EPR

Page 1

CHEMICAL MOTIONS MONITORED BY SPIN-SPIN EXCHANGE AND ELECTRON PARAMAGNETIC RESONANCE SPECTROSCOPY LAB PREPARATION Read the attached publication [J. Chem. Ed. 59, 677-679 (1982)] for the theory of spin exchange and diffusion. EXPERIMENT SUMMARY Molecules possessing a magnetic moment, either nuclear or electronic, can exchange their moment with a corresponding molecule upon collision. This phenomena is referred to as spin-spin exchange and it can be utilized to study the rates of chemical reactions and motions. The rate of spin-spin exchange between molecules can be determined in NMR or EPR experiments. In this experiment, we will use electron paramagnetic resonance (EPR), also called electron spin resonance (ESR), spectroscopy to measure the collision frequency of nitroxide free radicals in solution and compare the result to that expected from a simple diffusion process. If the spin exchange is governed by the rate of collisions, the observed rates of spin exchange can give kinetic information for reactions. The nitroxide EPR spectrum originates from a single unpaired electron which couples magnetically to a nitrogen atom. A collision between nitroxide molecules may allow an exchange of electron spins, but not the nitrogen nuclear spins. As the collision rate increases, the electron-nuclear coupling information is lost and the spectrum broadens. The collision rate will be varied in this experiment by changing the concentrations of nitroxide in solution. The corresponding changes in the EPR linewidth will be measured to give the spin exchange rate. THEORY For nitroxide free radicals, the single unpaired electron (S = 1/2, mS = ±1/2) usually has a significant density at the 14N nucleus (I = 1, mI = -1, 0, +1), i.e. the electron spend a fraction of time at N. This electron-nuclear contact gives rise to a hyperfine coupling, A, between S and I, with energy, Ehf = AmSmI. The electron spends most of its time on O, but 16O has no nuclear moment (I=0) and therefore no hyperfine interaction. The total energy of a state is given by the sum of the three terms: electron Zeeman, nuclear Zeeman, and hyperfine, respectively, E = geµBBmS - gNµNBmI + AmSmI

EPR

Page 2

where µB = 9.274 x 10-24 J/T and 1T (Telsa) = 104 G (Gauss).The following diagram gives the relative positions of the states mS, mI and their energies for a nitroxide free radical. ½geµBB - gNµNB + A/2 ½geµBB ½geµBB + gNµNB - A/2

-½geµBB + gNµNB + A/2 -½geµBB -½geµBB - gNµNB - A/2 The EPR spectrum can be predicted from this energy level diagram. To observe a resonance, the microwave energy, hν, of the radiation must match one of the above energy differences, ∆E. In addition, resonances are only allowed in accordance with the selection rules, ∆mS = ±1, ∆mI = 0. These rules means that only the three transitions shown by arrows in the above figure are allowed. The energy differences of these transitions are ∆E = geµBB - A ∆E = geµBB ∆E = geµBB + A

(mI = -1) (mI = 0) (mI = +1).

In most EPR experiments, ν is fixed near 10 GHz and B is varied. Using ∆E = hν, we expect to find three resonances at magnetic field values of B1 = (hν - A)/geµB

B2 = hν/geµB B3 = (hν + A)/geµB

and the corresponding spectrum is shown below.

EPR

Page 3

The hyperfine constant A (in MHz) can be determined from the field separation between any two adjacent resonance lines a (in Gauss). The conversion is A = 2.802 a. Read the attached publication [J. Chem. Ed. 59, 677-679 (1982)] for the theory of spin exchange and diffusion. Your objective in this experiment is to compare the spin exchange rates observed by EPR spectroscopy with the theoretical diffusion rates, and explain any differences.

EXPERIMENT Obtain a solid sample of α,α’-diphenyl-β-picrylhydrazyl (DPPH) in an EPR tube and find the resonance. Follow the operating instructions given in Appendix EPR (page Error! Bookmark not defined.). DPPH is a widely used calibration standard for EPR spectroscopy. It is a stable radical and is commercially available from Aldrich.

DPPH Magnetic field calibration. The g-value of DPPH is 2.0036. Use this value to calibrate the center magnetic field of the instrument. From the resonance condition, hν = gµBB, the magnetic field at resonance is given by B = 714.5 ν/g (ν in GHz, B in Gauss) The microwave frequency can be read from the instrument’s dial. Obtain sample of CaO in an EPR tube. CaO usually has a Mn2+ impurity of much less than 0.1%, however such concentrations can easily be detected. Mn has a nuclear spin of I=5/2, thus we expect to see 2I+1 = 6 resonances. Find the 6-line spectrum of Mn and use it to calibrate the magnetic field range of the instrument. The hyperfine splitting is a=8.5 mT. PADS experiment. Obtain no more than 3 mL stock solutions of PADS and K2CO3. The free radical has a finite self life in solution. Determine the concentration of the stock solution spectrophotometrically (ε is given in the JCE paper), diluting it by a factor of approximately 25. Make 8 samples by diluting an aliquot of the PADS stock solution into K2CO3, covering the concentration range of approximately 5-60 mM. Be sure to accurately determine the dilution factors you have used. Record the spectra of the 8 different concentrations of PADS solutions and measure the linewidth, ∆H, of the central

EPR

Page 4

resonances. You will need to expand the central resonance to accurately measure the width. To determine ∆H0, dilute the 5 mM sample by a factor of 10 and record its spectrum and width. Follow the analysis of the data as given in the JCE paper. The correction factor f* is due to the electrostatic repulsion of the charged radical ions. For a species without charge, f* = 1, but for our experiment with potassium nitrosodisulfonate [PADS, (KSO3)2NO], f* < 1. You can determine f* for your particular concentrations by interpolating between the values given in the table of the JCE paper. Plot f* vs. PADS concentration and fit this with a second order polynomial. You do not need to calculate f* as suggested in the paper. For the value of ∆H0, use the lesser of your value and that given in the JCE paper. Since your sample was not deoxygenated, the linewidth may not reflect a true minimum value. For a solution in water at 20ºC, the viscosity, η, is 0.001 Pa-s. Use MKS units for the calculations of the spin exchange rate and diffusion rates. Put units on your numbers and make sure they cancel properly to give either sec or sec-1. Note the typo in the JCE paper: γ = 1.76 x 10+7 sec-1Gauss-1 (not x 10-7). Your objective in this experiment is to compare the spin exchange rates observed by EPR spectroscopy with the theoretical diffusion rates, and explain any differences. Estimate the error in the measurements as they are recorded so that you can determine whether or not the difference between the two rates is within experimental uncertainties. If not, is there some theoretical reason for the difference?

EPR

Page 5

EPR

Page 6

EPR

Page 7

DNA THERMODYNAMICS

Page 1

THERMODYNAMICS OF DNA DUPLEX FOMATION LAB PREPARATION Read the attached publication [J. Chem. Ed. 77, 1469-1471 (2000)] and download the journal article Nucleic Acids Research, (28), No. 23 4762-4768 (2000). Use these papers as a guide for the determination of ∆H°, ∆S°, and ∆G° during the transformation of a DNA duplex to form two single stranded DNA monomers. EXPERIMENT In this lab you will measure thermodynamic properties of a short DNA duplex by melting the ordered native structure (duplex or double helix) into the disordered, denatured state (single strands) while monitoring the transition using ultraviolet (UV) spectrophotometry. As the ordered regions of stacked base pairs in the DNA duplex are disrupted, the UV absorbance increases. This difference in absorbance between the duplex and single strand states is due to an effect called hypochromicity. Hypochromicity, which simply means “less color”, is the result of nearest neighbor base pair interactions. When the DNA is in the duplex state, interactions between base pairs decrease the UV absorbance relative to single strands. When the DNA is in the single strand state the interactions are much weaker, due to the decreased proximity, and the UV absorbance is higher than the duplex state. The profile of UV absorbance versus temperature is called a melting curve; the midpoint of the transition is defined as the melting temperature, Tm. The dependence of strand concentration on the Tm of a melting transition can be analyzed to yield quantitative thermodynamic data including ΔH°, ΔS°, ΔG° for the transition from duplex to single strand DNA. Thermodynamic analyses of this type are done extensively in biochemistry research labs, particularly those involved in nucleic acid structure determination. In addition to providing important information about the conformational properties of either DNA or RNA sequences (mismatched base pairs and loops have distinct effects on melting properties), thermodynamic data for DNA are also important for several basic biochemical applications. For example, information about the Tm can be used to determine the minimum length of a oligonucleotide probe needed to form a stable double helix with a target gene at a particular temperature. PROCEDURE You will melt a duplex formed by two complementary synthetic DNA oligomers: Five separate samples with different concentrations (indicated on cuvettes) have been prepared for you in buffer (1M NaCl, 10 mM sodium phosphate pH 7.0, 0.1 mM EDTA). The buffer was carefully degassed by bubbling nitrogen through it before the samples were made. Oxygen dissolved in the sample will form bubbles at higher temperatures, which will scatter light and affect the absorbance measurements. The samples (0.4 mL each) have been placed in 1 cm path length quartz cuvettes that are sealed with teflon stoppers. One additional 1 cm cuvette filled with buffer will also be provided to act as a reference cell for the spectrophotometer.

