Dependence of Refractive Index on Concentration and Temperature in

Article pubs.acs.org/jced

Dependence of Refractive Index on Concentration and Temperature in Electrolyte Solution, Polar Solution, Nonpolar Solution, and Protein Solution Chan-Yuan Tan and Yao-Xiong Huang* Department of Biomedical Engineering, Key Laboratory of Biomaterials, Jinan University, Guang Zhou, China ABSTRACT: The dependence of refractive index on the concentration and temperature in six kinds of aqueous solutions was investigated. The six solutions were three electrolyte solutions (NaCl, KCl, and CaCl2), a polar solution (glucose solution), a nonpolar solution (ethyl acetate solution), and a protein solution (bovine serum albumin solution). It was found that dn/dc decreased with temperature and dn/dT decreased with concentration in polar, nonpolar, and electrolyte solutions. While in protein solution, both of the derivatives showed an opposite behavior due to the thermal aggregation effect of proteins. According to the experimental results, the empirical expressions of the refractive indices in terms of both concentration and temperature were derived for the six solutions. By having the derivatives of refractive index n with respect to concentration c (dn/dc) and with respect to temperature T (dn/dT) respectively from the expressions, the dependence of (dn/dc) on temperature and that of (dn/dT) on concentration were obtained.

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Table 1. Sample Descriptiona

INTRODUCTION Refractive index is one of the most important physical properties of solutions. By measuring the refractive index of a binary solution, one can determine the composition of the solution.1−4 So measurements of refractive index are widely used in many industrial and research applications to determine the concentration of solutions. However, refractive index at the same time also varies with temperature, pressure, and wavelength.3,5,6 Though in most of the time, by performing the measurement in atmosphere and using specific wavelength, one can eliminate the influences of pressure and wavelength; the effect of temperature should be always considered. Numerous theoretical and experimental studies have focused on the temperature and concentration dependence of the refractive indices of aqueous solutions.1−3,6,7 The Lorentz− Lorenz equation is the most often used formula to estimate the temperature and concentration dependence of the refractive index. But it does not provide reliable estimation for the derivatives of the refractive index with respect to temperature and concentration. So the values for the coefficients of temperature and concentration, or the derivatives of the refractive index n with respect to temperature T and concentration c: (∂n/∂T) and (∂n/∂c), have to come from experiments. After obtaining the temperature coefficient (AT) or concentration coefficient (Ac), one can give out the empirical expressions for the temperature or concentration dependence of the refractive index. Most of the empirical expressions available nowadays only give either the concentration coefficient or temperature coefficient and express as n = n0 + AT, or n = n0 + Ac.8,9 However, in some applications, both temperature and concentration change at the same time. © 2015 American Chemical Society

chemical name

source

sodium chloride (NaCl)

Guangzhou Chemical Reagent Factory Tianjin Kermel Chemical Reagents Development Centre Guangzhou Chemical Reagent Factory Guangzhou Chemical Reagent Factory Guangzhou Rui Special Biological Technology Co., Ltd. Tianjin DaMao Chemical Factory

potassium chloride (KCl) calcium chloride (CaCl2) glucose (C6H12O6, α-D-glucopyranose) BSA ethyl acetate (C8H16O4)

purity (mass fraction) AR (99.5 %) AR (99.5 %) AR (99.5 %) AR (99 %)

AR (99 %)

a

All of the employed samples were of AR purity and did not need further purification in the experiment.

For example, in a number of microbioreactors and microfluid devices, it is required to monitor the concentration variation in some points/regions of a solution by measuring its refractive index. Since in a streamflow, the mass diffusion is usually accompanied by thermal diffusion so temperature also changes with time from place to place. Therefore, it is needed to know by what extent the detected refractive index variation reflects the real change in concentration under temperature fluctuation. For this reason, empirical expressions with both the concentration coefficient and the temperature coefficient are necessary for Received: January 6, 2015 Accepted: August 25, 2015 Published: September 1, 2015 2827

DOI: 10.1021/acs.jced.5b00018 J. Chem. Eng. Data 2015, 60, 2827−2833

Journal of Chemical & Engineering Data

Article

varies with concentration or (∂n/∂c) varies with temperature is significant in numerous properties and physicochemical processes in materials such as the Soret coefficient, the glass transition temperature, and the thermal lens effect of solutions.11−13 Accordingly in this study, we investigated the simultaneous dependence of refractive index on concentration and temperature in some typical aqueous solutions. The investigated solutions include electrolyte solution, polar solution, nonpolar solution, and protein solution. We deduced the empirical expressions which include both the concentration and the temperature coefficients for the six solutions, and then investigated the dependence of (∂n/∂c) on temperature and that of (∂n/∂T) on concentration in the solutions.

