MATHCAD A TOOL FOR NUMERICAL CALCULATION OF SQUARE-WAVE VOLTAMMOGRAMS

Transcription

1 Bullein of he Chemiss and Technologiss of Macedonia, Vol. 8, No., pp (999) GHTMDD 328 ISSN Received: November 6, 998 UDC: : Acceped: March 9, 999 Professional paper MATHCAD A TOOL FOR NUMERICAL CALCULATION OF SQUARE-WAVE VOLTAMMOGRAMS Insiue of Chemisry, Faculy of Sciences and Mahemaics, The Sv. Kiril & Meodij Universiy, POB 62, 9000 Skopje, Republic of Macedonia An alernaive approach for numerical calculaion of he square-wave volammograms using he mahemaical programming package MATHCAD is presened. A quasi-reversible redox reacion is considered and a mahemaical model is developed under condiions of he square-wave volammery (SWV). Applicaion of he mahemaical model in MATHCAD is discussed and he file used for numerical simulaion is presened. The relaionships beween he properies of he SW response and he parameers of boh he quasireversible redox reacion and he exciaion signal are discussed. Key words: MATHCAD; square-wave volammograms; numerical calculaion INTRODUCTION Numerical simulaion of he volammeric response of paricular volammeric echnique is a common approach in he developmen of he volammeric mehod's heory. Sudying he numerically simulaed daa one can predic he behavior of he volameric experimen. Moreover, comparing and fiing he experimenal and heoreical daa, imporan kineic parameers of he invesigaed redox sysem, such as he sandard rae consan k s and he coefficien of elecron ransfer α, can be esimaed. For all hese reasons, a large number of scienific papers are dedicaed o his imporan subjec [l]. Addiionally, numerous specialized programming packages for simulaion of he volammeric response of various echniques are already available on he marke. I is, however, sill of ineres o develop simple and flexible mehods for calculaion of he heoreical response of various volammeric echniques. Nowadays, mos chemiss are familiar wih he general purpose muliasking programming packages such as EXCEL, QPRO, LO- TUS, MATHCAD, ec. Therefore, i is very useful o find a way for calculaion of he volammeric responses using hese programming packages. In his paper an alernaive approach for calculaion of he square-wave volammeric response of a quasi-reversible redox reacion, using he programming package MATHCAD, is presened. MATHCAD is one of he bes general purpose mahemaical programming packages [2]. The program is user-friendly, fas and precise. I should be emphasized ha he program provides various numerical mehods which are necessary for developmen of volammeric mehod's heory. As was menioned previously, he volammeric curves are numerically simulaed under condiions of he square-wave volammery, which is one of he mos advanced elecroanalyical echniques [3]. The square-wave volammery is a complex, muli-sep chronoamperomeric mehod. The exciaion signal used in SW volammery is a rain of cahodic and anodic pulses superposed on a saircase poenial ramp (see Fig. a). One poenial cycle in he SWV is presened in he Fig. b. The parameers of he signal are: he square-wave frequency f, which is he inverse value of he duraion of he poenial cycle f = /τ (see Fig. b), he SW ampliude E sw, which is he half of he peak-o-peak heigh, and he scan incremen de, which is he sep of he saircase ramp. The curren is measured a he end of each poenial pulse. All he currens measured a he end

2 58 V. Mireski, R. Gulaboski and I. Kuzmanovski of he cahodic pulses creae he forward (cahodic) componen Ψ f, while he currens measured a he end of he anodic pulses, creae he backward (anodic) componen Ψ b of he SW response (see Fig. c). The ne-response Ψ ne is calculaed as a difference beween he wo successive cahodic and anodic currens: Ψ ne = Ψ f Ψ b. τ τ /2 de i f E sw E /V E s /s i b (a) (b) (c) Fig.. (a) The scheme of he exciaion signal used in he square-wave volammery. (b) One poenial cycle in he squarewave volammery. (c) A ypical square-wave dimensionless volammeric response for a reversible redox reacion. E s he saring poenial; E sw he SW ampliude; de he scan incremen; τ he duraion of one poenial cycle; i f and i b he forward and he reverse real currens; Ψ f, Ψ b, Ψ ne he forward, he backward and he ne dimensionless componens of he SW response. Bull. Chem. Technol. Macedonia, 8,, (999)