DNA THERMODYNAMICS

Page 2

NOTE: the quartz cuvettes are expensive and fragile. Please treat them very carefully. During your melting experiment you will monitor the change in absorbance at 254 nm over the temperature range 10°C to 70°C. Your instructor will give you a set of detailed instructions 2 concerning the spectrophotometer parameters that you will be using. Please carefully record in your notebook the parameters used to collect your data. To record the most accurate data in a research laboratory, melting curves of this type would generally be done slowly (over several hours) at small temperature increments to ensure complete temperature equilibration at each point. However, your experiment has been designed to fit into a two lab periods by minimizing the amount of time necessary to equilibrate at each temperature by the choice of particular DNA duplex and the use of small sample volumes. Nonetheless, you should be aware that incomplete temperature equilibration could be a source of error in your measurements. CALCULATIONS Your first step will be to make a single graph of temperature versus absorbance that contains the four melting curves. Melting curves of DNA are commonly described using standard helix-tocoil transition theory. In our case the "helix" is duplex DNA and the "coil" is the disordered single DNA strands. The transition from helix to coil is monitored in our experiment as a function of temperature by UV absorbance. This can be done because the percentage of hyperchromicity (increase in absorbance as the duplex is melted) varies linearly with the number of unstacked bases. Thus our melting curve relates the absorbance to the fraction of paired bases (f) as the temperature is increased. The Tm is the temperature where f = 0.5. The steep part of the melting curves reflects the double strand (AB) to single strand (A+B) equilibrium. A + B ⇔ AB

(eq. 1)

The treatment we used assumes a two-state (all-or-none) model. In a two-state model, f is the fraction of fully based-paired strands since there are no partially base-paired intermediates in the melting process. The two-state model has been shown to be a very good approximation for short (< 12 base pairs) DNA duplexes. Using this model we must adjust the absorbance data to a normalized scale so that the values range from 0 to 1 (we will call these values relative absorbance). Then a relative absorbance of 0 occurs when all of the bases are paired (all in the duplex state) and a relative absorbance of 1 occurs when all of the bases are un-paired (all in the single strand state). At a relative absorbance of 0.5 half of the strands are paired and half are unpaired, thus f = 0.5 and the temperature at this point is Tm. You will obtain thermodynamic data from the concentration dependence of the Tm for each of your curves. Next make a van't Hoff plot of (1/Tm) versus ln(Ct) where Ct is the sum of the molar concentrations of each single strand. Using the following relationship. (eq 2.)

1

𝑻𝒎

𝑅

= ∆𝐻° ln(𝐶𝑡 ) +

(∆𝑆°−𝑅𝑙𝑛4) ∆𝐻°

DNA THERMODYNAMICS Calculate ΔH° and ΔS°. Finally, calculate ΔG° at 25°C. Things to include on your lab report. • • • • • •

Normalilzed melting curves. van't Hoff plot. ΔH° and ΔS° for helix formation. ΔG° at 25°C for helix formation. Literature values for ΔH°, ΔS°, and ΔG°. All appropriate errors.

Page 3

In the Laboratory

Thermodynamics of DNA Duplex Formation

W

A Biophysical Chemistry Laboratory Experiment Kathleen P. Howard Department of Chemistry, Swarthmore College, Swarthmore, PA 19081; [email protected]

The goal of this experiment is to measure thermodynamic properties of a short DNA duplex by melting the ordered native structure (duplex) into the disordered, denatured state (single strands) while monitoring the transition using ultraviolet (UV) spectrophotometry. Heating Cooling

As the ordered regions of stacked base pairs in the DNA duplex are disrupted, the UV absorbance increases. The profile of UV absorbance versus temperature is called a melting curve and the midpoint of the transition is defined as the melting temperature, Tm. The dependence of Tm on strand concentration can be analyzed to yield quantitative thermodynamic data (∆H °, ∆S °, ∆G °). Thermodynamic analyses of this type are routine in many biophysical research labs, particularly those involved in nucleic acid structure determination (1, 2). Thermodynamic information derived from DNA melts has helped in understanding the sequence dependence and polymorphism in the secondary structure of nucleic acids (mismatched base pairs and loops have distinct effects on melting properties) (3). In addition to providing important information about conformation, knowledge of thermodynamic data for DNA is essential for several basic biochemical applications. For example, one must know the Tm to determine the minimum length of a probe oligonucleotide needed to form a stable double helix with a target gene at a particular temperature. The experiment described here was devised for a newly developed course in Biophysical Chemistry, which is part of the Physical Chemistry requirement for our Biochemistry majors. The goal of this course is to demonstrate the important role physical chemistry plays in understanding the structures and properties of biological macromolecules. A challenge for us has been to incorporate real biophysical experiments into the laboratory component of the course despite the lack of such experiments in traditional physical chemistry textbooks. A few biophysical laboratory experiments have been published recently (4, 5), but there is clearly a need for more. Methods

Equipment Any standard commercial UV spectrophotometer with temperature control can be equipped to measure melting curves. We used a Cary 1 UV–vis spectrophotometer equipped with a multicell block and transport apparatus. This enabled us to complete the melts of four different samples at the same time and have each student collect his or her own full set of

data to analyze. Without a multicell block, arrangements can be made to have different students collect different concentration points and have the class share data. A set of quartz cuvettes that can be sealed to prevent evaporation during heating is also required. Plotting and analysis of the melting curves can be done using a graphing software program like Kaleidagraph (6 ).

DNA Melt The DNA samples we used were two complementary synthetic DNA oligomers, dCA7G and dCT7G. Four separate samples with different concentrations (each single strand between 10 and 75 µM) were prepared in degassed buffer (1 M NaCl, 10 mM sodium phosphate pH 7, 0.1 mM disodium EDTA).1 The samples (0.4 mL each) were then placed in 1-mm pathlength quartz cuvettes and sealed with Teflon stoppers. The denaturation of DNA duplex was monitored by measuring the change in absorbance at 260 nm over the temperature range 10 to 70 °C. The melting-experiment data were collected on a spectrophotometer equipped with a multicell block and transport apparatus, which allowed the simultaneous measurement of the four samples. To record the most accurate data in a research laboratory, DNA melts of this type are often done slowly (over several hours) at small temperature increments, to ensure complete temperature equilibration at each point (2). However, this single-lab-period experiment was designed to minimize the amount of time necessary to equilibrate at each temperature by the choice of a short DNA duplex and the use of small sample volumes. After the melting experiment, the samples were cooled to the starting temperature and equilibrated for at least 10 minutes, and the absorbance at 260 nm was recorded for each of the four samples. This value was compared to the absorbance at 260 nm at the start of the experiment to see if there was a rise in absorbance, which could indicate evaporation of the buffer or hydrolysis of the sample. Data Analysis Melting curves of DNA are commonly described using standard helix-to-coil transition theory. In this case the “helix” is duplex DNA and the “coil” is the disordered single DNA strands. The transition from helix to coil is monitored in our experiment as a function of temperature by UV absorbance. This can be done because the percentage of hyperchromicity (increase in absorbance as the duplex is melted) varies approximately linearly with the number of unstacked bases (7). Thus the melting curve relates the absorbance to a profile of fraction of bases paired ( f ) versus temperature. The Tm is the temperature at which f = 0.5. The steep part of the melt curves reflects the doublestrand (A2) to single-strand (A) equilibrium A2 2A

JChemEd.chem.wisc.edu • Vol. 77 No. 11 November 2000 • Journal of Chemical Education

1469

In the Laboratory

3.23

1.00

(1/Tm) / (10−3 K−1)

Normalized Absorbance

3.22 0.96

0.92

0.88

22 µM 40 µM 58 µM 104 µM

0.84

3.21

3.20

3.19

3.18

3.17 0.80 10

20

30

40

50

60

3.16 −11.0

70

Temperature / °C

−10.5

−10.0

−9.5

−9.0

ln Ct

Figure 1. Melting curves showing absorbance versus temperature for four total concentrations (Ct) of dCA7G and dCT7G in 1 M NaCl, 10 mM sodium phosphate pH 7.0, 0.1 mM sodium EDTA. All the curves are normalized to an absorbance of 1 at 65 °C.

Figure 2. van’t Hoff plot of 1/Tm vs ln Ct where Tm is the melting temperature and Ct is the total single-strand concentration. The line shown (y = 2.88 – 0.032x) is the best least-squares fit to the data. R 2 = .999.

The treatment we used assumes a two-state (all-or-none) model. (In a two-state model, f is the fraction of fully basepaired strands, since there are no partially base-paired intermediates in the melting process.) The two-state model has been shown to be a very good approximation for short ( 0) are termed paramagnetic. If a paramagnetic material is placed in a magnetic field it will experience an attraction to the field due to the alignment of the permanent paramagnetic moment of the substance with that of the applied field. As a result paramagnetic materials will appear to weigh more in the presence of an applied magnetic field. By definition, transition metals have incompletely filled d- or f-shells in at least one of their possible oxidation states. As a result, transition metals may be diamagnetic in one oxidation state and paramagnetic in another. In fact biological systems take advantage of the paramagnetic nature of transition metals to catalyze a variety of chemical reactions with would be spinforbidden for a diamagnetic material. For example superoxide dismutase isolated from the

MAGNETIC SUSCEPTIBILITY

Page 2

mitochondria (SOD2) has a mononuclear Mn-active site which follows a classic ‘pingpong’ mechanism. This reaction is named because the enzymatic active site oscillates between two active forms, the oxidized SOD1 which contains Mn(III) and the reduced form with contains Mn(II). (A) (B)

𝐴

𝑆𝑂𝐷𝑀𝑛(𝐼𝐼𝐼) + 𝑂2−• → 𝑆𝑂𝐷𝑀𝑛(𝐼𝐼) + 𝑂2 𝐵

𝑆𝑂𝐷𝑀𝑛(𝐼𝐼) + 𝑂2−• → 𝑆𝑂𝐷𝑀𝑛(𝐼𝐼𝐼) + 𝐻2 𝑂2

By exploiting these opposing forces generated by diamagnetic and paramagnetic substances on a permanent magnet one can measure the mass magnetic susceptibility (χg) of a given substance. From this the molar magnetic susceptibility (χM), and the spin-only or effective magnetic moment (µeff) can be determined. Since µeff is proportional to the number of unpaired electrons (Spin, S) in a given substance this method can be very useful in determining the electronic structure of a new or unknown material. Therefore a variety of techniques have been developed to measure the bulk magnetic susceptibility of a substance. In this experiment we will use the Evans method to determine the molar magnetic and the spin-only magnetic moment of a variety of transition metal complexes. EXPERIMENT

the microscale laboratory Literature Cited 1. Flash, P, Galle, F.; Radil, M. J. Cham.Edw. IWS,66,958. 2. Hel&,G. K; Johnaan,H.W., Jr.Sel&edEqwimntain C7~anicChomislry,3rd ed.; Freeman:New York. 1983;pp 96100. 3. Mayo, D. W : Pike, R. M.; Butcher, S. S. M i w m d e Organic Ldomlory, 2nd ed.: Wfiey: New Y d , 1989:pp 15&155. 4. Roberts, R. M.; Oilberf J. C.; Rodewald, L. 8.;Wi.sOi.so,A. S. M d e m Ewwimontol oganv chamlsfry,4th ed.: Saunders: NewYork, 1985,pp337-145. 6. Sayed, Y ; Ahlmar*, C. A,; Mad", N. H. J Cham. Educ. 1988,66,174. 6. Pavia, D.L.;Lampman, G . M.; &, G. S.;Engel. R.G. InlmducflonLo OrwicLab. omtory 'Ikhniqus;Sauldrrs: Philadelphia, 1990:p 149.

