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Analysis of the Infrared Spectra of Diatomic Molecules

W

Richard W. Schwenz Department of Chemistry and Biochemistry, University of Northern Colorado, Greeley, CO 80639 William F. Polik* Department of Chemistry, Hope College, Holland, MI 49423; *[email protected]

One of the most common physical chemistry laboratory exercises involves the infrared spectrum of a heteronuclear diatomic molecule, usually HCl, from which the internuclear separation, the vibrational force constant, and higher-order molecular constants can be determined (1–5). The reasons for frequent use of this exercise include the relevance to the harmonic oscillator and rigid rotor models taught in quantum mechanics, the high accuracy of the spectroscopic measurements (achieving essentially literature values for the line positions), the existence of several isotopic variants of HCl allowing tests of the Born–Oppenheimer approximation, and the opportunity for students to perform a rigorous data reduction and error analysis to determine molecular parameters from spectroscopic data. It is this last feature with which this paper is concerned. In particular, several methods of data reduction are examined and an improved method using the multiple linear regression feature available on most microcomputer spreadsheet programs is presented that eliminates some of the problems inherent in other methods. Theory

displacement of the bond as vibrational energy increases. While some of the spectroscopic literature includes more molecular constants, these five are sufficient to describe the infrared spectra recorded for most diatomic molecules in the undergraduate physical chemistry laboratory. Thus, a complete expression for the rovibrational energy levels of a diatomic molecule is given by ∼

E (v, J ) = ωe(v + 1⁄ 2 ) +Be J( J + 1) – ωexe(v + 1⁄ 2 )2 – De J 2( J + 1)2 – αe(v + 1⁄ 2 ) J( J + 1)

(3)

where all the constants have wavenumber units. The lines in an absorption or emission spectrum arise from transitions between two energy levels. The wavenumber ν∼ of a transition can be obtained by subtracting the wavenumber energy of the lower state from that of the upper state ∼ ∼ ν∼(v′, J′,v′′, J′′) = E (v′, J ′) – E (v′′, J ′′) ∼

(4a)

∼

E (v′, J ′) – E (v′′, J ′′) = ωe[(v′ + 1⁄ 2) – (v′′ + 1⁄ 2 )] + Be[ J′( J′ + 1) – J ′′( J ′′ + 1)] –

All textbooks, laboratory manuals, and original spectroscopy papers treat the infrared spectra of diatomic molecules similarly. The molecules are first treated in the harmonic oscillator–rigid rotor approximation with energy levels given by E(v,J ) = [(h/2π)(k/µ)1/2](v + 1⁄ 2 ) + [h 2/8π2Ie ] J( J + 1) (1) where v is the vibrational quantum number, J is the rotational quantum number, h is Planck’s constant, k is the vibrational force constant, µ is the reduced mass, and Ie = µre2 is the moment of inertia with re being the internuclear separation. Dividing through by hc yields the molecular energy and parameters in wavenumber units (cm{1), which is indicated by the use of a tilde, ∼

E(v,J )/hc = E (v, J ) = ωe(v + 1⁄ 2 ) + Be J( J + 1)

(2a)

ωe = (k/µ)1/2/2 πc

(2b)

Be = h/8 π 2cI e

(2c)

ωexe[(v′ + 1⁄ 2)2 – (v′′ + 1⁄ 2)2] – De[ J ′2(J′ + 1)2 – J ′′2(J′′ + 1)2] – αe[(v′ + 1⁄ 2 )J′( J′ + 1) – (v′′ + 1⁄ 2)J ′′( J ′′ + 1)]

(4b)

where the single prime indicates the upper state and the double prime indicates the lower state. In order to apply eq 4b, appropriate quantum numbers (v ≥ 0, J ≥ 0) and selection rules (∆J = ±1) must be used. For the HCl molecule, the infrared absorption spectrum consists of about 20 pairs of closely spaced lines for both the fundamental (v′′ = 0 → v′ = 1) and first harmonic (v′′ = 0 → v′ = 2) bands for each of the H and D isotopic variants. The

where and are the harmonic vibrational frequency and rotational constant, respectively. Three correction terms are then usually introduced to this energy level expression. The first of these, the anharmonicity constant ωexe, arises from the departure of the vibrational energy levels from the harmonic approximation, in part to allow for bond dissociation. The second, the centrifugal distortion constant De, arises from the stretching of the molecule as it rotates faster. The third, the vibration–rotation interaction constant α e, arises from the increase in mean 1302

Figure 1. Infrared absorption spectrum of the fundamental band for HCl.

Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu


spacing within a pair arises from the slight difference of the molecular constants for the 35Cl and 37Cl isotopic variants. Figure 1 presents the fundamental band of HCl. The overall appearance of the spectrum is due to the ∆J = ±1 rigid rotor selection rules, which result in a P branch (∆J = {1) and an R branch (∆J = +1) of almost equally spaced pairs of lines surrounding the band origin. The spacing between each pair of lines is approximately 2Be. The relative intensity of each pair of lines arises from the 2J + 1 degeneracy of the rigid rotor energy levels, the Boltzmann distribution of molecules in the ground state rotational levels, and the rotational absorption intensity factors (squared dipole transition moment) (6 ). The relative intensity of the lines within each pair is due to the relative isotopic abundance of 35Cl and 37Cl. Analysis of the Spectrum

Average Spacing The simplest method for analyzing the infrared spectrum of a diatomic molecule relies on the harmonic oscillator–rigid rotor energy level expression of eq 2. In this approximation, the spacing between all the lines is averaged, this average is set equal to 2Be, and the moment of inertia Ie and internuclear separation re are calculated. The force constant k for the bond is simply calculated from the wavenumber of the band center. This method has the virtue of simplicity. It also has the disadvantage of being vastly oversimplified, as it neglects the additional terms present in eq 3 that do not occur in eq 2. This method is suitable for homework problems in textbooks, but not for the quantitative analysis of an actual spectrum. Successive Differences It is readily apparent from the spectrum of HCl that the lines are not equally spaced, but that the spacing between lines in the P branch expand and lines in the R branch contract as J increases. The method of successive differences found in laboratory manuals (2, 5, 6 ) accounts for this observation by recognizing that the transition wavenumbers in the spectrum can be approximated as a parabolic function of an index variable m where m = J ′′ + 1 for the R branch and m = {J ′′ for the P branch. Neglecting the effect of De (since it contributes significantly only at high J ) and using this substitution, the transition wavenumbers in the fundamental band are found to be given by ν∼(m) = ωe – 2ωexe + (2Be – 2αe)m – αem2

(5)

Typically, a successive difference plot of the wavenumber difference between adjacent transitions versus m is constructed allowing the determination of Be and αe from a linear fit to ∆ν∼(m)

= ν∼(m

+ 1)

– ν∼(m)

= (2Be – 3αe) – 2αem

(6)

The fundamental band origin ω10 = ωe – 2ωexe may be calculated by rearranging eq 5 to isolate ωe – 2ωexe and substituting the values of Be and αe along with the observed transition wavenumbers for several values of m near the band center. The intercept (m = 0) of a plot of ν∼(m) – (2Be – 2αe)m + αem2 versus m yields the best estimate of the fundamental band origin ω10. Analysis of the first overtone band is similar, with the transition wavenumbers given by ν∼(m) = 2ωe – 6ωexe + (2Be – 3αe)m – 2αem2

(7)

and the difference between adjacent transitions by ∆ν∼(m) = ν∼(m + 1) – ν∼(m) = (2Be – 5αe) – 4αem

(8)

The fundamental band origin ω10 may be combined with knowledge or measurement of the first overtone ω20 = 2ωe – 6ωexe to determine the values of ωe and ωexe. The successive differences method readily introduces the concept that vibration–rotation interaction is responsible for the unequal spacing of spectroscopic transitions in the HCl spectrum. However, this method results in an inaccurate value of Be due to neglect of the centrifugal distortion term De, often yields inconsistent values of B e and αe from the independent analyses of the fundamental and overtone bands, and does not correctly calculate the uncertainties in Be and α e due to statistical correlation between the coefficients of m and m2. It should be noted that the successive difference method can be extended to determine De by retaining it in the energy level expression to yield an equation for ν∼(m) that is cubic in m (6, 7 ). This equation can be analyzed by least squares fitting to cubic polynomial or by taking the second successive difference ∆2ν∼(m); however, this method is not used in any physical chemistry laboratory manual owing to its complexity.

