A PREVIOUSLY UNREPORTED INTENSE ABSORPTION BAND AND THE pK, OF PROTONATED TRIPLET METHYLENE BLUE

Abstract— Excitation by a Q-switched giant ruby laser (1.2 J output at 694 nm ?50 ns flash) of 2–10 µM solutions of methylene blue in water, 30% ethanol in water or 50 v/v% water-CH₃CN at pH values in the range 2.0–9.3 converted the dye essentially completely to its T₁ state. The absorption spectrum of T₁ dye was measured in different media at pH 2.0 and 8.2 by kinetic spectrophotometry. Previously reported T-T absorption in the violet in acidic and alkaline solutions and in the near infrared in alkaline solution was confirmed. Values found for these absorptions in the present work with 30% ethanol in water as solvent are λ_max - 370nm, ε_max, - 13,200 M^-1 cm^-1 at pH 2 and λ_max,?420nm, ε_max 9000 M^-1 cm^-1, λ_max, - 840 nm, ε_max - 20,000 m ^-1 cm^-1 at pH 8.2. Long-wavelength T-T absorption in acidic solution is reported here for the first time: λ_max, ? 680 nm, e_max? 19,000 M cm^-1 in 30% ethanol in water at pH 2. Observation of a pH-independent isobestic point ? 720 nm confirms that the long-wavelength absorptions are due to different protonated states of the same species, MB⁺(T₁) and MBH²⁺(T₁). The pK_a of MBH²⁺(T₁) in water was determined from the dependence on pH of absorption at 700 and 825 nm to be 7.1₄± 0.1 and from the kinetics of decay of triplet absorption to be 7.2. The specific rate of protonation of MB⁺(T₁) by H₂PO₄ in water at pH 4.4 was found to be 4.5 ± 0.4 times 10⁸M^-1s^-1.     

Reduction Kinetics of Thionine in Aerobic Condition with D-Galactose

Reduction kinetics of thionine (Th) with D-galactose (RH) was observed on a UV/Visible 1601 Shimadzu spectrophotometer at λmax 599 nm. The results showed that the initially slow reduction kinetics got enhanced and proceeded to completion within a few minutes. A pseudo first order kinetics was observed when influence of different parameters like concentration of dye and reductant, ionic strength and temperature was investigated. A significant shift in wave length from 599 to 517 nm was observed at alkaline pH whereas addition of a small amount of acid caused a shift in equilibrium. This resulted in the generation of oxidized form of thionine which was pragmatic in the presence of atmospheric oxygen. Change in ionic strength at elevated temperature lied to decrease in the rate constant. Thermodynamics activation parameters like Ea reflects a high amount of energy required for reduction of Th with RH whereas entropy of activation (∆S!) and free energy of activation (∆G!) show the highly solvated states of transient complex which was less disorderly arranged than the oxidized form of dye. A mechanism consistent with above findings has been discussed in the relevant section of paper.     

The T3 (π, π) State of acridine

The 00 band maximum of the transition T₃(π, π^*) ← T₁ (π, π^*) of acridine occurs at ≈ 10200 ± 20 cm^?1 in inert (n-hexane, benzene, CCl₄), at 10220 ± 20 cm^?1 in polar (acetonitrile) and at 10170 ± 50 cm^?1 in hydrogen-bonding (methanol, 2-propanol and alkaline water) solvents. Based on the solvent-independent energy of T₁ (π, π^*), the T₃(π, π^*) state of acridine is estimated at 26050 ± 50 cm^?1 in all the solvents.     