Microscale Techniques For Determination of Magnetic Susceptibility John ~ o o l c o c k 'and Abdullah Zafar Indiana University of Pennsylvania Indiana, PA 15705 There are several traditional techniques for determining the magnetic susceptibility of a transition metal complex: the Gouy method, the Faraday method, and the NMR method (13). The table compares the Farady and NMR methods as well as a modified Gouy device previously described in this Journal (4).

It does not require a separate magnet or power supply. It plugs into a standard 115-Voutlet. It has a digital readout that provides quick and accurate readings with a sensitivity comparable to that of traditional methods. This device has been advertised as measuring the mass susceptibility of solid samples as small as 100 mg and determining the magnetic susceptibility of liquids and solutions. Since the features of the MSB-1 seem to make it ideal for use by students, we have examined the minimum sample size required for accurate measurements of magnetic moments with this balance. We have compared this technique to the NMR method and to the modified Gouy balance (4). Theory and Operation of the MSB-1 Balance The MSB-1 has the same basic equipment configuration as the Gouy method. However, instead of measuring the force exerted by the magnet on the sample, it measures the equal and opposite force exerted by the sample on a pair of suspended permanent magnets (5). The following general expression in cgs units of an3g-' for the mass susceptibility X, can be derived for the MSB-1 balance in much the same way as the relationship used in the Gouy method is obtained.

Comparison of the Maln Techniques Used To Determine Magnetic Susceptibility Modified Gouy Types of Solids Only Samples Used

Faraday

NMR

MSB-1

Solutions Solids. Solids, Liquids, Liquids, and Only and Solutions Solutions 10-300 0.1-50 1 4 0 mg mg mg

175-500 mg Solid Sample Size Solution Not Applicable 0.20 mL Sample Size Cost of $100 $17,000 Instrument (ref 5) (Cahn Instruments) (Source)

0.0250.50 mL

0.070.30 mL

$56,000 $3,000 (Varian (Johnson EM-360) Matthey)

Although the Gouy method is the most common of these techniques, it often requires the largest sample size. The Faraday and NMR method each use delicate or relatively expensive equipment, and the NMR method can be used only with solution samples. More recently a new type of magnetic susceptibility balance, the MSB-1, has been developed by D. F. Evans of Imperial College, London (marketed by Johnson Matthey) (5). This balance has several convenient features and is easily portable. It is compact (11.8 in. long x 8.7 in. wide x 5.3 in. tall) and lightweight (7.3lh). 'Author to whom correspondence should be addressed. 2Presented in part at the 200th ACS National Meeting. Washington, DC, August 1990,and the 11th Biennial Conference on Chemical Education, Atlanta, GA, August 1990. A176

where L is the sample length in an;m is the sample mass in grams; C is the balance calibration constant; R is the digital display reading when the filled sample tube is in place; R, is the digital display reading for the empty sample tube; x', is the volume susceptibility of air (0.029 x lod cgs); and A is the cross sectional area of the sample. The second term,x'B, is usually ignored with solid samples. Although the correction in these cases is small, it must be included to determine mass susceptibilities of liquids (x,) and solutions (x.) For these samples eq 1can be modified to

Journal of Chemical Education

where d, is the density. If the density is not known, i t can be calculated h m the mass and volume of a solution placed in the sample tube using a microliter syringe. The cross-sectional area we use in this equation is calculated from the inside diameter (the sample diameter) provided by the manufacturer. For the standard 3.23-mm-i.d. MSB-1 tube, the cross-sectional area is 0.0819 em2. The solution susceptibility X, in cgs units is then converted to solute mass susceptibility xgusing the Weidmann additivity relationship.

where m lis the mass of the sample in grams, m. is the mass of the solvent in grams, and X, is the mass susceptibility of the solvent. The mass of solute and solvent should be measured as the solution is being prepared. Once the mass susceptibility of the sample has been found by one of the above methods, the molar susceptibility, XM, and the effective magnetic moment in Bohr magne-

tons (BM) are then calculated in the usual manner. All the calculations described above have been automated using a computer spreadsheet template. Thus, the students need only enter their data into labelled cells on the sheet from which the program will automatically calculate x,, &, XM, and ua. ~l;'followin~ representative student data were obtained for solid HelcotSCN~~l usinn the 2.00-mm-i.d. tube. This compound c i a common magnetic susceptibility standard. R, = -70 R=42l length = 2.81 em temperature = 25.3 'C C = 1.30 mass = 0.1139 g

.. .

This data yields a mass susceptibility value of 1.61 x lo5 cgs (0.45% error) and a kffof 4.39 BM. Errors in the calculated values of mass susceptibility using the MSB-I with solid or solution samples range from about 1% to 2%when compared to literaturi values~Allkfi values lie within the expected ranges for a particular transition metal ion. Negative values for R immediately indicate that the sample is diamagnetic. Limits for Solid and Solution Sample Sizes using the MSB-1 Balance

The two sizes of sample tubes commonly used with the MSB-1 have inside diameters of 2.00 and 3.23 mm, respectively. These are advertised as using a minimum of 100 and 250 mg of solid sample, respectively. We used 1.00mm4.d. tubes (donated by JohnsonMatthey) in anattempt to further decrease the minimum sample mass. By placing increasing amounts of Hg[Co(SCN)41 i n each of these tubes, we found that the calculated value of kffincreased to the literature value of 4.4 BM as the mass of the sample, the sample length, and the value of R increased. This is shown in the figure for the 1.00-mm tube. From these results we have determined that the minimum sample length required for each tube is about 0.9-1.0 cm, which corresponds to the point a t which the sample completely fills the volume in the tube between the two disk magnets. Since adding more solid will place the sample outside the field between the magnets, it is not surprisine that R and the effective mametic moment are unaffe&d by increasing the sample l>hpast this point (also seen in the fiewe). This minimum s a m* ~ l elene-th wrresponds to a mikimum sample mass of 10 mg for the 1.00-mm-i.d. tube 40 mg for the 2.00-mm-i.d. tube 100 mg for the 3.23-mm-i.d. tube

As indicated above, the minimum sample size is also the maximum sample size required by this device. In addition to using the 1.00-mm4.d. tube to decrease the amount of solid required, this can also be achieved by measuring the mass susceptibility of a paramagnetic compound in solution. Since both concentration and sample volume determine the minimum mass of solute reauired, we examined the effect of each of these quantities on p.m. Usinr! aaueous solutions of c o ~ p e d l lsulfate ) pentahvdrate in t h i 3123-mm4.d. sample tube with a constant sample volume of 250 pL, accurate values of bewere obtained only for concentrations of 0.2 M (32 mg/mL) or greater. As with

The effect of increasing sample mass on the calculated value of for H~[CO(SCN)~] in the MSB-1 balance using the 1.00-mm-1.d.sample tube. solid samples, the value ofR increased with concentration until 0.2 M was reached. After this, R remained constant. The effect of sample volume on the calculated value of kff was also determined by measuringR while 10-mL aliquots ofthe 0.2 M CuS04solution were added to the 3.23-mm4.d. tube. As the volume was increased, R and kffdecreased until a volume of 70 pL was reached. At this point these quantities become constant with an expected ktf of 1.98 BM. Avolume of 70 pL corresponds to the minimal sample volume observed for solids (0.9 cm x 0.0819 cm2). These results show that the minimum amount of solute needed to obtain accurate values of kffis about 2.3 mg. Thus, to give a comfortable margin, it should be possible to use about 5 mg of any solute to prepare a 100-pL solution for use with the MSB-1. The values of both R and decrease with increasing sample volume because an empty 3.23-mm4.d. tube has an R value that is less negative (about -31) than a tube filled with water (R = -80). Since the diamagnetism of the solvent increases faster with increasing volume than the paramagnetism of the solute, the value ofR should become more and more negative until the diamagnetism of the full sample tube is measured. Thus, when the volume of solvent is less than 70 pL,the R value for the solution is higher than it should be. Under these conditions eq 3 will compute a solute paramagnetism that is too large. We have found that only manganese(I1) sulfate solutions, with five unpaired electrons for each metal ion and with concentrations of 0.2 M or larger, give accurate results with the 2.00-mm-id. tube. Also, no solutions give ac(Continued on page A178)

Volume 69 Number 6 June 1992

A177

t h i microscale laboratory with the 1.00-mm4.d.tube. Thus, the curate values of 3.23-mm-i.d. tube is best for measuring solution susceptibilities. Other Methods for Determining Magnetic Susceptibility

As noted earlier there are other methods that have been used to determine mass susceptibility, including a modified Gouy method previously described in this Journal (4). We were interested in determining the minimum sample size for this device because it has a significant cost advantage over the MSB-1 and othermagnetic susceptibility systems. We constructed our apparatus as described by the authors using disk magnets (Edmund Scientific; catalog no. as the standard to exam530,962). We used H~[CO(SCN)~] ine the effect of increasing sample mass on the value of h~ for copper(I1) sulfate pentahydrate. Our results indicate that the minimum sample size for this device is approximately 175 mg. Although this is much higher than the MSB-1, it is still within the range of many micmscale pmcedures. We also tried to measure the magnetic susceptibility of 1M solutions of CuSOa.We were unsuccessful because the change in weight of the magnets and the yoke must be at least10 mg toobtain an accurate value of ia.Only concentrated (1 M) manganese(I1) sulfate solutions changed the weight enough to give an accurate value f o r k .