Combination Differences An analysis technique used by spectroscopists prior to modern computational abilities was the combination differences method (6, 8–10). In this method, it is recognized that certain pairs of spectral lines share a common upper or lower state, allowing an energy difference between states to be calculated. For example, the R( J ′′ = 0) and P( J ′′ = 2) transitions both have J ′ = 1 in the upper state; thus, the difference in transition energies for the R( J ′′ = 0) and P( J ′′ = 2) lines is a direct measurement of the energy separation between J ′′ = 0 and J ′′ = 2 in the lower state. Formulas involving the rotational constants of the upper and lower vibrational states and the centrifugal distortion constant can be derived by taking the expression for spectroscopic transitions, substituting upper state quantum numbers using the rigid rotor selection rules, identifying appropriate pairs of combination difference states, and eliminating the upper or lower state energy (1, 3, 4, 6 ). The two simplest resulting formulas are ν∼R( J ′′) – ν∼P( J ′′) = 2(2J ′′ + 1)Bv′ – 4[( J ′′ + 1)3 + J ′′3]De (9)

and ν∼R( J ′′) – ν∼P( J ′′ + 2) =

2(2J ′′ + 3)B0 – 4[( J ′′ + 1)3 + ( J ′′ + 2)3]De

(10)

where Bv ′ = Be – αe(v′ + 1⁄ 2 ) is the rotational constant of the upper vibrational state and B0 is the rotational constant of the lower vibrational state. A formula involving the band origin can be similarly derived: ν∼R( J ′′) + ν∼P( J ′′ + 1) =

2[v ′ωe – v ′(v ′ + 1)ωexe] + 2( J ′′ + 1)2(Bv ′ – B0)

(11)

To determine the values of B1 and De from the fundamental band, for example, eq 9 is linearized by dividing through by 2(2J ′′ + 1) and a plot of [ν∼R( J ′′) – ν∼P( J ′′)]/[2(2J ′′ + 1)] versus {4[(J ′′ + 1)3+ J ′′3]}/[2(2J ′′ + 1)] is constructed. The intercept equals B1 and the slope is {De. Similar linearized plots of eqs

JChemEd.chem.wisc.edu • Vol. 76 No. 9 September 1999 • Journal of Chemical Education

1303


10 and 11 yield B0, another determination of De, ω10, and the difference between upper and lower state rotational constants B1 – B0. Analysis of the first overtone band yields B2, two additional values of De, another value of B0, ω20, and B2 – B0. The molecular constants Be and αe are then determined from the intercept and slope of a plot of Bv ′ versus v ′ + 1⁄ 2 , and De is taken to be its average determined value. The constants ωe and ωexe are determined from the values of ω10 and ω20. The difficulties encountered by students using this method are legendary. Problems include understanding the derivation of the combination difference formulas, the immense amount of data that must be handled to produce the three graphs per vibrational band per isotope required to determine the molecular constants, and resolution of inconsistent values of molecular constants determined from independent fits. In addition, spectroscopists recognize that the combination differences method introduces errors when missing or blended lines occur, alters the weights of lines of differing J values because of the form of the linearized equation, and completely neglects the statistical correlation between the molecular constants in the upper and lower states (11). A New Method of Analysis We suggest here that a different analysis method be used by physical chemistry students, which takes advantage of the relative simplicity of the energy expression in eq 3 and the ability of a microcomputer spreadsheet program to perform multiple simultaneous regression on a large amount of data. The preceding data analysis methods can be frustrating to students because of the repetitive calculations, the graph production, and the indirect method of obtaining molecular constants. In addition, the methods themselves suffer from inherent problems in determining molecular constants when the energy expression coefficients are highly correlated. The new method dramatically simplifies the effort of data reduction, provides a more direct connection between experimental data and molecular constants, and yields results that are more accurate and precise than the traditional methods.

Fundamental and Overtone In our method, the wavenumber and the associated quantum numbers of each transition are entered into a spreadsheet row. The data occupy five columns of the spreadsheet, one for the transition wavenumber, one each for the upper state vibrational and rotational quantum numbers, and one each for the lower state vibrational and rotational quantum numbers. Thus, five pieces of data are entered for each observed transition: ν∼, v′, J ′, v′′, J ′′. The same information is entered into a new row for each transition observed in the spectrum. Five columns are subsequently computed that allow the determination of the molecular constants directly via multiple linear regression. These five columns are computed from the five terms in the wavenumber transition expression, eq 4b. Each term in eq 4b consists of a molecular constant multiplied by a combination of quantum numbers for the upper state minus the same combination of quantum numbers for the lower state. For example, Be is multiplied by J ′( J ′ + 1) – J ′′( J ′′ + 1). The five computed columns consist of the quantum number expressions that multiply the five molecular constants that are being determined. Thus, these five columns will contain formulas for (v′ + 1⁄ 2) – (v′′ + 1⁄ 2 ), J ′(J ′ + 1) – 1304