binary solutions. They are also important for light scattering experiments of nonequilibrium fluctuations in a liquid mixture and can improve the fundamental understanding of binary mixtures as described previously.1,10 Furthermore, whether (∂n/∂T) Table 2. Experimental Values of Refractive Index (nD) at Temperature T, Mass Fraction w (kg/kg), and Pressure p = 0.1 MPa for the Solutions of Glucosea nD w

293.15 K

298.15 K

303.15 K

308.15 K

313.15 K

318.15 K

0.01 0.05 0.10 0.15 0.20 0.30 0.40 0.50

1.3360 1.3412 1.3481 1.3553 1.3623 1.3775 1.3947 1.4119

1.3355 1.3406 1.3475 1.3548 1.3617 1.3768 1.3939 1.4106

1.3349 1.3401 1.3469 1.3542 1.3610 1.3763 1.3932 1.4100

1.3345 1.3395 1.3463 1.3535 1.3604 1.3757 1.3925 1.4098

1.3337 1.3388 1.3455 1.3527 1.3596 1.3747 1.3919 1.4086

1.3329 1.3379 1.3447 1.3519 1.3588 1.3739 1.3907 1.4073

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EXPERIMENTAL PROCEDURE

Three electrolyte solutions (NaCl, KCl and CaCl2) of different mass fractions (kg/kg, 1 %, 5 %, 10 %, 15 %, 20 %, 25 %) were prepared by dissolving the electrolytes (Guangzhou Chemical Reagent Factory, Analytical Reagent) in distilled water.

a

Standard uncertainties u are u(p) = 10 kPa, u(T) = 0.05 K, u(w) = 0.0005, and u(nD) = 0.0003.

Figure 1. Refractive indices of the six solutions as functions of solution concentration at different temperatures. (a) sodium chloride; (b) potassium chloride; (c) calcium chloride; (d) glucose; (e) BSA; (f) ethyl acetate. In each of the six plots, ■, 293.15 K; ●, 298.15 K; ▲, 303.15 K; □, 308.15 K; ○, 313.15 K; △, 318.15 K. 2828



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Glucose solution of different mass fractions (kg/kg, 1 %, 5 %, 10 %, 15 %, 20 %, 30 %, 40 %, 50 %) were prepared by dissolving the glucose (Guangzhou Chemical Reagent Factory, AR) in distilled water. Protein solution of different mass fractions (kg/kg, 0.5 %, 1 %, 1.5 %, 2 %, 2.4 %, 2.9 %, 3.4 %, 3.8 %) were prepared by dissolving bovine serum albumin (BSA, Guangzhou Rui Special Biological Technology Co., Ltd.) in distilled water. Ethyl acetate solution of different mass fractions (kg/kg, 0.9 %, 2.7 %, 4.5 %, 6.4 %, 8.2 %) were prepared by dissolving ethyl acetate (Tianjin DaMao Chemical Factory, AR) in distilled water. Table 1 gives the description of the samples used in this investigation. The refractive indices of the solutions at different temperatures, (293.15, 298.15, 303.15, 308.15, 313.15, and 318.15) K, were measured using a AHC5-2WAJ Abbe refractometer with a temperature controller (Shanghai Optical Instrument Factory, China) at the wavelength of 589.3 nm. The refractometer can measure refractive index ranging from 1.3000 to 1.7000 with an accuracy of ± 0.0001. It was calibrated using distilled water and ethanol before each measurement. Considering all the six solutions were aqueous solutions, we also measured the temperature dependence of RI in distilled water as listed in Table 2 for reference.

Table 3. Experimental Values of Refractive Index (nD) at Temperature T, Mass Fraction w (kg/kg), and Pressure p = 0.1 MPa for the Solutions of NaCl, KCl, CaCl2, and Watera nD w