A178

Journal of Chemical Education

Another method commonly used to determined the mass susceptibility of solutions a t the microscale level is the NMR method developed by Evans (6). According to Lijlinger and Scheffold, a concentration of 2 x lo3 M and a volume of 25 pL is all that is needed for microscale NMR measurements (7).The lowest concentration that we could measure conveniently was 4.2 mgImL (2.63 x 10" M) with Av = 5.8 Hz when we used an 80.13-MHz Bruker NRBOAF NMR spectrometer and solutions of copper(I1) sulfate in DzO of decreasing concentration. (The indicator species was 1 4 % t-butyl alcohol.) With a sample volume of 25 pL placed in the capillary insert, as little as 0.1 mg of solute is required with this method. No references for the NMR method discuss the use of FTNMR instruments. We have found that locking an FTNMR spectrometer requires the following replacements for the liquids that are typically used in both tubes of the coaxial cell: DzO for water, CDC13 for chloroform, and CDBCOCDB for acetone (7,8). However both t-butyl alcohol and TMS can still be used as the indicator species. The tube should be spun as fast as possible, and the sweep width on the FT-NMR should be expanded fmm about -10 to +10 ppm so that there will be little or no "fold over" (9) of the spinning sidebands that can make determination of Av difficult. There seem to be no literature values for the mass susceptibility of deuterated solvents in common references

(1,2, 10). Using the MSB-1, we have determined the diamagnetic susceptibility of deuterium oxide and deuterated and -4.97 + 0.05 x 10-7cgs chloroform: 4 . 3 7 f 0.13 x respectively, compared to -7.21 x 10" and -4.97 x 10" cgs for water and chloroform. Also, high-field spectrometers with superconducting magnets require a special variation of the oripinal eauation piven bv Evans (lZ).Finallv. it is importantto measure the temp&ature of the ~ ~ l t ' p r o b e at each session. As discussed in this Journal (11,. chanees in solution density with temperature can lead to sign%cant errors in the value of kff. Conclusions The MSB-1 balance can easily perform magnetic susceptibility measurements on both solid and solution samples with awiderange of sample sizes. As little as 10mgof solid sample is needed for measurements of ireff,and as little as 3-5 mg of solid is required for a solution. As an alternative. the NMR method and the use of the modified Gouy device can complement each other well: The marmetic suscevtibilitv of solutions can be measured with thcformer, andsolidscan be measured with the latter. As little as 0.1 me of solute is needed for the \ 3 l R method. while the modked Gouy method requires about 175 mg of solid. Using either of these approaches, students can easliy measure both the solid and solution magnetic susceptibilities of microscale samples and with a spreadsheet tem-

plate quickly calculate hff.If the lab instructor selects transition metal compounds that have different magnetic properties in the solid and solution state, students will learn that these measurements can provide important structural information as well as determine the electron configuration of the transition metal (13). Acknowledgment We would like to thank Jeff Lucht of Johnson Matthey and Dr. Steven Bogdanski of Sherwood Scientific Ltd. for providing additional technical information regarding the MSB-1 device and for suppling the l-mm-i.d. sample tubes. Literature Cited 1. e g i s , B.N.; Leuis, J. In Modrrn C m r d i ~ t i o nChamiahy; Lewia, J.;Willdna, R.G., Eda.; W h y : New York, 1960: p p 4004% 2. Selwood. P W. Mogmtoehmisfry, 2nd ed.: Jntersdence Publishera, Wiky: New Vnrk ... .., ,966 . ....

3. Jolly, W. L. SnythesiaandChomdrtiaafionofhorgonieCompounds;Renti-Hall: Engelwood C M s , NJ. 1970: p p 389-385. 4. Eeton, S. S.; Eaton, G. R.J. Chem. Edue. ISTS, 56, 170-171. 5. Mognotie Suseepfibility Bolanwlnstnrefion Monunl,JohnsonMatthey: Wsyne, PA, 3-0 A""".

6. Evans,V. F J. Chem Soe. 1968.2W3-ZW5. 7. Ldliger, J.:Seheffold,R. J Chem. Edue. 19?P,49, 646647. 8. Crawford, T. H.; Swanson, J. J Chem. Edue. 1811,48,332386. 9. Sehsffer, C; Yoder, C. Infrodvctlan l o Multinuclear NMR; BenjaminiCummings: Menlo Park, CA, 1987: pp 6 1 4 2 . 10. CRC H o n d h d of Ckmislry and Phyaicp, 49th ad.; We&, R. C., Ed.; CRC: Boea Raton. FL. 1968: p E-115E-132. 11. Ostfeld, V.; Cohen,I. A. J Chem Educ. 1872,49,829. 12. Schubert. E. M. J C k m . Educ 1892,69,62. 13. Szafran, 2.: Pike, R. L.; Smgh, M . M . Mieraseole Inorgonie Chemisrry: A Comp m k n s u e h b o m f o r y E I P I ~ ~ NWfiey: P : New Yo*, 1991; p p 231-235: 257-260.

Volume 69 Number 6 June 1992

A179

In the Classroom

Diamagnetic Corrections and Pascal’s Constants Gordon A. Bain† and John F. Berry* Department of Chemistry, University of Wisconsin–Madison, Madison, WI 53706; *[email protected]

Laboratory experiments involving measurement of magnetic susceptibilities (χ, the ability of a substance to be attracted to or repelled by an external magnetic field) have had longstanding success in the undergraduate curriculum. Many experiments suitable for laboratory courses in physical chemistry or inorganic chemistry focus on the determination of the number of unpaired electrons in various transition-metal salts (1–6). These experiments present students with a special set of challenges, one of the most confusing and frustrating of which is the use of tabulated diamagnetic susceptibilities or empirical Pascal’s constants that are used to correct for the fundamental or underlying diamagnetism of a paramagnetic compound. Many sources (1, 2, 7–15) contain selected (i.e., incomplete) tabulated data, and often conflicting values are given in different sources owing to the different interpretations of diamagnetic susceptibilities that arose in the early 20th century (8). In this article we present an explanation for the origin of the diamagnetic correction factors, organized tables of constants compiled from other sources (1, 2, 7–17), a link to a new interactive online resource for these tables, a simple method for estimating the correct order of magnitude for the diamagnetic correction for any given compound, a clear explanation of how to use the tabulated constants to calculate the diamagnetic susceptibility, and a worked example for the magnetic susceptibility of copper acetate. Unlike paramagnetism (attraction of a substance to a magnetic field, a property of compounds having nonzero spin or orbital angular momentum), diamagnetism (repulsion from a magnetic field) is a property of all atoms in molecules. Whereas paramagnetism arises from the presence of unpaired electrons in a molecule, all electrons, whether paired or unpaired, cause diamagnetism. It is the conflict between paramagnetism and diamagnetism that defines the overall (measured) magnetic susceptibility, χmeas , which is positive for paramagnetic substances and negative for diamagnetic substances. Paramagnetic contributions to the measured susceptibility, or paramagnetic susceptibility χP, are positive and temperature-dependent (for a Curie paramagnet, χP is proportional to 1/T where T is temperature). Diamagnetic susceptibilities, χD, are temperature independent and are negative. The total measured magnetic susceptibility, χmeas, is defined as the sum of these contributions:

D meas  D P DD

(1)

Thus a compound having unpaired electrons but with an abundance of other paired electrons, such as a metalloprotein (18), may display diamagnetism at room temperature in a bulk measurement. For room temperature magnetic susceptibility measurements carried out in an undergraduate laboratory, the goal is determination of χP using DP  D meas  DD (2) The paramagnetic susceptibility can be related to the number of unpaired electrons in the molecule by eq 3a, a specialized form †Current address: Thermo Fisher Scientific, 5225 Verona Rd., Madison, WI 53711.

532

of the more general eq 3 that assumes g = 2, N g 2 C2 DP T  A ©« S S 1 ¸º 3k

(3)

B

1 n n 2

(3a) 8 where T is absolute temperature, NA is Avogadro’s constant, g is the Landé factor or electronic magnetogyric constant, β is the Bohr magneton, kB is the Boltzmann constant, S is the overall spin state of the molecular substance, and n is the number of unpaired electrons. Values of χD are obtained from literature sources that may list data for whole molecules, fragments of molecules, or individual atoms, ions, or bonds. It is important to pay close attention to the sign of tabulated data, for example, ref 16 lists diamagnetic susceptibilities for organic molecules as ‒ χ; that is, the table contains positive values but they must be treated as negative values for use in eq 2. The same volume, however, lists magnetic susceptibilities for elements and inorganic compounds as χ, so these values should be used as given. For convenience, we compile here tables of diamagnetic susceptibilities from various sources in Tables 1–6 (1, 2, 7–17); where data from two or more sources conflicted, we have generally chosen the more precise value. DP T 

Contents of the Tables In his original publications (19–21), Pascal proposed that the diamagnetism of a molecule could be determined in an additive fashion using values for the diamagnetic susceptibility of every atom (χDi) and bond (λi) in the molecule:

DD 

¥ DD i i



¥ Mi i

(4)

The values of χDi and λi became known as “Pascal’s constants”. One source of confusion about Pascal’s constants stems from the fact that the values of these constants were often revised during Pascal’s lifetime (22, 23), leading to the propagation of several conflicting values cited in different texts. The main reason for revising the original constants was to remove the λ “constitutive corrections” by introducing specialized χDi values (e.g., χDi for an O atom in a carbonyl group is often cited as +1.7 × 10‒6 emu mol‒1). Since many of these revised values can be derived directly from Pascal’s original set of constants (e.g., χD(O) + λ(C=O) = +1.7 × 10‒6 emu mol‒1), the values cited in Tables 1 and 2 are kept as close as possible to Pascal’s original premise (eq 4). This formulation of Pascal’s constants also allows for their greater versatility, at the expense of some insignificant accuracy. Tables 3–6 contain pre-determined χDi values for important groups of atoms or ions. Specifically, Table 6 contains χDi values for cations assuming that they are present in purely ionic compounds (note that the atomic χDi values in Table 1 assume that the atoms are present in purely covalent species). Consequently, various values in Table 1 and Table 6 are seldom used (e.g., the values for alkali metals in Table 1 and the value for the

Journal of Chemical Education  •  Vol. 85  No. 4  April 2008  •  www.JCE.DivCHED.org  •  © Division of Chemical Education 

In the Classroom

influence the determination of the number of unpaired electrons in a molecule. When determining χD from the values in Tables 1–6, we therefore round our derived values at the decimal. P N

Using the Tables Tables 3–6 are the easiest to use; these values may be included in eq 4 as given. Values of χDi for species not included in Tables 3–6 may be determined from data in Tables 1 and 2 by adding up the values for all constituent atoms as given in Table 1 and for all bonds given in Table 2. For diamagnetic molecules or ligands in routine use, the reader may extend Table 4 or Table 5 by either calculating values as described above or simply obtaining χmeas for the pure substance experimentally. These tables are available at JCE ChemInfo: Inorganic, where readers can submit and archive χDi values for substances of their own interest (24). As a measure of the efficacy of eq 4, we have measured χD at room temperature for two common ligands listed in Table 4, 2,2´-dipyridyl (bipy) and triphenylphosphine (PPh3), shown in Figure 1, using an Alfa Aesar magnetic susceptibility balance Mark 1. The measured values were χD (bipy) = ‒91 × 10‒6 emu mol‒1 and χD (PPh3) = ‒160 × 10‒6 emu mol‒1 and the tabulated values are χD (bipy) = ‒105 × 10‒6 emu mol‒1 and χD (PPh3) = ‒167 × 10‒6 emu mol‒1, which are in reasonable agreement with experiment (Note: the unit emu is not an SI unit but is the most widely used unit for magnetic susceptibility; 1 emu = 1 cm3).