3 MATHCAD a ool for numerical calculaion of square-wave volammograms 59 MATHEMATICAL MODEL A quasireversible redox reacion of wo chemically sable species is considered: Ox + ne = Red I is assumed ha he mass ranspor occurs hrough planar, saionary, and semi-infinie diffusion model. The redox reacion (I) is described mahemaically wih he following se of differenial equaions: (I) δc Ox /δ = D(δ 2 c Ox /δx 2 ) () δc Red /δ = D(δ 2 c Red /δx 2 ) (2) For he meaning of he symbols and abbreviaions see he Table I. For simpliciy, he diffusion coefficiens of boh species Ox and Red are supposed o be equal. A he very beginning of he experimen, only he Ox form of he redox couple is presen in he elecrolye soluion. Hence, he above differenial equaions are solved wih he following iniial and boundary condiions: = 0: c Ox = c* Ox ; c Red = 0 (a) > 0, x : c Ox c* Ox ; c Red 0 (b) > 0, x = 0: D(δc Ox /δx) x=0 = D(δc Red /δx) x=0 = i/(nfs) (c) Since he redox reacion is parly conrolled by he charge ransfer rae, a he elecrode surface, he following condiion is valid: i/(nfs) = k s exp( αφ) [(c Ox ) x=0 (c Red ) x=0 exp(φ)] (3) where φ is dimensionless relaive elecrode poenial: φ = (nf)(e E 0 Ox/Red)/(RT). The soluions which relae he concenraions of boh species Ox and Red a he elecrode surface wih he curren, were obained applying Laplace ransforms: T a b l e I Ox Red c Ox c Red Lis of symbols and abbreviaions oxidized form of elecroacive species reduced form of elecroacive species concenraion of Ox species anywhere in he soluion concenraion of Red species anywhere in he soluion c* Ox concenraion of Ox species in he bulk of he soluion (c Ox ) x=0 concenraion of Ox species a he elecrode surface (c Red ) x=0 concenraion of Red species a he elecrode surface x i n F S k s α T R E E 0 D Ψ Ψ j Ψ f Ψ b Ψ ne K f E sw de E ime disance from he elecrode curren number of elecrons Faraday consan elecrode surface area sandard kineic rae consan of he redox reacion coefficien of elecron ransfer hermodynamic emperaure universal gas consan elecrode poenial sandard redox poenial of he Ox/Red couple diffusion coefficien dimensionless curren a he very firs ime incremen dimensionless curren a he j-h ime incremen dimensionless forward curren dimensionless backward curren dimensionless ne curren dimensionless kineic parameer SW frequency SW ampliude scan incremen poenial inerval E s saring poenial vs. E 0 * i ( cox ) x= 0 = cox ( τ ) dτ (4) π n F S D 0 2

4 60 V. Mireski, R. Gulaboski and I. Kuzmanovski i ( c d x= = 2 Re ) 0 ( τ ) dτ π 0 n F S D (5) A combinaion of he eqs. (3) (5) gives he following inegral equaion: i n F S α φ * i i = k s e cox 2 φ ( τ ) dτ e ( τ ) 2 dτ 0 n F S D π 0 n F S D π (6) The above inegral equaion relaes he curren wih ime a a cerain poenial. The soluion of he las inegral equaion under condiions of he SW volammery was obained by he numerical mehod of Nicholson and Olmsead [l]. Boh he ime variable and dimensionless curren Ψ = i(nfsc* Ox ) (fd) /2 are discreized. To each = jd, where d is he ime incremen, a cerain Ψ j can be ascribed. The numerical soluion is represened wih he following recursive formulae: α φ K e Ψ = + 2 K (50 π ) 2 φ ( α φ + e ) e (7) Ψ j = K e α φ j 2 (50 π ) 2 + e + 2 K (50 π ) 2 + e φ j φ j j Ψi S ji+ i= α φ j e (8) where S =, S k = (k) /2 (k ) /2 while K = k s / (fd) /2 is dimensionless kineic parameer. For his calculaion, he ime incremen d = (50f ) was used, which means ha each SW half-period τ/2 (see Fig. b) was divided in 25 incremens. SOLVING THE MATHEMATICAL MODEL USING THE PROGRAMMING PACKAGE MATHCAD The MATHCAD file used for calculaion of he dimensionless SW volammograms is given in he Fig 2. A he very beginning of he file, all he consan parameers which are needed for numerical calculaions, are defined. For numerical inegraion, he enire ime of he SW exciaion signal is divided in he finie number of ime incremens. In he previous chaper, i was menioned ha he ime incremen d is relaed o he SW frequency hrough he formula d = (50f ). I means ha each poenial cycle in he SW volammery is divided in 50 incremens. The number of he poenial cycles depends on he poenial inerval E and he scan incremen de. The oal number of he poenial cycles is equal o he raio E/dE. Therefore, he oal number of he ime incremens is ( E/dE) 50, while he ordinary number of each ime incremen is ranged wihin he inerval of o ( E/dE) 50 (see equaion (I) in Fig. 2). Bull. Chem. Technol. Macedonia, 8,, (999)