Kinetic and redox characteristics of semireduced species derived from phenosafranine in homogeneous aqueous and sodium dodecyl sulfate micellar media

One‐electron reduction of phenosafranine (PS⁺ 3,7‐diamino‐5‐phenylphenazinium chloride), a phenazinium dye, has been studied in homogeneous aqueous and sodium dodecyl sulfate (SDS) micellar media, using the pulse radiolysis technique. The various reducing radicals employed for the study in homogeneous aqueous medium were e_aq^?, H^˙, CO_2,^˙?, and isopropyl ketyl radicals (CH₃)₂ ^˙COH. Semireduced species generated by these reactions have been characterized by their absorption spectra, decay kinetics, and pK_a. The one‐electron reduction potential of PS⁺ was determined at pH 7 in homogeneous aqueous solution employing nitrobenzene (NB/NB^˙?) as the standard couple. One‐electron reduction in SDS micellar medium and a detailed spectrophotometric investigation of the parent dye in this surfactant system was carried out in order to understand the dye–surfactant interactions in the micellar and premicellar media.© 2001 John Wiley & Sons, Inc. Int J Chem Kinet 34: 56–66, 2002     

Pulse radiolysis study of redox reactions of safranine T in sodium dodecylsulphate (SDS) micellar medium

Reaction of hydrated electrons with safranine T (SF⁺), a phenazine dye useful as sensitizer in photogalvanic cell and the transient semireduced species formed by this reaction have been studied in SDS micellar medium using the technique of pulse radiolysis. Thee _aq ⁻ reaction with SF⁺ in the micellar environment was only marginally slower (5.1 × 10⁹ dm³ mol⁻¹ s⁻¹) as compared to that in homogeneous aqueous medium (2.2 × 10¹⁰ dm³ mol⁻¹ s⁻¹) explicable on the basis of our finding that although a large fraction of the dye gets localized near the micelle Stern layer where the molecule experiences a dielectric.constant of ≈40, a small but significant concentration of the dye exists in the aqueous bulk as charge pair complex with the anionic surfactant monomer (association constant for the formation of the complex being 2.8 × 10⁴ dm³ mol⁻¹). The transient semireduced absorption band observed in the micellar medium showed a red shift of ≈ 50 nm and also the decay of the transient, which was very fast with 2k = 1 × 10⁹ dm³ mol⁻¹ s⁻¹ in aqueous medium, was stable in the SDS micellar medium over a few tens of milliseconds suggesting that the radical is incorporated deeper than the parent molecule in the SDS micelle. The effect of this stability on the photogalvanic conversion needs to be examined.     

Chemically modified porous silicon for laser desorption/ionization mass spectrometry of ionic dyes

Desorption/ionization on silicon (DIOS) mass spectra of model ionic dyes methylene blue (MB⁺Cl^?) and methyl orange (Na⁺MO^?) were studied using p⁺ type‐derived porous silicon (PS) free layers. As‐prepared PS (PS‐H), the PS thermally oxidized at 300 °C (PS‐OX), PS with chemically grafted cation‐exchanging alkylsulfonic acid (PS‐SO₃H) and anion‐exchanging propyl‐octadecyldimethylammonium chloride (PS‐ODMA⁺Cl^?) groups was tested as ionization platforms. Two mechanisms of the methylene blue desorption/ionization were found: (1) the formation of [MB + H]^+? ion due to the reduction/protonation of MB⁺, which is predominant for PS‐H and PS‐OX platforms and (2) direct thermal desorption of the MB⁺ cation, prevailing for PS‐SO₃H. The fragmentation of the cation is significantly suppressed in the latter case. The samples of PS‐SO₃H and PS‐ODMA⁺ Cl^? efficiently adsorb the dyes of the opposite charge from their solutions via the ion‐exchange. Consequent DIOS MS studies allow to detect only low fragmented ions (MB⁺ and MO^?, respectively), demonstrating the potential of the ion‐exchange adsorption combined with DIOS MS for the analysis of ionic organic compounds in solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Yields of trapped electrons in pulse irradiated aqueous LiCl Glasses at temperatures down to 6K

Pulse radiolysis of deuterated aqueous LiCl glasses at temperatures in the range 6 K to 70 K show that the yield G(e^?_IR) of infrared absorbing electrons (e^?_IR) increases sharply as the temperature is lowered when [LiCl] ? 10 M. Under these conditions the yield of visible absorbing electrons (e^?_vis) decreases, but to a lesser extent. When [LiCl] ? 8 M, G(e^?_IR) and G(e^?_vis) are both much less dependent on temperature. These data suggest that at very low temperatures e^?_IR are not trapped exclusively in a purely aqueous environment.     