N

bipy

PPh3

Figure 1. Molecular structures of bipy and PPh3.

N5+ ion in Table 6), but are included for the sake of completeness. Table 3 contains the corresponding χDi values for anions in completely ionic environments. Tables 4 and 5 contain χDi values for whole molecules that may be present as ligands or solvents of crystallization. Note that no λi values are presented for bonds to metals in coordination compounds. Coordination complexes are assumed to behave as ionic species such that the value of χD will be determined by the sum of the ionic contributions of the metal ion(s) from Table 6 and the corresponding values for the ligands from Tables 3 or 4. Precision of the data in Tables 1–6 is an issue worth discussing since the number of significant figures in the data varies drastically from one to three or four. Many of the data are thus imprecise, and, as will be discussed later, the improved precision of some values is not entirely necessary since even a 10% change in χD does not have a significant influence in the derived χP values for paramagnetic species (~1%) and will not

Table 1. Values of χDi for Atoms in Covalent Species Atom

Ag Al As(III) As(V) B Bi Br C

χDi /(1 x emu mol–1) –31.0 –13.0 –20.9 –43.0 –7.0 –192.0 – 30.6 – 6.00

10–6

Atom

C (ring) Ca Cl F H Hg(II) I K

χDi /(1 x 10–6 emu mol–1) –6.24 –15.9 – 20.1 –6.3 –2.93 –33.0 –44.6 –18.5



Atom



Li Mg N (ring) N (open chain) Na O P Pb(II)

χDi /(1 x 10–6 emu mol–1) –4.2 –10.0 –4.61 –5.57 –9.2 –4.6 –26.3 –46.0

Atom

S Sb(III) Se Si Sn(IV) Te Tl(I) Zn

χDi /(1 x 10–6 emu mol–1) –15.0 –74.0 –23.0 –13 –30 –37.3 –40.0 –13.5

Table 2. Values of λi for Specific Bond Types λ i /(1 x emu mol–1) C=C +5.5 +0.8 C≡C C=C–C=C +10.6 +3.85 Ar–C≡C–Arb CH2=CH–CH2–(allyl) +4.5 C=O +6.3 COOH –5.0 COOR –5.0 C(=O)NH2 –3.5 N=N +1.85 C=N– +8.15 +0.8 –C≡N +0.0 –N≡C N=O +1.7 –NO2 –2.0 C–Cl +3.1

λ i /(1 x 10–6 λ i /(1 x 10–6 λ i /(1 x 10–6 Bond Bond emu mol–1) emu mol–1) emu mol–1) Cl–CR2CR2–Cl +4.3 Ar–Br –3.5 Imidazole +8.0 R2CCl2 +1.44 Ar–Cl –2.5 Isoxazole +1.0 RCHCl2 +6.43 Ar–I –3.5 Morpholine +5.5 C–Br +4.1 Ar–COOH –1.5 Piperazine +7.0 Br–CR2CR2–Br +6.24 Ar–C(=O)NH2 –1.5 Piperidine +3.0 C–I +4.1 R2C=N–N=CR2 +10.2 Pyrazine +9.0 Ar–OH –1 +0.8 Pyridine +0.5 RC≡C–C(=O)R Ar–NR2 +1 Benzene –1.4c Pyrimidine +6.5 Ar–C(=O)R –1.5 Cyclobutane +7.2 α- or γ-Pyrone –1.4 Ar–COOR –1.5 Cyclohexadiene +10.56 Pyrrole –3.5 Ar–C=C –1.00 Cyclohexane +3.0 Pyrrolidine +0.0 –1.5 Cyclohexene +6.9 Tetrahydrofuran +0.0 Ar–C≡C Ar–OR –1 Cyclopentane +0.0 Thiazole –3.0 Ar–CHO –1.5 Cyclopropane +7.2 Thiophene –7.0 Ar–Ar –0.5 Dioxane +5.5 Triazine –1.4 Ar–NO2 –0.5 Furan –2.5 aOrdinary C–H and C–C single bonds are assumed to have a λ value of 0.0 emu mol–1. bThe symbol Ar represents an aryl ring. cSome sources list the λ value for a benzene ring as –18.00 to which three times λ(C=C) must then be added. To minimize the calculations involved, this convention was not followed such that λ values given for aromatic rings are assumed to automatically take into account the corresponding double bonds in the ring. Bonda

10–6

Bond

© Division of Chemical Education  •  www.JCE.DivCHED.org  •  Vol. 85  No. 4  April 2008  •  Journal of Chemical Education

533

In the Classroom Table 3. Values of χDi for Anions χDi /(1 x 10–6 emu mol–1)



Anion



AsO33– AsO43– BF4– BO33– Br– BrO3– Cl– ClO3– ClO4– CN– aThe

–51 –60 –37 –35 –34.6 –40 –23.4 –30.2 –32.0 –13.0



χDi /(1 x 10–6 emu mol–1)

Anion C5H5– C6H5COO– CO32– C2O42– F– HCOO– I– IO3– IO4– NO2– NO3–

–65 –71 –28.0 –34 –9.1 –17 –50.6 –51 –51.9 –10.0 –18.9



χDi/(1 x 10–6 emu mol–1)

Anion NCO– NCS– O2– OAc– OH– PO33– PtCl62– S2– SO32– SO42–

–23 –31.0 –12.0a –31.5 –12.0 –42 –148 –30 –38 –40.1

Anion S2O32– S2O82– HSO4– Se2– SeO32– SeO42– SiO32– Te2– TeO32– TeO42–



χDi /(1 x 10–6 emu mol–1) –46 –78 –35.0 –48b –44 –51 –36 –70 –63 –55

value of χDi for O2– is reported as –6.0 in some sources. bThis value is uncertain.

Table 4. Values of χDi for Common Ligands

Ligand



Acac– Bipy CO C5H5– En

χDi /(1 x emu mol–1) –52 –105 –10 –65 –46.5

10–6



Ligand



Ethylene Glycinate H2O Hyrdazine Malonate

χDi /(1 x 10–6 emu mol–1) –15 –37 –13 –20 –45

Ligand

NH3 Phen o-PBMA Phthalocyanine PPh3

χDi /(1 x 10–6 emu mol–1) –18 –128 –194 –442 –167



Ligand



Pyrazine Pyridine Salen2– Urea

χDi /(1 x 10–6 emu mol–1) –50 –49 –182 –34

Note: Abbreviations: acac = acetylacetonate, bipy = 2,2’-dipyridyl, en = ethylenediamine, phen = phenanthroline, PBMA = phenylenebisdimethylarsine, salen = ethylenebis(salicylaminate)

Table 5. Values of χDi for Common Solvents of Crystallization Solvent

CCl4 CHCl3 CH2Cl2 CH3Cl CH3NO2 CH3OH CCl3COOH

CF3COOH

χDi /(1 x 10–6 emu mol–1) –66.8 –58.9 –46.6 –32.0 –21.0 –21.4 –73.0

Solvent

CH3CN 1,2-C2H4Cl2 CH3COOH CH3CH2OH HOCH2CH2OH CH3CH2SH CH3C(=O)CH3

χDi /(1 x 10–6 emu mol–1) –27.8 –59.6 –31.8 –33.7 –38.9 –44.9 –33.8

Solvent

–43.3





2 DD Nring 8 DD H

2 M pyridine M Ar–Ar

 10 DD Cring

 ©«10 6. 24 2  4. 61 8 2. 93

2 0. 5 0. 5 ¸º t 10 6 emu mo l 1   95 t 10 6 em mu mol 1





DD PPh 3  DD P 18 DD Cring







15 DD H 3 M benzene



 ©« 26 . 3 18 6. 24 15 2. 93

3 1. 4 ¸º t 10 6 emu mol 1  187 t 10 6 emu mol 1 534

CH3C(=O)OC(=O)CH3 CH3CH2CH2CN CH3C(=O)OCH2CH3 CH3CH2CH2CH2OH CH3CH2OCH2CH3 Pentane o-Dichlorobenzene

Benzene

Values of χD for bipy and PPh3 may also be determined from Pascal’s constants (Tables 1 and 2) as follows: DD bipy



χDi /(1 x 10–6 emu mol–1) –52.8 –50.4 –54.1 –56.4 –55.5 –61.5 –84.4

χDi /(1 x 10–6 emu mol–1) Cyclohexane –68 Hexane –74.1 Triethylamine –83.3 Benzonitrile –65.2 Toluene –65.6 Isooctane –99.1 Naphthalene –91.6

Solvent

–54.8

As a check, the computed value of χD should be close to the value estimated by the following equation

DD { 

MW 6 10 emu mo l 1 2

(5)

where MW is the (unitless) molecular weight of the substance (9). In fact, for situations where great accuracy is not needed (such as for a room temperature measurement in an undergraduate lab), use of eq 5 can be sufficient for determining χD. For comparison, the χD values for bipy and PPh3 estimated from eq 5 are ‒78 × 10‒6 emu mol‒1 and ‒131 × 10‒6 emu mol‒1. For magnetic susceptibility measurements of metalloenzymes, in which χmeas is dominated by χD, an accurate method for determining χD is necessary. This is typically achieved by measuring the susceptibility of the protein without the metal included or on a homolog of the protein having a diamagnetic metal center in place of the paramagnetic metal (18). This technique emphasizes the fact that the most accurate χD values can only be obtained by measurement.