J ′′(J ′′ + 1), (v′ + 1⁄ 2 )2 – (v′′ + 1⁄ 2 )2, J ′2(J ′ + 1)2 – J ′′2(J ′′ + 1)2, (v′ + 1⁄ 2 )J ′(J ′ + 1) – (v′′ + 1⁄ 2 )J ′′(J ′′ + 1), in which each formula is calculated from the quantum numbers in the first five columns of the spreadsheet. The most straightforward procedure is to create these formulas in the first row and then copy them into the rest of the rows. The remaining step is to initialize and perform the multiple linear regression. The regression variables must be defined in the spreadsheet. The y (or dependent) variable is the wavenumber of the transitions. The x (or independent) variables consist of the five computed columns defined in the previous paragraph, which contain the quantum number expression associated with each molecular constant. Since the form of the energy expression does not contain a constant term, the regression should have a defined zero value for the intercept. Once the regression is performed, the output will include five x coefficients with their associated errors and the standard error of the regression analysis. The five regression coefficients are ωe, Be, { ωexe, {De, and { αe, which correspond directly to the energy expression coefficients of eq 3. The coefficient errors are the errors in each molecular constant. The standard error is a measure of how well the observed transitions fit eq 4b. The equilibrium internuclear separation, vibrational force constant, and their associated uncertainties are easily calculated from the regression output using eqs 2b and 2c.

Fundamental Only The preceding treatment assumes that measurements have been made on both the fundamental and first overtone bands of HCl. Should measurements be made only on the fundamental band, then information is not present to determine both the harmonic frequency ωe and the anharmonicity ωexe, and the matrix inversion in the regression routine will fail. In this case, the known vibrational quantum numbers, v′ = 1 and v′′ = 0, should be incorporated into the expression for transition wavenumbers, eq 4b, resulting in ν∼(v′=1, J ′,v′′=0, J ′′) = ωe – 2ωexe + Be[ J ′( J ′ + 1) – J ′′( J ′′ + 1)] –

De[ J ′2( J ′+ 1)2 – J ′′ 2( J ′′+ 1)2] – αe[ 3⁄2 J ′( J ′+ 1) – 1⁄2 J ′′( J ′′ + 1)]

(12)

Three data columns are needed for the values of ν∼, J ′, and J ′′. Three formula columns are used to compute J ′( J ′ + 1) – J ′′( J ′′ + 1), J ′ 2( J ′ + 1)2 – J ′′ 2( J ′′ + 1)2, and 3⁄ J ′( J ′ + 1) – 1⁄ J′′( J ′′ + 1). Since eq 12 contains a constant 2 2 term, that is, a term not multiplied by a quantum number expression, the regression is performed with three independent variables that correspond to the three formula columns and the intercept must be allowed to vary. The three regression coefficients will correspond to Be, {De, and { αe, and the intercept will correspond to the vibrational origin, ω10 = ∼ ∼ E(v′=1, J ′=0) – E (v′′=0, J ′′=0) = ωe – 2ωexe. The coefficient errors are the errors in each molecular constant, and the intercept error is the error in the vibrational origin. An alternate method of fitting the constant term in eq 12 is to include a column of ones for it in the spreadsheet and perform the regression with four independent variables while defining the intercept to be zero. The vibrational origin will then correspond to the coefficient for the column of ones. This



alternate method is particularly useful if the spreadsheet program does not output the error for the intercept.1

Calculation of Residuals Regardless of whether data from the overtone band are used, it is critical to use two more columns in the spreadsheet to calculate the predicted transition wavenumber from the determined constants and to calculate the residual difference between observed and predicted wavenumber for each transition. This not only illustrates how to use molecular constants to calculate spectroscopic transitions, but also serves as an essential check for data entry or calculational errors. The regression method is numerical rather than graphical, and data entry errors are not easily caught before performing the regression analysis. Since line positions are determined with high precision (typically 0.2 cm{1 or better) with modern FTIR instrumentation and eq 3 fits the observed energy levels extremely well over the observed quantum number range, the calculated residuals should be on the order of the precision of the spectrometer. It is very easy to pick out a transition with a data entry error, as its residual will be much larger than the standard error of the regression. Also, errors in the first-order spectroscopic constants (ωe and Be) may easily be as small as 1 part in 105 with data taken on a FTIR instrument, so the existence of a data entry or calculational error is flagged by an unreasonably large error for a regression constant.