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RESULTS AND DISCUSSION A. Effect of Concentration on the Refractive Index of Solution. Figure 1 shows the refractive indices of the six solutions versus solution concentration at different temperatures. We can see that the refractive indices in all the six solutions increase linearly with concentration. Though with increasing temperature the refractive indices decrease, the slopes of the refractive indices changing with concentration are almost the same at all the temperatures for the six kinds of solutions. Especially in the solutions of sodium chloride, calcium chloride, and glucose, the variation of temperature from (293.15 to 318.15) K only induce slight decrease of the refractive indices. The simulation analysis of the experimental data shows that, except for the ethyl acetate solution which is of the order of 10−4, the concentration coefficients of the refractive indices in all the other five solutions are of the order of 10−3. Table 2−5 give the detailed data of how the refractive indices in all of the six solutions vary with mass fraction. A comparison of our experimental results with the available literature data shown in Figure 2 indicates that our data are quite consistent (deviation is from 0.0003 to 0.0015 for NaCl solution and just 0.0003 for glucose solution) with those reported previously by other authors.14−19 B. Effect of Temperature on the Refractive Index of Solution. Figure 3 illustrates the dependence of refractive index on temperature in all of the six solutions at different concentrations. It indicates that, in all the solutions, the refractive indices decrease linearly with temperature in the range of (293.15 to 318.15) K. All of the temperature coefficients deduced from the experimental data of the six solutions are of the order of 10−4. C. Empirical Expressions. According to the concentration and temperature dependences of the refractive indices in the six kinds of solutions, it is desired to have empirical equations with both the concentration and temperature coefficients so that the composition changes of the solution would not be overestimated by the refractive index variation. By using a data analysis program called OriginPro (OriginLab, USA) with the method of ordinary

293.15 K

298.15 K

0.01 0.05 0.10 0.15 0.20 0.25

1.3365 1.3434 1.3521 1.3610 1.3702 1.3795

1.3359 1.3428 1.3514 1.3604 1.3692 1.3788

0.01 0.05 0.10 0.15

1.3349 1.3405 1.3486 1.3577

1.3345 1.3401 1.3480 1.3571

0.01 0.05 0.10 0.15 0.20 0.25

1.3356 1.3442 1.3568 1.3689 1.3818 1.3945

1.3353 1.3437 1.3561 1.3686 1.3814 1.3943

1.3335

1.3330

303.15 K NaCl 1.3354 1.3422 1.3508 1.3596 1.3683 1.3780 KCl 1.3340 1.3396 1.3476 1.3566 CaCl2 1.3349 1.3431 1.3555 1.3680 1.3807 1.3940 Water 1.3324

308.15 K

313.15 K

318.15 K

1.3346 1.3414 1.3502 1.3590 1.3678 1.3773

1.3339 1.3407 1.3495 1.3580 1.3670 1.3767

1.3333 1.3398 1.3485 1.3572 1.3661 1.3757

1.3335 1.3390 1.3470 1.3560

1.3330 1.3384 1.3464 1.3554

1.3320 1.3379 1.3460 1.3550

1.3344 1.3426 1.3551 1.3671 1.3802 1.3933

1.3339 1.3420 1.3546 1.3667 1.3795 1.3926

1.3331 1.3414 1.3539 1.3662 1.3789 1.3920

1.3320

1.3316

1.3312

a


least-squares, we obtained the temperature and concentration coefficients for all the six kinds of aqueous solutions and established empirical expressions for them as follows: NaCl solution: n = 1.3373 + (1.7682 ·10−3)c − (5.8 ·10−6)c 2 − (1.3531 · 10−4)(T − 273.15) − (5.1 · 10−8)(T − 273.15)2 (1)

KCl solution: n = 1.3352 + (1.6167 · 10−3)c − (4.0 · 10−7)c 2 − (1.1356 · 10−4)(T − 273.15) − (5.7 · 10−9)(T − 273.15)2 (2)

CaCl2 solution: n = 1.3339 + (2.5067 ·10−3)c − (3.9 ·10−8)c 2 − (1.1122 · 10−4)(T − 273.15) − (4.0 · 10−8)(T − 273.15)2 (3)

glucose solution: n = 1.3356 + (1.5333 ·10−3)c − (9.0 ·10−5)c 2 − (1.2647 · 10−4)(T − 273.15) − (4.0 · 10−8)(T − 273.15)2 (4)

protein solution: n = 1.3384 + (1.5985 ·10−3)c + (3.1 ·10−5)c 2 − (1.3939 · 10−4)(T − 273.15) + (2.4 · 10−7)(T − 273.15)2 (5) 2829