Journal of Chemical Education  •  Vol. 85  No. 4  April 2008  •  www.JCE.DivCHED.org  •  © Division of Chemical Education 

In the Classroom Table 6. Values of χDi for Cations Cation

χDi /(1 x 10–6 emu mol–1)

Cation

χDi /(1 x 10–6 emu mol–1)

Cation

χDi /(1 x 10–6 emu mol–1)



Ag+

–28



Ir4+

–29



Rh4+

–18



Ag2+

–24a



Ir5+

–20



Ru3+

–23



Al3+

–2



K+

–14.9



Ru4+

–18



As3+

–9a



La3+

–20



S4+



As5+

–6



Li+



S6+

–1



Au+

–40a



Lu3+



Sb3+

–17a



Au3+

–32



Mg2+



Sb5+

–14



B3+

–0.2



Mn2+

–14



Sc3+

–6



Ba2+

–26.5



Mn3+

–10



Se4+

–8



Be2+

–0.4



Mn4+

–8



Se6+

–5



Bi3+

–25a



Mn6+

–4



Si4+



Bi5+

–23



Mn7+

–3



Sm2+

–23



Br5+

–6



Mo2+

–31



Sm3+

–20



C4+

–0.1



Mo3+

–23



Sn2+

–20



Ca2+

–10.4



Mo4+

–17



Sn4+

–16



Cd2+

–24



Mo5+

–12



Sr2+

–19.0



Ce3+

–20



Mo6+

–7



Ta5+

–14



Ce4+

–17



N5+

–0.1



Tb3+

–19



Cl5+

–2



NH4+

–13.3



Tb4+

–17



Co2+

–12



N(CH3)4+

–52



Te4+

–14



Co3+

–10



N(C2H5)4+

–101



Te6+

–12



Cr2+

–15



Na+



Th4+

–23



Cr3+

–11



Nb5+

–9



Ti3+

–9



Cr4+

–8



Nd3+

–20



Ti4+

–5



Cr5+

–5



Ni2+

–12



Tl+

–35.7



Cr6+

–3



Os2+

–44



Tl3+

–31



Cs+

–35.0



Os3+

–36



Tm3+

–18



Cu+

–12



Os4+

–29



U3+

–46



Cu2+

–11



Os6+

–18



U4+

–35



Dy3+

–19



Os8+

–11



U5+

–26



Er3+

–18



P3+

–4



U6+

–19



Eu2+

–22



P5+

–1



V2+

–15



Eu3+

–20



Pb2+

–32.0



V3+

–10



Fe2+

–13



Pb4+

–26



V4+

–7



Fe3+

–10



Pd2+

–25



V5+



Ga3+

–8



Pd4+

–18



VO2+

–12.5



Ge4+

–7



Pm3+

–27



W2+

–41



Gd3+

–20



Pr3+

–20



W3+

–36



H+

0



Pr4+

–18



W4+

–23



Hf4+

–16



Pt2+

–40



W5+

–19



Hg2+

–40.0



Pt3+

–33



W6+

–13



Ho3+

–19



Pt4+

–28



Y3+

–12



I5+

–12



Rb+

–22.5



Yb2+

–20



I7+

–10



Re3+

–36



Yb3+

–18



In3+

–19



Re4+

–28



Zn2+

–15.0



Ir+

–50



Re6+

–16



Zr4+

–10



Ir2+

–42



Re7+

–12



Ir3+

–35



Rh3+

– 22

aThis

–1.0 –17 –5.0

–6.8

–3

–1

–4

value is uncertain.

© Division of Chemical Education  •  www.JCE.DivCHED.org  •  Vol. 85  No. 4  April 2008  •  Journal of Chemical Education

535

In the Classroom

A Practical Example: Copper(II) Acetate Hydrate A possible experiment for an undergraduate inorganic chemistry lab or physical chemistry lab is to determine the magnetic susceptibility of dimeric copper(II) acetate hydrate, Cu2(OAc)4(H2O)2, and relate this value to the number of unpaired electrons per copper atom. Here, we will use Tables 1–6 to determine the diamagnetic correction factor (χD) for Cu2(OAc)4(H2O)2. First, we estimate the value of χD from eq 5 [the molecular weight of Cu2(OAc)4(H2O)2 is 399.3]. DD ©«Cu 2 OAc 4 H2 O 2 ¸º 399 . 3   t 10 6 emu mol 1 2  200 t 10 6 em u mol 1 A more accurate value of χD can be calculated from the following values using Table 3, 4, and 6: DD ©«Cu 2 OAc 4 H2 O 2 ¸º



 2 DD Cu

2

4 D OAc 2 D H O



D

D



2

 ©«2 11 4 31 . 5

2 13 ¸º t 10 6 emu mol 1  174 t 10 6 emu mol 1

Using an Alfa Aesar magnetic susceptibility balance Mark 1, we determined χmeas for a sample of Cu2(OAc)4(H2O)2 to be +1.30 × 10‒3 emu mol‒1 at a temperature of 296.5 K. We now determine χP using the χD value we determined from Tables 3, 4, and 6:

DP  D meas  DD  ©«1300  174 ¸º t 10 6 emu mol 1  1470 t 10 6 emu mol 1

To relate this value to the number of unpaired electrons in the molecule, we may determine the value of χPT or, alternatively, μeff (effective magnetic moment), which is an older convention but still widely used:





DP T  1470 t 10 6 emu mol 1 296 . 5 K  0.4436 emu K mol

N eff  

3 kB

NA C 2



1

DP T



8 1470 t 10 6 emu mol 1 296 . 5 K

 1. 87 N B

Note that the value of μeff is given in units of μB or Bohr magnetons. Students should see that Cu2(OAc)4(H2O)2 has somewhere between one and two unpaired electrons, since the

536

unpaired electrons on each copper atom are antiferromagnetically coupled. If the approximated value of χD, ‒200 × 10‒6 emu mol‒1, is used in the above calculations, values of χPT = 0.445 emu K mol‒1 and μeff = 1.89 μB are obtained, which differ from the more accurate values by only 1–2% and therefore do not affect the conclusions that may be drawn by the students. Literature Cited 1. Girolami, G. S.; Rauchfuss, T. B.; Angelici, R. J. Synthesis and Technique in Inorganic Chemistry, 3rd ed.; University Science Books: Sausalito, CA, 1999. 2. Jolly, W. L. The Synthesis and Characterization of Inorganic Compounds, Waveland Press, Inc.: Prospect Heights, IL, 1970. 3. Bain, G. A. Chemistry 311 Laboratory Manual; Department of Chemistry, University of Wisconsin: Madison, WI, 2007. 4. Blyth, K. M.; Mullings, L. R.; Phillips, D. N.; Pritchard, D.; van Bronswijk, W. J. Chem. Educ. 2005, 82, 1667. 5. Malerich, C.; Ruff, P. K.; Bird, A. J. Chem. Educ. 2004, 81, 1155. 6. Evans, W. J. Chem. Educ. 2004, 81, 1191. 7. Adams, D. M.; Raynor, J. B. Advanced Practical Inorganic Chemistry; John Wiley & Sons, Ltd.: London, 1965. 8. Selwood, P. W. Magnetochemistry, 2nd ed.; Interscience Pub.: New York, 1956. 9. Kahn, O. Molecular Magnetism; Wiley-VCH: New York, 1993. 10. Carlin, R. L. Magnetochemistry; Springer-Verlag: Berlin, 1986. 11. Earnshaw, A. Introduction to Magnetochemistry; Academic Press: London, 1968. 12. Mulay, L. N. Magnetic Susceptibility; Interscience Pub.: New York, 1963. 13. Magnetic Susceptibility Balance Instruction Manual; Johnson Matthey Fabricated Equipment: Wayne, PA. 14. Drago, R. S. Physical Methods in Chemistry; W. B. Saunders Company: Philadelphia, 1992. 15. Figgis, B. N.; Lewis, J. The Magnetochemistry of Complex Compounds. In Modern Coordination Chemistry Principles and Methods; Lewis, J., Wilkins, R. G., Eds.; Interscience Pub. Inc.: New York, 1960. 16. Handbook of Chemistry and Physics, [Online]; 86th ed., Lide, D. R., Ed.; Taylor and Francis: Boca Raton, FL, 2007. 17. Gupta, R. R. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Madelung, O., Ed.; Springer-Verlag: Berlin, 1986; Vol 16. 18. Moss, T. H. Meth. Enzymol. 1978, 54, 379. 19. Pascal, P. Ann. Chim. Phys. 1910, 19, 5. 20. Pascal, P. Ann. Chim. Phys. 1912, 25, 289. 21. Pascal, P. Ann. Chim. Phys. 1913, 28, 218. 22. Pacault, A. Rev. Sci. 1946, 84, 1596. 23. Pascal, P.; Pacault, A.; Hoarau, J. Compt. Rend. 1951, 233, 1078. 24. JCE ChemInfo: Inorganic. http://www.jce.divched.org/JCEDLib/ ChemInfo/Inorganic/ (accessed Feb 2008); see Holmes, J. L. J. Chem. Educ. 2008, 85, 590 for a description of this column.

Supporting JCE Online Material

http://www.jce.divched.org/Journal/Issues/2008/Apr/abs532.html Abstract and keywords Full text (PDF) with links to cited URLs and JCE articles JCE ChemInfo: Inorganic Collection of the JCE Digital Library http://www.jce.divched.org/JCEDLib/ChemInfo/Inorganic/

Journal of Chemical Education  •  Vol. 85  No. 4  April 2008  •  www.JCE.DivCHED.org  •  © Division of Chemical Education 

ENZYME KINETICS: OXIDATION OF L-LACTIC ACID LAB PREPARATION Read SGN Experiment #22 and this handout. 1. Derive integrated rate laws for the oxidation of l-lactate that are first order and zeroth order in lactate. 2. Assume you are monitoring the concentration of lactate as a function of time during which it undergoes an oxidation to form pyruvate. What quantities would you plot to determine whether the reaction is zeroth order or first order in lactate? EXPERIMENT

1. Theory

The reaction studied in this experiment, the enzyme-catalyzed oxidation of lactate, illustrates the simplest mechanism for a catalytic reaction. The substrate lactate S reacts reversibly with the enzyme E to form an unstable complex ES which can revert back to E and S or can decompose to the product P and release the enzyme for use in another cycle. E

+

S

k1 k-1

ES

k2

E

+

P

The forward and reverse reactions in the first step are very fast and the rate-controlling step forming product is very much slower. The concentration of complex is always very small, with the consequence that a minute amount of enzyme can catalyze conversion of an indefinitely large amount of substrate to product. Though there is in general no direct relation between the kinetic order of a chemical reaction and the overall stoichiometry, in some simple cases, as in this one, the stoichiometry does correspond to the reaction mechanism and the kinetic rate equations can be written by inspection: i.e. d[S] dt = -k1[E][S] + k-1 [ES]