Table 1. Transition Wavenumbers and Assignments for Fundamental and First Overtone Bands of H 35 Cl v ′′

v′

J ′′

J′

∼ ν/cm{1

0

1

0

1

2905.995

0

1

1

2

2925.581

0

1

2

3

2944.577

0

1

3

4

2962.955

0

1

4

5

2980.689

0

1

5

6

2997.788

0

1

6

7

3014.202

0

1

7

8

3029.941

0

1

8

9

3044.965

0

1

9

10

3059.234

0

1

10

11

3072.771

0

1

11

12

3085.600

0

1

12

13

3097.550

0

1

13

14

3108.914

0

1

14

15

3119.418

0

1

15

16

3129.099

0

1

1

0

2864.834

0

1

2

1

2843.315

0

1

3

2

2821.249

0

1

4

3

2798.641

0

1

5

4

2775.499

Comparison of Methods

0

1

6

5

2751.817

0

1

7

6

2727.624

The regression method of data analysis is computationally superior to both the successive-difference and combination-difference methods. To demonstrate this, an H35Cl data set recorded with a Nicolet 730 FTIR spectrometer is presented in Table 1. This data set has been analyzed using each of the methods described above. A spreadsheet with these analyses is available on the World Wide Web (12, 13). Implementation of the regression method using Mathcad is also possible (14, 15). The results of these analyses are compared to accepted literature values (16, 17) in Table 2. The regression analysis results are both more precise and more accurate than the successive-difference and combination-difference methods. The successive-difference method neglects the centrifugal distortion constant De, and its effect is therefore incorporated into the determination of the other constants. This is apparent in the successive difference plot of Figure 2, in which a systematic curvature is observed resulting in a significant underestimate of Be when the slightly curved data set is fit to a straight line. The combination-difference method results from a linear least squares analysis with equal weights are also given in Table 2. The numerical difficulties of this method are apparent from the deviation from linearity near the intercept in the plot of Figure 3 (the result of fitting eq 10) and from the inconsistent De values among plots despite the fact that all values should be the same at this level of theory. Owing to division by 2(2J ′′ + 3) when linearizing eq 10, for example, the y uncertainties for low J ′′ data points are greater than high J ′′ data points, resulting in larger deviation of the data from the best fit near the intercept in Figure 3. The determination of De by averaging its value from four separate analyses causes an unrealistically large error for De in Table 2. The regression method simultaneously determines all the molecular constants without neglecting any terms. In addition,

0

1

8

7

2702.907

0

1

9

8

2677.697

0

1

10

9

2651.932

0

1

11

10

2625.689

0

1

12

11

2598.979

0

1

13

12

2571.861

0

1

14

13

2544.220

0

1

15

14

2516.141

0

2

0

1

5687.494

0

2

1

2

5705.926

0

2

2

3

5723.158

0

2

3

4

5739.109

0

2

4

5

5753.793

0

2

5

6

5767.262

0

2

6

7

5779.441

0

2

7

8

5790.312

0

2

8

9

5799.833

0

2

9

10

5808.141

0

2

1

0

5646.969

0

2

2

1

5624.896

0

2

3

2

5601.612

0

2

4

3

5577.185

0

2

5

4

5551.571

0

2

6

5

5524.865

0

2

7

6

5496.971

0

2

8

7

5467.968

0

2

9

8

5437.895


1305

Information • Textbooks • Media • Resources Table 2. H35Cl Molecular Constants Determined by Various Methods Value (SD) /cm{1 Quantity Average Spacing ωe 2860(37)

Successive Difference 2989.215(23)

2989.206(55)

2989.281(34)

2989.74

2990.97424

Be ωexe

10.41(12)

10.58982(85)

10.58919(89)

10.5909

10.593404

9.8(31)

Combination Difference

Multiple Regression

Ref 16

Ref 17

—

51.763(8)

51.765(18)

51.796(12)

52.05

52.84579

De αe

—

—

0.0005041(181)

0.0005206(29)