5 MATHCAD a ool for numerical calculaion of square-wave volammograms 6 Fig. 2. The MATHCAD file creaed for numerical simulaion of he SW volammograms of a quasi-reversible redox reacion

6 62 V. Mireski, R. Gulaboski and I. Kuzmanovski Creaing he file, a crucial sep is developmen of a funcion which simulaes he poenial waveform used in he SW volammery. MATHCAD has a daabase of various mahemaical funcions, however, neiher of hem resembles he shape of he SW exciaion signal. Neverheless, he problem can be solved if he mahemaical sep funcion is appropriaely combined wih he logical if funcion. The new funcion, called poenial, is defined by he eq. (II) in he Fig. 2 and is of exacly he same form as he SW exciaion signal (see he Plo in he Fig. 2). This funcion represens he relaive elecrode poenial applied o he working elecrode under condiions of he SW volammery. Using his funcion, one can readily defines he dimensionless poenial used for numerical calculaions φ = nf(e E 0 )/(RT). The calculaion of he dimensionless response under condiions of he SW volammeric experimen was carried ou by he recursive formulae (IV) and (V) in he Fig. 2. The eq. (V) calculaes he dimensionless curren Ψ k a each poenial pulse applied o he working elecrode. Above hese formulae, he S k facor needed for numerical inegraion is defined. Afer processing of he formulae (IV) and (V), a new plo is creaed (see Plo 2 in he Fig.) which shows he variaion of he curren wih ime a each SW poenial pulse. As i was menioned previously, according o he curren-sampling procedure used in he SW vol- ammery, only he curren obained a he end of a single poenial pulse is measured. The reason is o discriminae he capaciive curren during he measuremen and o increase he sensiiviy of he echnique [3]. Hence, all he currens measured a he end of all he cahodic pulses creae he forward branch Ψ f of he SW volammogram. The backward branch Ψ b conains he currens measured a he end of each anodic pulse. The ne curren Ψ ne is defined as a difference beween he forward and backward curren. Therefore, one needs o selec only he currens obained a he end of he each SW pulse. This can be done wih he se of formulae from (VI) o (VIII) given in he Fig. 2. Finally i should be noed, ha in he SW volammery, he curren daa are ploed versus he poenial values of he saircase ramp. The las formulae of he file (IX) defines he poenial of he saircase ramp. Plo 3 in he Fig. 2 represens he numerically calculaed SW volammeric response of he quasireversible redox reacion. The processing ime for calculaion of a single SW volammogram depends on he performance of he used processor, poenial inerval and he scan incremen. Wih he processor PC 486DX2/66 MHz wih 8 MB RAM memory, poenial inerval E = 0.3 V and scan incremen is de = 5 mv, processing ime akes abou 5 min. The processing ime can be markedly decreased wih increase of he scan incremen. DISCUSSION OF THE NUMERICALLY CALCULATED DATA The MATHCAD file was uilized for numerical simulaion of abou hundred SW volammograms, in order o invesigae he relaionships beween he properies of he response and he parameers of boh he redox reacion and he exciaion signal. As can be seen from he Plo 3 in he Fig. 2, he SW volammograms are curren-poenial bellshaped curves characerized wih he dimensionless peak curren Ψ p, he peak poenial E p and he half-widh of he peak E p/2. The number of poins consiuing a single volammogram depends on he scan incremen de. The poin wih he highes curren value deerminaes he peak curren Ψ p, while is posiion a he poenial axis defines he peak poenial E p. The widh of he peak a is half heigh, expressed in Vols, is called he half-widh of he peak E p/2. According o he eq. (8), he dimensionless SW response of he quasi-reversible redox reacion (I) is mainly dependen on he kineic parameer K and he elecron ransfer coefficien α. The apparen reversibiliy of he redox reacion enirely depends on he kineic parameer K = k s / (Df) /2. The influence of his parameer o he dimensionless peak curren is presened in he Fig. 3. If he redox reacion appears eiher irreversible (logk.5), or reversible (logk 0.75), he dimensionless peak curren does no depend on he kineic parameer (see Fig. 3). Wihin he region logk 0.3 he redox reacion appears quasi-reversible, and he dimensionless peak curren depends linearly on he kineic parameer K. The slope of his linear porion is deerminaed by he paricular value of he ransfer coefficien α (see Fig. 3). Bull. Chem. Technol. Macedonia, 8,, (999)