(1)

d[ES] = -k-1[ES] + k1[E][S] - k2[P] dt

(2)

dP dt = k2[ES]

(3)

Here [...] denotes the concentration of a reagent. The system of kinetic equations can be solved easily if an approximation is made. SGN (p. 265) employ the steady-state approximation: i.e. that the small concentration of complex ES is effectively constant so that d[ES]/dt is zero. Combining this condition and the conservation of enzyme, [E] + [ES] = [E]o = constant

(4)

with eq. 3, they obtain, after some algebra, the Michaelis-Menten equation for the rate of oxidation of lactate: d[P] k 2 [E]o [S] v = dt = K m + [S]

(5)

where the Michaelis constant Km = (k-1 + k2)/k1. The rate of loss of S (or the rate of accumulation of P) is measured, and the initial concentration of enzyme [E]o is known. Then Km is obtained by fitting the data to eq. 5. An alternative simplifying approximation that leads to the same result, and seems a bit more direct, is the assumption that the reaction step that produces the product does not appreciably perturb the equilibrium among E, S, and ES. Then we write [E] [S] Km = [ES]

(6)

so that [ES] = [E] [S] Km. Eliminating [E] between eqs. 4 and 6, solving for [ES], and substituting in eq. 3, gives the desired result, eq. 5. Looking at eq. 5 we see that v =

k 2 [E]o [S] Km

(7)

at the limit with [S] > Km. Thus at the first limit, v is first-order with respect to [S]; and at the second it is zero-order with respect to [S] (i.e. independent of [S]).

(8)

This behavior corresponds to physical intuition. If most of the enzyme molecules are unoccupied, the rate of formation of P is increased proportionally by adding more substrate; and when all enzyme molecules are combined with substrate, adding more substrate does nothing. This behavior is shown schematically in the plot of v versus [S] on p. 265 of SGN. The usual approach in studying enzyme kinetics is to measure the initial rate vo of the reaction, so that the concentrations [E] and [S] do not change substantially from the initial values [E]o and [S]o: i.e. vo =

∆[P] k [E] [S] = 2 o ∆t K m + [S]o

(9)

Traditionally, eq. 9 is rearranged in the Lineweaver-Burke (L-B) form Km 1 1 1 = + k 2 [E]o k2[E]o [S]o vo

(10)

showing that the intercept of a linear plot of 1/vo versus 1/[S]o is 1/k2[E]o and the slope is Km/k2[E]o. Hence from these two quantities Km is determined: Km = slope/intercept. A more recent innovation is the Eadie-Hofstee (E-H) plot of vo/[S]o versus vo, which corresponds to another rearrangement of eq. 9:

vo vo k [ E] = 2 o [S]o Km Km

(11)

Here again slope and intercept of a straight line determine the Michaelis constant Km.. With perfect data conforming exactly to eq. 9, the information from the L-B and E-H plots would be exactly equivalent. However, the E-H method is said to have some practical advantages in analyzing real (imperfect) data. The rate constant k2, which can also be determined from the initial-rate data is called the turnover number. It is the number of substrate molecules reacted per second per molecule of enzyme when the enzyme is saturated with substrate: i.e. [S]o >> Km so that eq. 8 holds and vo has its maximum value. Inhibition of Enzymes: In enzymatic reactions, the activity of the enzyme can be decreased through noncovalent binding of inhibitors. Studies of this type can often help to elucidate the mechanism of the overall enzymatic reaction.

The most common types of reversible inhibition are competitive, non-competitive, uncompetitive, and mixed. Only the first two cases will be of significance in this experiment. Competitive inhibition occurs when another molecule which resembles the substrate can compete for the enzyme’s active site with the substrate. This has the effect of preventing the substrate’s access to the catalytic site, thus raising the observed Km (K’m). Since the inhibitor is not altering the enzymes active site, the rate of the reaction once substrate is bound does not change. Therefore the Vmax is unaltered in the case of competitive inhibition. Mechanistically, the incorporation of a competitive inhibitor into the Michaelis-Menten formulation is written:

E

+

+

S

k1 k-1

ES

k2

E

+

P

I k3

k-3 EI

From this, the new apparent Michaelis-Menten constant K’m is given by:

 [I]   K' m = K m 1 +  Ki 

(12)

where [I] is the inhibitor concentration and Ki is the dissociation constant for the enzymeinhibitor complex. Following a similar logic as for Km, Ki is defined as:

Ki =

k − 3 [E][I] = k3 [EI]

(13)

In the case of competitive inhibition, the Lineweaver-Burke and Eadie-Hofstee equations take the form: L-B:

K  1  1 [I]  1  + = m  1 + v o Vmax  [S]  K i  Vmax

(14)

E-H:

 v  [I]   + Vmax v o = −K m  o 1 +  [S]  K i 

(15)

Noncompetitive Inhibition occurs when the inhibitor does not compete with the substrate for the active site. Both inhibitor and substrate can bind to the enzyme simultaneously to form a temporary complex. This can be accomplished by (1) a permanent (irreversible) modification of the enzyme; (2) reversible binding of the inhibitor to the enzyme but not within the active site; (3) reversible binding of the inhibitor to the enzyme substrate complex. Typically, this form of inhibition does not affect the enzyme’s affinity for the substrate (Km), but instead lowers the Vmax. Because a portion of the enzyme molecules are effectively inactivated, the maximum velocity of the reaction is decreased, but the binding constant of the functional enzymes remains unchanged. Mechanistically, the incorporation of a noncompetitive inhibitor into the Michaelis-Menten formulation is written: E

+

S

+

k1

ES

k-1

+ S

I k3

k-3 EI

k'-3 +

S

k4 k-4

EIS

k'3

k2

E

+

P

In the case of noncompetitive inhibition, the Lineweaver-Burke and EadieHofstee equations take the form: L-B:

1  Km  1  1  [I]  1 +  =   + v o  Vmax  [S]  Vmax  K i 

(16)

E-H:

 v  [I]   + Vmax v o = −K m  o 1 +  [S]  K i 

(17)

2. Laboratory Procedure: Lactate dehydrogenase is a hydrogen transfer enzyme which catalyzes the 2 electron oxidation of l-lactic acid to pyruvate. β-Nicotinamide adenine dinucleotide (NAD+) functions as the electron acceptor and serves as a cofactor for the enzyme. The reduction of NAD+ is easily observed at 340 nm using a uv-vis spectrophotometer. (You’ll need to look up the extinction coefficient for NADH at this wavelength prior to making your calculations). CH3 H

NAD

OH

+

Lactate Dehydrogenase

H3C

+

pH 8.8 to 9.8

O

O

O

O

NADH

O

Michaelis-Menten and Enzyme Inhibition Kinetics: Necessary Reagents: • • • • • •

1.0 M lactic acid solution, 50 mM CHES, pH 9.2 5 µM lactate dehydrogenase (LDH), 50 mM CHES, pH 9.2 20 mM β-NAD, 50 mM CHES, pH 9.2 50 mM CHES buffer, pH 9.2 250 mM Borate solution, 50 mM CHES, pH 9.2 20 mM EDTA solution, 50 mM CHES, pH 9.2 (30-mL maximum)

+

H

Protocol: From the 1000 mM stock lactic acid solution, prepare a series of 6 samples with a lactate concentration (in the cuvet) of 1, 5, 10, 50, 100, and 250 mM. Each sample should have 1.5-mL of NAD solution, 10-µL of enzyme, and a ratio of stock lactate to buffer solutions to give the appropriate substrate concentration. Also, all the samples should have a final volume of 3.0-mL. Use the pH 9.2 50 mM CHES buffer provided to dilute the lactate stock solution. Verify that the samples are actually at a pH of 9.2 prior to starting. Using both the stock 250 mM borate solution, and 1000 mM lactic acid solution, prepare another set of 6 samples of lactic acid at their previous concentrations. Into each sample, spike in an appropriate volume of borate solution such that the final borate concentration is 20 mM in each vial. Again, use the buffer solution to dilute the solutions to their final volumes as done in the previous step. Using both the stock 20 mM EDTA solution, and 1000 mM lactic acid solution, prepare another set of 6 samples of lactic at their previous concentrations. Into each solution spike in the appropriate volume of EDTA solution such that the final EDTA concentration is 2 mM in each vial. Again, use the buffer solution to dilute the samples to their final volume of 3.0-mL. The spectrophotometer should be blanked vs. 1.5-mL of 20 mM NAD + 1.5-mL of the buffer. Once the instrument is blanked, each sample run should start with 1.5-mL of the 20 mM NAD solution and 1.49-mL of the lactic acid solution prepared. The addition of 10 µL of the enzyme solution starts the reaction. Make sure to measure the absorbance at 340 nm prior to the addition of the enzyme for a t=0 reading. This value should be subtracted from all subsequent readings within that run. Measure the absorbance of the sample after the addition of the enzyme solution at t = 0, 15, 30, 45, 60, 120, 180, and 240 seconds. Repeat the above procedure for all 6 concentrations of lactic acid and in the presence of each inhibitor. Analyze your kinetic data on the enzymatic oxidation of lactate by the method of initial rates. Make Lineweaver-Burke (1/vo vs. 1/[S]o) and Eadie-Hofstee (vo vs. vo/ [S]) plots before you leave the lab. (Make sure to calculate the initial rate using the slope of the linear portion of the [S] vs. time curve). From the Lineweaver-Burke and EadieHofstee plots of the enzymatic reaction in the presence of borate and EDTA you should be able to determine the type of inhibition for both borate and EDTA.