0.00053

0.000532019

—

0.30421(118)

0.30318(57)

0.30167(10)

0.3019

0.307139

ω10

—

2885.689(8)

2885.676(18)

2885.690(41)

—

2885.9775

ω20

—

5667.852(5)

5667.823(7)

5667.789(97)

—

5667.9841

Deviation a

49

1.431

0.172

0.046

—

—

aRoot-mean-square

residual.

of the energy level expression, eq 3, and the concept that spectroscopic transitions arise from transitions between energy levels, eq 4, over the details of a data analysis method that is specific to the analysis of diatomic rovibrational spectra. Third, it gives students experience with the technique of multiple linear regression. Physical chemistry students get much experience fitting data to a straight line, y = mx + b (linear regression), but almost no experience fitting data to an equation of the form y = m1x1 + m2x2 + m3x3 + … + b (multiple linear regression). This latter form is critical when dealing with data that depend on more than one variable, as is the case for most real data sets. Finally, it introduces students to a method that roughly corresponds to the data analysis method used by practicing spectroscopists, namely, multiple nonlinear regression (nonlinear least squares fitting) to the energy level expression (Hamiltonian matrix) of a molecule. Conclusion Analysis of the infrared spectra of diatomic molecules by multiple linear regression in a spreadsheet dramatically simplifies the reduction of spectroscopic data compared to

2 2 J ′′ + 3

νR J ′′ – νP J ′′ + 2

/ cm{1

no skewing of the weights is introduced from linearizing fitting equations, correlation between the molecular constants in the upper and lower state constants is accounted for, and each observed transition is weighted equally. It should be noted that equal weighting is appropriate for data acquired with an FTIR instrument, as the wavenumber error in line positions is constant over the entire spectral range rather than varying with wavelength, as occurs with grating or prism instruments. In all cases, care must be taken when comparing fit values and literature values to make sure they use a consistent set of molecular constants and the same range of rotational and vibrational levels in the analyses. Inclusion of additional constants or more molecular states will affect the values obtained in any analysis. There are also pedagogical advantages to using the regression method to analyze the infrared spectrum of diatomic molecules. First, it replaces the tedium of preparing and fitting multiple plots with a much easier and far less time-consuming use of computer technology. Students learn the concepts underlying the experiment and gain experience analyzing data using spreadsheets, rather than worrying about the details of graph preparation. Second, the method emphasizes the form

3

{4 J ′′ + 1 + J ′′ + 2 2 2 J ′′ + 3

Figure 2. Successive difference plot, eq 6, for HCl fundamental band with least squares linear fit. The systematic curvature of the data points is caused by neglect of the centrifugal distortion constant De. The lowered intercept of the least squares fit results in an underestimate for the value of the rotational constant Be.

1306

3

/ cm{1

Figure 3. Combination difference plot, eq 10, for HCl fundamental band with least squares linear fit. Inclusion of the centrifugal distortion constant De results in a more linear plot than the successive difference plot in Figure 2. However, low J ′′ data points near the y-axis have larger uncertainties and should be weighted less than high J ′′ data points.



the traditional successive difference and combination difference methods. It also provides a more direct connection between experimental data and the molecular constants. The resulting molecular constants are shown to be both more accurate and more precise than the results of traditional analysis methods. Acknowledgments This work was supported by National Science Foundation grants CHE-8851651 (RWS) and CHE-9157713 (WFP). Note W

Supplementary materials for this article are available on JCE Online at http://jchemed.chem.wisc.edu/Journal/issues/1999/Sep/ abs1302.html. 1. After submission of this paper, an alternative spreadsheet-based analysis method was published which uses the Solver tool in Excel to perform a nonlinear least squares optimization of a transition wavenumber expression to a data set (18). This method could be adapted to optimize eq 4 or eq 12 and should give the same results as the multiple linear regression method. The Solver method has the advantage of working for more complex, nonlinear functional forms. However, it is possible for the method to converge at a local minimum giving an incorrect result, and the determination of errors in fitting parameters is not performed as easily as with the multiple linear regression method.

3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14.

15.

Literature Cited

16.

1. Daniels, F.; Williams, J. W.; Bender, P.; Alberty, R. A.; Cornwell, C. D.; Harriman, J. E. Experimental Physical Chemistry, 7th ed.; McGraw-Hill: New York, 1970; pp 247–256. 2. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in

17. 18.

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