance measurements. The concentration of the re-suspended sample was determined by the dry weight method. First, 5 mL of the microalgae suspension was placed on a filter paper and washed twice with 5 mL of tap water. After filtering, the filter paper containing the microalgae cells was dried at 105°C (MA35, Satorius, Germany) and weighed. The re-suspended sample was further diluted with 0.9% NaCl solution to obtain subsamples with different concentrations (see Table 2). Pipettes and falcon tubes were used for these dilutions. In the preparation of the suspensions of P. tricornutum and C. vulgaris, the suspensions were prepared individually according to the above procedures.

Analytical standards of two pigments, chlorophyll a (purity ≥97%) and fucoxanthin (purity ≥95%), were purchased from Sigma-Aldrich (Germany). The pigment solutions (chlorophyll a and fucoxanthin) were prepared separately as described below. First, 1 mg of the pigment powder (pre-weighted by the company) was dissolved in 1 mL of pure ethanol in the glass vial provided by the manufacturer. Then, the pigment solution with a concentration of 1 g L^-1 was diluted stepwise in falcon tubes to obtain subsamples with the designed concentrations. The experimental setups of the pigment solutions are almost the same as those of the microalgae suspensions, but with smaller proportions of the concentrations: chlorophyll a at 10% and fucoxanthin at 2.5% of the concentrations shown in Table 2.

In this work, the absorbance measurements were conducted using a HITACHI U-2900 spectrophotometer with a spectral bandpass of 1.5 nm. The spectrophotometer was calibrated every year, including wavelength accuracy, absence of stray light, baseline deviation, and photometric linearity. To ensure the accuracy and consistency of measurements, the spectrophotometer was warmed up for at least 30 min before measuring. The measurements were performed at a room temperature of 20°C–22°C. The sample cuvettes used were quartz cuvettes (100-QS, Hellma, Germany) with four different light path lengths: 1, 0.5, 0.2, and 0.1 cm.

The spectral measurements for the different materials used in this work were conducted separately. The standard procedure for a measurement run is described as follows. First, the cuvettes were cleaned with 3 M H₂SO₄ for at least 20 min and rinsed 5 times with double-distilled water. After cleaning, the cuvettes were kept to air dry and then filled with the medium solvent to perform a blank measurement with the setups shown in Table 3. The cuvettes were then cleaned again with 3 M H₂SO₄ and double-distilled water and kept to air dry. The subsamples (Table 1 or Table 2) were then loaded into the cuvettes and sealed with parafilm to prevent evaporation. Spectral measurements were then conducted using the setups described in Table 3. For each subsample, this spectral scan was repeated three times. For potassium dichromate solutions, three measurement runs were conducted as each potassium dichromate solution was repeatedly prepared three times. For microalgae suspensions and pigment solutions, one measurement run was conducted. Each run of the aforementioned experiment was conducted on the same day.

The measured data were first pre-processed to perform a regression with the BLB law and with the proposed model. The first step of pre-processing was to subtract blank measurements to remove the absorption contributed by the medium. The second step was to average the repeated measurements to obtain a single representative spectrum for each subsample array. As mentioned earlier, three sets of potassium dichromate solutions were prepared separately, and the measurement was repeated three times for each set. Therefore, each subsample of potassium dichromate solution in Table 1 contains a total of nine absorption spectra, which were averaged for later use. For microalgae suspensions and pigment solutions, each subsample in Table 2 has three absorption spectra that were also averaged. These representative spectra were then used to fit the equations of the BLB law and the proposed model, as in Eqs. 2, 3.

Bouguer-Lambert-Beer (BLB) law: A λ = A λ , s a m p l e − A λ , b l a n k = ε λ ∙ c ∙ l (2)

Here λ is the wavelength of the input light, A λ the apparent absorbance at the wavelength of λ , A λ , s a m p l e the apparent absorbance of the sample, A λ , b l a n k the apparent absorbance of the medium, ε λ the apparent specific absorbance, c the sample concentration, and l the light path length.

The proposed model: A λ = A λ , s a m p l e − A λ , b l a n k = ε λ ′ ∙ c α λ ∙ l β λ (3)

Here λ , A λ , A λ , s a m p l e , A λ , b l a n k , c and l are as above. In addition, ε λ ′ is the effective specific absorbance at the wavelength of λ , α λ the correction coefficient of concentration, and β λ the correction coefficient of the path length.