环丙沙星-钨硅多金属氧酸盐的制备及热分解动力学

A new polyoxometalate （CPFX·HCl）3H4SiW12O40·10H2O was prepared from ciprofloxacin hydrochloride and H4SiW12O40·nH2O in aqueous solution, and characterized by elemental analysis, IR spectra and DTA-TG-DTG techniques. The IR spectrum confirmed the presence of Keggin structure and the characteristic functional group for ciprofloxacin in the compound. The TG-DTA-DTG curves showed that its thermal decomposition was a four-step process consisting of simultaneous collapse of Keggin type structure. The residue of decomposition was the mixture of WO3 and SiO2, confirmed by X-ray diffraction and IR spectroscopy. The decomposition mechanism and nonisothermal kinetic parameters of the polyoxometalate were obtained from an analysis to the TG-DTG curves by the single scanning methods （the Achar method and Coats-Redfern method） and the multiple scanning methods （the Kissinger method, Flynn-Wall-Ozawa method and Starink method）. The results indicate that the kinetic equationswith parameters describing the thermal decomposition reaction are dα/dt=6.65×10^6[3（1-α）^2/3]e^-10495.5/T with E=87.26 kJ/mol and A=6.65×10^6 s^-1 for the second step,dα/dt=7.01×10^9（1-α）e^-18770.7/T with E=156.06 kJ/mol and A=7.01×10^9 s^-1 for the third step,dα/dt=9.77×10^43[（1-α）^2]e^-88980.0/T with E=739.78 kJ/mol and A=9.77×10^43 s^-1 for the fourth step.     

Kinetics of the methylene blue oxidation by cerium(IV) in sulphuric acid solutions

The oxidation of methylene blue (MB⁺) by cerium(IV) was studied in 0.1–5 M H₂SO₄. The reaction proceeds via MB radical (MB^2+•) formed by one electron transfer to the oxidant. The radical is observed spectrophotometrically by a very intense absorbance at λ_max = 526 nm and by the e.p.r signal at g = 2.000. The kinetics of the fast radical formation are two orders of magnitude slower than its decomposition, which were examined using a stopped-flow method at 298 K under pseudo-first order conditions. The rate laws for the both steps were determined and a likely mechanism reported.     

The first two excited 1Σg+ states of Cs2

Laser-induced fluorescence Of Cs₂ molecules in the infrared region (4000–9000 cm^?1) has been observed using several exciting wavelengths from an argon-ion laser and from a ring dye laser. Accurate molecular constants for the first two excited ¹Σ_g⁺ electronic states are derived from spectra recorded at high resolution by Fourier transform spectroscopy. Main molecular constants are: (2)¹Σ_g⁺: T_c = 12114.090 cm^?1, ω_e = 23.350 cm^?1, B_c = 7.4.5 × 10^?3 cm^?1, R_c = 5.8316 Å; (3)¹Σ_g⁺: T_e = 15975.450 cm^?1, ω_e = 22.423 cm^?1 , B_e = 8.23 × 10^?3 cm^?1, R_c = 5.5569 Å.     

Kinetic Spectroscopy of Erythrosin Phosphorescence and Delayed Fluorescence in Aqueous Solution at Room Temperature*

The photophysics and polarization of the phosphorescence and delayed fluorescence of erythrosin in conditions compatible with the current biological applications of the dye (aqueous buffers at pH 7.4 at ambient temperatures) and in ethanol have been studied as a function of dye concentration (10 ^?7-10^?5M) and temperature (245–333K). The emission decay is strictly single exponential and the detailed kinetic analysis of all the rate processes connected with the emitting T₁ state showed that (1) the lowering of the emission lifetime at the higher temperatures is due to a very efficient self-quenching process, (2) the back intersystem crossing rate T_x S₁ is temperature dependent (δE_TS7 kcal mol^?1) but the T₁S₀ is not (E_a0.1 kcal mol^?1) and (3) both intersystem crossing processes are very sensitive to solvent polarity, which accounts for the solvent dependence of the phosphorescence yield and lifetime. The high value of the phosphorescence anisotropy (r₀= 0.25 lt 0.006) is independent of the excitation and emission wavelengths, and its evolution in time accurately reflects the rotational restrictions in solid solutions. The relevance of these findings to studies with protein-dye conjugates is also outlined to facilitate the design and interpretation of phosphorescence depolarization experiments that probe the (μs-ms dynamics of biomolecules and supramolecular systems.     