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Table 4. Experimental Values of Refractive Index (nD) at Temperature T, Mass Fraction w (kg/kg), and Pressure p = 0.1 MPa for the Solutions of BSA (Bovine Serum Albumin)a nD

a

w

293.15 K

298.15 K

303.15 K

308.15 K

313.15 K

318.15 K

0.005 0.010 0.015 0.020 0.024 0.029 0.034 0.038

1.3356 1.3372 1.3376 1.3380 1.3388 1.3397 1.3404 1.3411

1.3350 1.3368 1.3373 1.3379 1.3384 1.3392 1.3400 1.3408

1.3345 1.3362 1.3367 1.3374 1.3380 1.3385 1.3396 1.3401

1.3339 1.3354 1.3360 1.3366 1.3373 1.3380 1.3388 1.3397

1.3333 1.3348 1.3352 1.3358 1.3365 1.3374 1.3382 1.3388

1.3323 1.3336 1.3340 1.3351 1.3357 1.3364 1.3375 1.3382


Table 5. Experimental Values of Refractive Index (nD) at Temperature T, Mass Fraction w (kg/kg), and Pressure p = 0.1 MPa for the Solutions of Ethyl Acetatea nD

a

w

293.15 K

298.15 K

303.15 K

308.15 K

313.15 K

318.15 K

0.009 0.027 0.045 0.064 0.082

1.3341 1.3352 1.3368 1.3384 1.3391

1.3335 1.3348 1.3364 1.3380 1.3388

1.3331 1.3342 1.3358 1.3377 1.3383

1.3327 1.3337 1.3353 1.3368 1.3378

1.3321 1.3332 1.3346 1.3358 1.3373

1.3317 1.3326 1.3342 1.3352 1.3365


Figure 2. Comparison between our results and the reported data about the refractive index of NaCl and glucose solutions at 293.15 K and 298.15 K. In a, ●, our experimental data of NaCl solution at 293.15 K; ⧫, our experimental data of NaCl solutions at 298.15 K; □, data of NaCl at 298.15 K from ref 14;14 ◇, data of NaCl at 298.15 K from ref 15;15 ○, data of NaCl at 293.15 K from ref 16.16 In b, ⧫, ▲, our experimental data of aqueous glucose solution at 293.15 K and 298.15 K, respectively; ○, data of glucose at 298.15 K from ref 17;17 △, data of glucose at 298.15 K from ref 18;18 ◇, data of glucose at 293.15 K from ref 19.19 The dashed lines in (a) and (b) were plotted according to eqs 1 and 4, respectively for 293.15 K, whereas the solid line in (a) and (b) were plotted according to eqs 1 and 4, respectively for 298.15 K.

ethyl acetate solution:

it was proven that the result obtained from each of them was highly consistent with that of the Lorentz−Lorenz equation (see Figure 4). D. Variation of dn/dc with Temperature. Figure 5 shows the variation of dn/dc vs temperature. We can see that, in all the six solutions, the derivatives of the refractive index n with respect to concentration just slightly change with temperature and the depending coefficients are in the order of 10−6−10−8. Except that of the protein (BSA) solution, in all the other five solutions, no matter it is polar, nonpolar, or electrolyte solution, dn/dc decreases with temperature. The reason for the protein solution to have an opposite behavior is probably due to the structural variation of the protein with temperature. For there is a thermal aggregation effect on BSA to lead the particle radii of the protein aggregates to increase with temperature,21,22 so the dn/dc of the solutions increases with temperature. Since protein

n = 1.3360 + (0.7125 · 10−3)c − (2.6 · 10−6)c 2 − (1.1566 · 10−4)(T − 273.15) − (1.7 · 10−7)(T − 273.15)2 (6)

where c is the concentration of the solution and T is temperature in Kelvin. The concentrations in all the equations are in kg/kg·100. We can see that most of the concentration coefficients (except the ethyl acetate solution) are in the order of 10−3, whereas all the temperature coefficients are in the order of 10−4. In a comparison of the six equations with the Lorentz− Lorenz equation:20 NA n2 − 1 1 = α(ρ , T , λ), 2 3ε0M n +2ρ

(7) 2830



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Figure 3. Dependence of refractive index on temperature in all the six solutions at different concentrations. (a) Sodium chloride; (b) potassium chloride; (c) calcium chloride; (d) glucose; (e) BSA; (f) ethyl acetate.

Figure 4. Fitting curves of our empirical equation (solid line) and the Lorentz−Lorenz equation (dashed line) for NaCl solution. Dots are the experimental data. The left one is the RI variation with concentration at 293.15 K; the right one is the RI variation with temperature at 5% concentration. Both of the curves fit with the experimental point quite well. Similar results were also found in the other five solutions. 2831



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Figure 5. Variation of dn/dc vs temperature.