The software Matlab R2022a (MathWorks, Uunited States) was used to solve the regression problem in this work, which contains three-dimensional data points (output: A λ , inputs: c , l ). The built-in function “fit” in Matlab with the fitting method “Non-linearLeastSquares” was used to obtain the coefficients in Eqs. 2, 3. The performance of the BLB law and the proposed model was then compared using the correlation coefficient ( R 2 ) and the normalized root mean square error ( N R M S E ). The quality of the absorbance measurements was evaluated with the coefficient of variation (CV). These statistical criteria are shown in the following equations. R 2 = n ∑ y λ , i y ^ λ , i − ∑ y λ , i ∑ y ^ λ , i 2 n ∑ y λ , i − ∑ y λ , i 2 n ∑ y ^ λ , i − ∑ y ^ λ , i 2 (4)

Here R 2 is the correlation coefficient for each wavelength λ , where n is the sample size, y λ , i the measured data, and y ^ λ , i the predicted data. N R M S E = 1 y ¯ λ ∑ i = 1 n y λ , i − y ^ λ , i 2 n (5)

Here N R M S E is the normalized root mean square error for each wavelength λ , where n , y λ , i , and y ^ λ , i are defined as above, and y ¯ λ = ∑ i = 1 n y λ , i n is the mean of the measured data. C V = σ λ y ¯ λ (6)

Here C V is the coefficient of variation for each wavelength λ , where σ λ = 1 n − 1 ∑ i = 1 n y λ , i − y ¯ λ 2 is the standard deviation of the measured data and y ¯ λ is defined as above.

Figures 1A, D show that the proposed model outperforms the BLB law for potassium dichromate solutions with an R 2 of more than 0.99 and an N R M S E of less than 0.03 over the entire wavebands, while the worst R 2 and N R M S E of the BLB law are around 0.94 and 0.15, respectively. As shown in Figures 2A, D, the correction coefficients of the path length ( α ) are close to one in all wavebands, confirming that the deviation of the BLB law in potassium dichromate solutions is caused by chemical equilibria, since the deviation is independent of path length and occurs only with changes in concentration (Buijs & Maurice, 1969; Burke et al., 1972; Burke & Mavrodineanu, 1977; Ingle & Crouch, 1988).

The equilibrium equation of the potassium dichromate solutions with weakly acidic media, as used in this work, is given in the equation below. 2 H C r O 4 − ⇌ C r 2 O 7 2 − + H 2 O (7)

Here, the dimerization of the monomer, H C r O 4 − , forms the dimer C r 2 O 7 2 − . Apparent deviations from the BLB law are expected at wavelengths where the molar absorption coefficients of H C r O 4 − and C r 2 O 7 2 − differ from each other (Burke et al., 1972; Burke & Mavrodineanu, 1977). The chemical deviation is not caused by the BLB law itself but by the non-linear chemical equilibria of the individual ions ( H C r O 4 − ; C r 2 O 7 2 − ) as shown in Eq. 7. Our proposed model can describe this non-linear behavior and it is valid as long as the BLB law remains valid for the ions involved. The relationship between total analyte concentration, path length, and apparent absorbance can be expressed by a much more complicated mathematical expression, as described in Ingle & Crouch, 1988. With the proposed model, neither the equilibrium equation nor the absorption spectra of individual ions are needed to describe the absorbance characteristics, which can be used as a practical tool in analytical chemistry. Furthermore, the proposed model can identify the chemical deviation when only β is close to one and α is not, which is similar but much simpler than the polynomial methods described in (Buijs & Maurice, 1969).

Potassium dichromate is used as a reference standard to calibrate the linearity of UV-vis spectrophotometers (Burke & Mavrodineanu, 1977). This linearity is only valid within the linear range and the path length should be constant. Another common application of potassium dichromate is serving as an oxidizing agent in various reactions, where the change of chromium form can be determined by UV-visible spectroscopy and the concentration of the target analyte can be calculated. One example of this application is the chemical oxygen demand (COD) analysis, in which the wavelength 440 (435) nm is widely used for the detection of dichromate and 610 (600) nm for Cr³⁺ (Thomas et al., 2017). As shown in Figure 2D, the selection of 440 (435) is a very smart choice, because both α and β are close to one. Therefore, the BLB law is valid here, which means that the calibration of the COD determination can be extended to other setups of concentration and path length. The choice of this setup can be determined based on different conditions, for example, smaller path length for higher concentration.