Solvatochromic Parameters and Preferential Solvation Behavior for Binary Mixtures of 1,3-Dialkylimidazolium Ionic Liquids with Water

Binary mixtures of 1,3-dialkylimidazolium based ionic liquids (ILs) and water were selected as solvent systems to investigate the solute-solvent and solvent-solvent interactions on the preferential solvation of solvatochromic indicators at 25℃. Empirical solvatochromic parameters, dipolarity/polarizability (π^*), hydrogen-bond donor acidity (α), hydrogen-bond acceptor basicity (β), and Reichardt's polarity parameters (E_T^N) were measured from the ultraviolet-visible spectral shifts of 4-nitroaniline, 4-nitroanisole, and Reichardt's dye. The solvent properties of the IL-water mixtures were found to be influenced by IL type and IL mole fraction (x_IL). All these studied systems showed the non-ideal behavior. The maximum deviation to ideality for the solvatochromic parameters can be obtained in the xIL range from 0.1 to 0.3. For most of the binary mixtures, the π^* values showed the synergistic effects instead of the E_T^N, α and β values. The observed synergy extent was dependent on the studied systems, such as the dye indicator and IL type. A preferential solvation model was utilized to gather information on the molecular interactions in the mixtures. The dye indicator was preferentially solvated on the following trend: IL >IL-water complex >water.     

Theoretical investigations of the electronic states of porphyrins. IV. Low-lying electronic states of bisammineporphinatoiron(II)

Ab initio CI calculations are reported on the lowest quintet, triplet, and singlet states of Fe^II(P) (NH₃)₂. The lowest singlet state has strong mixing between the configuration (d_xy² (d_π)⁴ and (d_xy)²(d_π)³e_gπ*. The lowest quintet is mixed between ⁶A_1g)d_π and (⁶A_1g)e_gπ*, where ⁶A_1g refers to the high-spin ferric configuration. We calculate many low-energy states as ³(π→π*) ring and metal triplet and quintet configurations [“triptriplets” and “tripquintets”]. The calculations also show low-energy charge-transfer configurations of ring anion excited quartets and ferric quartets and sextets [“quartquartets” and “quartsextets”]. The farthest red x,y-polarized bands of the experimental spectra of low-spin hemoproteins are identified as d_xy → e_gπ* or d_π → d mixed with d_π → d and the z-polarized bands are assigned as d_π → e_gπ*. The farthest red x,y-polarized bands of the high-spin hemoproteins are identified as excited quartsextet states. Picosecond transients observed in Fe^II(TPP) (pip)₂ are attributed to an initial ¹(d_π → e_gπ*) state, which inter-system crosses to high-spin states that lose one ligand.     

Reaction‐path dynamics and theoretical rate constants for the reaction CH4 + O3 → HOOO + CH3

We present a direct ab initio dynamics study of thermal rate constants of the hydrogen abstraction reaction of CH₄ + O₃ → HOOO +CH₃. The geometries of all the stationary points are optimized at MPW1K/6‐31+G(d,p), MPWB1K/6‐31+G(d,p), and BHandHLYP/6‐31+G(d,p) levels of theory. The energies are refined at a multi‐high‐level method. The extended Arrhenius expression fitted from the CVT/SCT and μVT/Eckart rate constants of ozonolysis of methane in the temperature range 200–2500 K are k^CVT/SCT(T) = 5.96 × 10^?29T^4.49e^(?17321.3/T) and k^μVT/Eckart(T) = 7.92 × 10^?29T^4.46e^(?17301.7/T), respectively. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Chelation versus Binucleation: Metal Complex Formation with the Hexadentate all‐cis‐N1,N2‐Bis(2,4,6‐trihydroxy‐3,5‐diaminocyclohexyl)ethane‐1,2‐diamine