E. Variation of dn/dT with concentration. The gradient of refractive index with respect to temperature dn/dT, is also a significant parameter in various applications, such as in the measurements of the glass transition temperature13 and thermal lens effect.11 Whether dn/dT depends on concentration should be concerned for these applications. Figure 6 illustrates the variation of dn/dT vs concentration. Similar to the case of dn/dc varying with temperature, except the protein solution, in all the other five solutions, dn/dT just slightly decreases (or increases in absolute value) with concentration, and the depending coefficients are in the order of 10−7−10−8 thus can be neglected in most cases. However, in the protein solution, dn/dT increases with concentration and the depending coefficient is in the order of 10−6. This phenomenon is also probably due to the thermal aggregation effect of proteins.

thermal aggregation is quite common in protein solutions, it is expected that the dn/dc of other protein solutions would also increase with temperature. As mentioned above that, the refractive index increment, dn/dc is significant in polymer solutions for determining molecular weight with the Zimm equation23 using the technique of static light scattering. Therefore, it is needed to know if the variation of dn/dc with temperature in the protein solution is within an acceptable range. According to our experimental results for the solutions, there were only very little changes in the dn/dc with temperature in all the six solutions so that they can be neglected.

CONCLUSIONS In this paper, we investigated the temperature and concentration dependences of refractive indices of six aqueous solutions and established empirical expressions with both concentration coefficient and temperature coefficient for the solutions. We also deduced the variation of dn/dc vs temperature and the variation of dn/dT vs concentration for the six solutions. We demonstrated that dn/dc decreases with temperature and dn/dT decreases with concentration in polar, nonpolar, and electrolyte solutions. While in protein solution, both of the derivatives show an opposite behavior due to the thermal aggregation effect of proteins. These findings can help to improve the fundamental understanding of binary mixtures, and are significant for the determination of concentration,

Figure 6. Variation of dn/dT vs concentration. (a) ⧫, NaCl solution; solution; (d) ethyl acetate solution.

■

■,

KCl solution and ▲, CaCl2 solution; (b) glucose solution; (c) protein

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Glycol + NaCl + H2O, Ethylene Glycol + KCl + H2O, Ethylene Glycol + RbCl + H2O, and Ethylene Glycol + CsCl + H2O at 298.15 K. J. Chem. Eng. Data 2010, 55, 1289−1294. (16) O'Donnell, P. B.; McGinity, J. W. Preparation of microspheres by the solvent evaporation technique. Adv. Drug Delivery Rev. 1997, 28, 25−42. (17) Yeh, Y.-L. Real-time measurement of glucose concentration and average refractive index using a laser interferometer. Optics and Lasers in Engineering 2008, 46, 666−670. (18) Dong, L.; Liu, M.; Li, G.; Wang, L.; Sun, L.; Wei, X.; Di, Y. Volumetric Properties and Refractive Indices ofN,N′-Hexamethylenebisacetamide in Aqueous Glucose and Sucrose Solutions. J. Chem. Eng. Data 2011, 56, 4031−4039. (19) Lide, D. R. CRC Handbook of Chemistry and Physics, 82nd ed.; The Chemical Rubber Company: Cleveland, 2001. (20) Lorentz, H. A. The theory of electrons; Dover: New York, 1952. (21) Honda, C.; Kamizono, H.; Samejima, T.; Endo, K. Studies on Thermal Aggregation of Bovine Serum Albumin as a Drug Carrier. Chem. Pharm. Bull. 2000, 48, 464−466. (22) Zhao, H.; Brown, P. H.; Schuck, P. On the distribution of protein refractive index increments. Biophys. J. 2011, 100, 2309−2317. (23) Zimm, B. H. Molecular Theory of the Scattering of Light in Fluids. J. Chem. Phys. 1945, 13, 141−145.

the Soret coefficient, the glass transition temperature, the molecular weight, and the thermal lens effect of solutions.

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AUTHOR INFORMATION

Corresponding Author

*Phone number: +86-20-85223742. Fax number: +86-2085223742. E-mail address: [email protected]. Present Address

(C-Y.T. and Y.-X.H.) Department of Biomedical Engineering, Ji Nan University, Guang Zhou, China 510632. Funding

This work was supported partly by the Chinese National Natural Science Foundation (Nos. 30940019 and 60377043) and Guang Dong Provincial Science and Technology Foundation. Notes

The authors declare no competing financial interest.

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REFERENCES

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NOTE ADDED AFTER ASAP PUBLICATION This paper originally posted ASAP on September 1, 2015. Due to a production error, Figures 1 and 3 were swapped in the paper. The corrected version was reposted on September 3, 2015.

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Dependence of Refractive Index on Concentration and Temperature in

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