Our proposed model also outperforms the BLB law for scattering microalgae suspensions, as shown in Figures 1B, C, E, F. Unlike the deviation caused by chemical equilibria, the values of both α and β for microalgae suspensions mostly deviate from one (see Figures 2E, F). The deviation of the BLB law also causes the error in using linear scaling or standardization of spectra, which is commonly used for pre-processing spectral data (Craig et al., 2006; Rinnan et al., 2009; Yao & Lewis, 2010; Gholizadeh et al., 2016; Mishra et al., 2020). To demonstrate this effect, we examined the spectra of scattering microalgae suspensions and non-scattering pigment solutions, the experimental setups of which are given in Table 2. We scaled the spectra of subsamples No. 5 to 8 by a factor of 2 and plotted them together with the spectra of subsamples No. 1 to 4 in Figure 6. This is because the concentrations of subsample No. 5 to 8 are each half that of subsamples No. 1 to 4, respectively. Moreover, the product ( c ∙ l ) of concentration ( c ) and path length ( l ) remains the same for subsamples No. 1 to 4, and No. 5 to 8, which means that all scaled and unscaled spectra should be identical if the BLB law is valid. However, Figures 6A, B show clearly that the use of linear scaling causes deviations in spectra of scattering microalgae suspensions. Therefore, one should be very careful when applying linear scaling in the spectra of scattering suspensions. In contrast, our model can solve the aforementioned problem in different ways. First, if scaling is needed, non-linear scaling based on the proposed model should be used instead of a linear one. Second, if the goal is to compare the optical properties of different suspensions, the coefficients of the proposed model can be used instead of individual scaled spectra.

Unlike the microalgae suspensions, the results in Figures 6C, D show that the BLB law is indeed valid for pure pigment solutions because all the spectra with the same c ∙ l are shown to be identical. In fact, UV-Vis spectroscopy is widely used in pigment analysis of the extracts of microalgae (Wellburn, 1994; Jeffrey et al., 1997; Dere et al., 1998). This kind of pigment analysis is based on the validity of the BLB law and the additivity of absorbance. For instance, the concentration of chlorophyll a can be determined by solving a linear equation with absorbances at two wavelengths. These wavelengths normally correspond to the local maximums of the spectrum to maximize the signal-to-noise ratio. These maximums of the spectrum are specie and solvent dependent. Our results in Figures 6C, D show the same maximums of absorbance for chlorophyll a (663 nm) and fucoxanthin (450 nm) as described in the literature (Wang et al., 2018). According to the results, we can confirm that if the pigment extracts of microalgae do not contain other scattering matters, the pigment analysis can be carried out with multiple component analysis. Moreover, the calibrations of this type of pigment analysis can be extended to other setups of concentration and path length by using the BLB law. It is noteworthy that there are some other rapid pigment analyses for the determination of chlorophyll a and fucoxanthin with apparent absorbance of microalgae suspension without extraction. In theory, the BLB law and the additivity of absorbance cannot be applied in this type of pigment analysis based on our results. However, there are still studies of pigment analysis using the same analysis principle for microalgae suspension as used for pigment extracts (Eriksen & Iversen, 1995; Wang et al., 2018). Because of the scattering effect, this type of calibration will be disturbed by the non-linearity of absorbance and cannot be extended to other setups of concentration and path length.

A closer look at R 2 and N R M S E in Figures 1B, C, E, F reveals a trend that the stronger the absorption of the pigments, the better the accuracy of the BLB law (linear model). This can be explained by the fact that the apparent absorbance in scattering suspensions consists of two parts: absorption and scattering. The pure absorption of pigments is linear, as shown earlier, and the non-linearity is caused by the scattering effect. Therefore, the non-linear behavior of the apparent absorbance is lower when the absorption fraction of pigments is higher, and vice versa. This phenomenon is very evident at about 450 nm and 690 nm, which are known as the absorption bands of carotenoids and chlorophylls (Mueller, 1973; Fernández et al., 1997). For both P. tricornutum and C. vulgaris, the accuracy of the BLB law is the best and is closer to the proposed model around these pigment wavebands. On the other hand, the accuracy of the BLB law is poorest at wavelengths above 750 nm, where there is no absorption of pigments. Meanwhile, our proposed model shows excellent performance even at 900 nm, where the BLB law has the worst accuracy.

It is known that the BLB law is only valid for single scattering. A common criterion for single scattering is the optical depth τ < 0.1 (Hulst and van de Hulst, 1981; Morel & Bricaud, 1981), corresponding to an apparent absorbance A < 0.04, where τ = A ⁡ ln ⁡ 10 . When the optical depth/apparent absorbance increases into the range where the scattering interactions occur between multiple cells, the BLB law can no longer be applied. In practice, the absorbance measurement of microalgae suspensions is usually outside this range, as shown by the results in Figures 6A, B. However, our proposed model is also applicable to multiple scattering when these interactions remain moderately weak. In other words, the model presented in this work extends the range of apparent absorbance with scattering effect that can be described by analytical equations. In particular, this expands the applications of absorption spectroscopy with photosynthetic microorganisms and provides a simple and powerful tool to improve the quality of spectral data preprocessing and the accuracy of calibrations such as biomass concentration. We anticipate that the model will also be very useful for the quantification of intracellular content in microorganisms.