7 MATHCAD a ool for numerical calculaion of square-wave volammograms 63 slighly wih he increase of he SW ampliude. I is imporan o noe ha he raio Ψ p / E p/2 reaches he maximum value for E sw = 90 mv, which means ha his ampliude is he mos suiable for analyical purposes. Fig. 3. The effec of he kineic parameer K on he dimensionless SW peak currens for divers values of he ransfer coefficien α. The condiions of he simulaion were: E sw = 0,025 V; de = 0,005 V; T = K; α = 0.3 (); 0.5 (2) and 0.7 (3) The posiion of he SW peak is also sensiive o he kineic parameer K. The relaionship beween he peak poenials E p and he logarihm of he kineic parameer K, for differen values of he ransfer coefficien α, is presened in he Fig. 4. If he redox reacion is close o he reversible region (logk 0.5), he peak poenial becomes almos independen on he kineic parameer. Only wihin he irreversible region, he peak poenial depends linearly on he logarihm of K. The slope of he linear porion is dependen on he ransfer coefficien α and i is defined by he following equaion: E p / logk = 2,303 RT/(αnF). Therefore, if he irreversibiliy of he redox reacion was reached experimenally, his equaion could be uilized for an esimaion of he ransfer coefficien. The half-widh of he SW peak gradually changes wih he aleraion of he kineic parameer K. If he redox reacion appears irreversible, he half-widh of is SW peak is solely deermined by he ransfer coefficien α, hrough he equaion: E p/2 = (90 ± 2)/(αn) mv. Wihin he quasireversible region, he half-widh of he peak decreases proporionally wih increase of he kineic parameer K, reaching a consan value for he reversible redox reacion. If he SW ampliude was E sw = 50 mv, he half-widh of he peak for reversible redox reacion is E p/2 = 25 mv. Numerical simulaions shown ha he SW peak curren depends linearly on he SW ampliude (see Fig. 5). The peak poenial remained virually unchanged wih he variaion of he ampliude from 2 o 00 mv. The half-widh of he peak enhances Fig. 4. The effec of he kineic parameer K on he SW peak poenials for divers values of he ransfer coefficien α. The condiions of he simulaion were: E sw = V; de = V; T = K; α = 0.4 (); 0.5 (2) and 0.7 (3) Fig. 5. The dependence of he peak currens on he SW ampliude. The condiions of he simulaions were: logk = 0.5; α = 0.5; de = 5 mv; T = K When paricular redox reacion is invesigaed experimenally, he variaion of he kineic parameer K, which is defined as K = k s /(Df) /2, can be aained by an aleraion of he frequency f of he exciaion signal. Therefore, he effec of he SW frequency on he SW response can be undersood hrough he previously discussed effec of he kineic parameer K.

8 64 V. Mireski, R. Gulaboski and I. Kuzmanovski CONCLUSION In his paper i is demonsraed ha he MATHCAD programming package can be successfully used as a ool for numerical calculaion of he square-wave volammograms. I is shown ha his program can easily generae a complex funcion which possesses exacly he same shape as he SW exciaion signal. The presened MATHCAD file reflecs he simpliciy in which one can communicae wih he program. Sudying he numerically simulaed volammograms one can undersand he behavior of he volammeric experimen and realize he relaionships beween he properies of he invesigaed redox reacion and he feaures of he volammeric response. REFERENCES [] ELECTROCHEMISTRY, Calculaions, Simulaion, and Insrumenaion, eds. James S. Mason, Harry B. Mark, Jr., and Huber C. MacDonald, Jr. Marcel Dekker Inc., New York 972. [2] MATCAD 3.0, User s Guide, MahSof Inc., 20 Broadway, Cambridge, Massachuses, 0239 USA. [3] J. G. Oseryoung and R. A. Oseryoung, Anal.Chem., 57, 0 A (985). MATHCAD!"#"#$ %$"&'$(!"##$% "&#$$()MATHCAD!" # $ %& ' %$ % $ (!" # & %& # & MATH- CAD %! "! ) $! " *$ % + % " & & # & MATHCAD' &! + &,! $ ())#$ % $ ( # ' $% -$) &.$&!!" %! &! ) &! ))# Bull. Chem. Technol. Macedonia, 8,, (999)