Determining the order of an enzymatic reaction: From the uninhibited 1.0 and 250 mM lactate solution data, determine whether the rates are zero or first order in substrate. Analyze your data from these runs with the use of integrated rate laws that are zero-order and first-order in pyruvate concentration. In you experimental write-up, you should discuss the results of the these plots, and compare your data to the results found in the Michaelis-Menten section. In particular, given you experimentally derived Km, does the observed order of the reaction for each substrate concentration make sense. Also, assuming that the Km for the pyruvate is ~25 µM, explain how measuring first order kinetics for lactate might be problematic and how this could be overcome. 3. Enzyme Lab Analysis Analysis of rate of lactate oxidation with time. Attempt to answer the question: is the reaction zeroth order or first order in substrate concentration? Zeroth order means that the rate is independent of substrate concentration: -d[S]/dt = +d[P]/dt = ko (the zeroth order rate constant) Integrating both sides over time yields: [P] = kot or [S] = [S]o – kot where [S]o is the concentration of substrate at zero time (remember, [P]o is zero). A plot of [S] versus time should give a straight line with slope –ko. First order means that the decay of the substrate or the formation of the product are single exponential in time, where the differential equation describing a first order reaction is as follows: d[S] − = k1 [S] dt where k1 is the first order rate constant. By rewriting the equation in order to separate variables, the equation takes the form: d[S] = − k1dt [S]

Then, through integration of both sides of the equation, the formula takes the form:

∫

d[S] = − k1 ∫ dt [S]

ln[S] = -k1t + C Where C is the constant of integration. In order to evaluate this the concentration has to be known at a particular time. For instance, if [S]o is the concentration at time zero, then it must satisfy the condition, ln[S]o = C. This implies that the concentration of the substrate decreases exponentially as a function of time. Therefore at time zero, ln[S]o = 1 and will decay in a linear fashion to zero as t → ∞ . Note: ln[S]∞ must also satisfy ln[S]∞ = C, therefore, you can also plot the following formula in terms of product: ln

[ P]∞ - [P]obs = − k1 t [P]∞

where [P]obs is the concentration of product at each time measured and [P]∞ is the asymptote for product formation. A plot of ln ([S]/[S]o) vs. t will generate a straight line with slope –k1 for first-order kinetics. (Note: You should derive these equations in your write-up) Plot your experimental data as [S] vs. t and ln([S]/[S]o) vs. t for both the 1.0 and 250 mM lactate runs. From these, you should be able to determine the enzymatic rate law for both high and low substrate concentration. Calculate both the zeroth and first order rate constants (ko and k1). Activity. Using data from , calculate specific activity which is given by: (# micromoles substrate oxidized) / [(minute)x(grams of enzyme)]. Turnover Number (# substrate molecules oxidized)/[(seconds)x(# of enzyme molecules)]. Use 134,000 g/mol as the molecular weight of the enzyme. Compare the value obtained from the fit to a zero order plot to the value of the turnover number that you obtain from the Lineweaver-Burke and Eadie-Hofstee plots (below). Analysis of Enzyme Kinetics via methods of initial rates. In this set of runs, you vary the initial substrate concentration, [S]o, and measure the initial rate, [P]/minute, at each [S]o. This data can be analyzed in (at least) two ways in order to determine both k2, the rate of dissociation of the enzyme-substrate complex (or the rate determining step in an enzyme catalyzed reaction) and the Michaelis-Menten constant (Km). Also, for the runs in the presence of an inhibitor, calculate the effective Michaelis-Menten constant (K’m) and inhibitor constant (Ki) based on equations (12) through (17). Make sure to identify the type of inhibition occurring in the presence of both EDTA and borate.

** Keep in mind that Vmax = k2[E]o ** Lineweaver-Burke: 1/vo = Km/Vmax(1/[S]) + 1/Vmax)

plot 1/v0 versus 1/[S0] slope = Km/Vmax y-intercept = 1/Vmax

k2 is the turnover number which can be obtained if [E]o is known. Report this value. The turnover number is the number of lactate molecules oxidized per second when the enzyme is saturated with substrate or, in other words, when [E]s = [E]o . Eadie-Hofstee Plot: vo = Km(vo/[S]) + Vmax

plot vo versus vo/[So] slope = -Km y-intercept = Vmax

Report Writing: One final note of clarification: Whether or not you choose to use a spreadsheet program such as EXCEL to do your calculations (and we strongly advise that you do), you must show a sample calculation for the operations that you are performing so that it is clear what equations you are using and what values you are plugging into them. This will allow both you and us to check your work. Only include tables that are relevant and included in the discussion. The tables that list your data must be numbered, with a title, and must be properly labeled showing correct units.

‘PARTICLE IN A BOX’ ESTIMATION OF CONJUCATED BOND DISTANCES LAB PREPARATION Read the attached publication [J. Chem. Ed. 74, 985 (1997)] EXPERIMENT Use a chemical drawing program to produce bond-line drawings of the molecules of interest for your report. All of the conjugated double bonds are in the “trans” configuration. Determine the particle in a box quantum numbers for each HOMO and LUMO. For each molecule, count the number of π-electrons in the conjugated system (the phenyl rings don’t count). Your bond-line drawing will help. Using this electron count, determine the 1-D particle in a box quantum number of the highest occupied molecular orbital (HOMO) for each molecule, assuming that a pair of electrons goes into each orbital. For example, if your system has two π-electrons, then the particle in a box quantum number for the HOMO is n = 1. The corresponding 1-D particle in a box quantum number of the lowest occupied molecular orbital (LUMO) is one more than that of the HOMO. Therefore, the particle in a box quantum number for the lowest unoccupied molecular orbital (LUMO) in this example is n = 2. Using the expression for the energy of a particle in a one dimensional box, derive an equation for the energy difference between two 1-D particle in a box energy levels. For each molecule, find the peak in the spectrum that corresponds to the HOMO-LUMO transition. Use the Planck-Einstein equation for the energy of a photon to find the ∆E corresponding to the absorption of light with this wavelength. Plug this ∆E and the relevant particle in a box quantum numbers from

The theoretical box length Determine the ‘through-bond’ phenyl-phenyl bond length. Note: this is not simply the through-space distance from one ring to the next. Work out the sum of each bond distance based on the known bond distances and angles. Include these values as your expected value for comparison to what you determine from the UV-visible absorption spectrum as illustrate in Table 1.

Table 1: Through space phenyl-phenyl distances Dye 1,4-diphenyl-1,3-butadiene 1,6-diphenyl-1,3,5-hexatriene 1,8-diphenyl-1,3,5,7-octatetraene

Experimental (Å)

Theoretical (Å)

Report

Start with a brief, simple introduction. Briefly recount how you took the spectra, including what concentration of each molecule you used and any other relevant explanation, such as what you used as a blank. Include publication quality bond-line drawings of the molecules of interest. Present the spectra in a publication quality figure in your report. Explain how you determined the HOMO and LUMO particle in a box quantum numbers for each molecule. Explain which peaks you used in your calculations and how the calculations were done. Present the results of your calculated experimental box lengths for each molecule. How do your experimental box lengths compare to the theoretical box lengths? End with a brief, coherent conclusion.

In the Laboratory

Alternative Compounds for the Particle in a Box Experiment Bruce D. Anderson Department of Chemistry, Muhlenberg College, Allentown, PA 18104

A common experiment in many undergraduate physical chemistry laboratories is the study of how the absorption spectra of polymethine dyes can be used to determine the box length in the one-dimensional particle-in-a-box model (1, 2). A new series of compounds is proposed to replace the polymethine dyes in this experiment that are less expensive and less hazardous, and the length of the box is easier for students to visualize. The compounds are 1,4diphenyl-1,3-butadiene; 1,6-diphenyl-1,3,5-hexatriene; and 1,8-diphenyl-1,3,5,7-octatetraene.

box lengths determined from the absorption spectra. The theoretical and experimental box lengths for the polymethine dyes are included in Table 1 for comparison. In conclusion, the experimental box lengths are in surprisingly good agreement with the theoretical box lengths. Further, the results are comparable to those obtained using the polymethine dyes, but the quantum mechanical box is much easier for students to recognize using the diphenyl compounds. Literature Cited

Experimental Box Length To begin, each compound is dissolved in cyclohexane to make a 10{6 M solution. Then the absorption spectrum of each solution is acquired from 300 to 425 nm as shown in Figure 1. For each spectrum, the wavelength of the lowest energy peak is determined and converted to energy. This energy is used in the eigenvalue expression shown below for the one-dimensional particle in a box.

∆E =

n 2f – n 2i h 8mL

1. Sime, R. J. Physical Chemistry: Methods, Techniques, and Experiments; Saunders College: Philadelphia, 1990; p 687. 2. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry; McGraw-Hill: New York, 1989; p 440.

2

2

To calculate the length, L, of the quantum mechanical (experimental) box, m is taken to be the mass of an electron, and n is the initial or final quantum level for the electronic transition. For each compound n is determined by counting the number of π electrons between the phenyl rings and then filling the energy levels with pairs of electrons. The lowest energy transition occurs from the highest occupied energy level to the lowest unoccupied energy level (1, 2). For example, 1,4-diphenyl-1,3-butadiene contains 4 π electrons between the phenyl rings. Thus, the two lowest quantum levels are filled and the lowest energy transition is from n i = 2 to n f = 3. Theoretical Box Length In this series of diphenyl compounds, the length of the box is taken to be the distance between the phenyl rings and the phenyl rings represent the walls of the box. In the polymethine dyes, the box length includes not only the distance between the rings, but extends into and beyond the rings (1, 2). As a result, students have great difficulty picturing where the walls of the box are and what length they are trying to measure. Students have a better understanding of the model when the diphenyl compounds are used. To calculate the theoretical box length for the diphenyl compounds, use 0.139 nm as the average bond length of a carbon–carbon bond in the conjugated system between the rings. The theoretical box lengths are calculated by determining the number of bonds between the phenyl rings and multiplying by the length per bond. The theoretical box lengths are given in Table 1 along with the experimental

Figure 1. Absorption spectra of the diphenyl compounds in cyclohexane: --- 1,4-diphenyl-1,3-butadiene; ... 1,6-diphenyl-1,3,5hexatriene; — 1,8-diphenyl-1,3,5,7-octatetraene.

Table 1. Experimental and Theoretical Box Lengths Experimental Theoretical Compound

(nm)

(nm)

Diphenyl Compounds 1,4-diphenyl-1,3-butadiene

0.726

0.695

1,6-diphenyl-1,3,5-hexatriene

0.889

0.973

1,8-diphenyl-1,3,5,7-octatetraene

1.040

1.251

1,1'-diethyl-2,2'-cyanine iodide

1.053

0.834

1,1'-diethyl-2,2'-carbocyanine chloride

1.285

1.112

1,1'-diethyl-2,2'-dicarbocyanine iodide

1.534

1.390

Polymethine Dyes

Vol. 74 No. 8 August 1997 • Journal of Chemical Education

985