The hexadentate ligand all‐cis‐N¹,N²‐bis(2,4,6‐trihydroxy‐3,5‐diaminocyclohexyl)ethane‐1,2‐diamine (L^e) was synthesized in five steps with an overall yield of 39 % by using [Ni(taci)₂]SO₄?4 H₂O as starting material (taci=1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol). Crystal structures of [Na_0.5(H₆L^e)](BiCl₆)₂Cl_0.5?4 H₂O ( 1 ), [Ni(L^e)]‐ Cl₂?5 H₂O ( 2 ), [Cu(L^e)](ClO₄)₂?H₂O ( 3 ), [Zn(L^e)]CO₃?7 H₂O ( 4 ), [Co(L^e)](ClO₄)₃ ( 5 c ), and [Ga(H_?2L^e)]‐ NO₃?2 H₂O ( 6 ) are reported. The Na complex 1 exhibited a chain structure with the Na⁺ cations bonded to three hydroxy groups of one taci subunit of the fully protonated H₆(L^e)⁶⁺ ligand. In 2 , 3 , 4 , and 5 c , a mononuclear hexaamine coordination was found. In the Ga complex 6 , a mononuclear hexadentate coordination was also observed, but the metal binding occurred through four amino groups and two alkoxo groups of the doubly deprotonated H_?2(L^e)^2?. The steric strain within the molecular framework of various M(L^e) isomers was analyzed by means of molecular mechanics calculations. The formation of complexes of L^e with Mn^II, Cu^II, Zn^II, and Cd^II was investigated in aqueous solution by using potentiometric and spectrophotometric titration experiments. Extended equilibrium systems comprising a large number of species were observed, such as [M(L^e)]²⁺, protonated complexes [MH_z(L^e)]^2+z and oligonuclear aggregates. The pK_a values of H₆(L^e)⁶⁺ (25 °C, μ=0.10 m ) were found to be 2.99, 5.63, 6.72, 7.38, 8.37, and 9.07, and the determined formation constants (log β) of [M(L^e)]²⁺ were 6.13(3) (Mn^II), 20.11(2) (Cu^II), 13.60(2) (Zn^II), and 10.43(2) (Cd^II). The redox potentials (vs. NHE) of the [M(L^e)]^3+/2+ couples were elucidated for Co (?0.38 V) and Ni (+0.90 V) by cyclic voltammetry.     

NMR- und schwingungsSpektroskopische Untersuchungen höherer Indiumtrialkyle

NMR and Vibrational Spectroscopic Investigations on Higher Indium Trialkyls Several isomeric indium tripentyles and trihexyles are synthesized by known methods. The chemical shifts δ of the ¹³C NMR spectra are used together with those of the corresponding alkanes for determining the increments Δδ(¹³C) = δ(InR₃)–δ(RH). By means of these increments and the Grant/Paul-method the chemical shifts δ(¹³C) of any indium trialkyl can be calculated. The vibrational spectra (IR and Raman) of most liquid trialkyls show very obvious rotameric splittings of the InC vibrations between 450–600 cm^?1. Both frequent conformations of the alkyl ligands with three and more C atoms consist of either a βH atom (notations P_H, S_HH, and T_HHH with vInC between 450–500 cm^?1) or a γC atom (P_C, S_CH, T_CHH with vInC between 550–600 cm^?1) in the transposition to indium. The InC stretching modes of all other more rare configurations can be observed between 500 to 550 cm^?1.     

Electronic Structure Modulation in an Exceptionally Stable Non‐Heme Nitrosyl Iron(II) Spin‐Crossover Complex

The highly stable nitrosyl iron(II) mononuclear complex [Fe(bztpen)(NO)](PF₆)₂ (bztpen=N‐benzyl‐N,N′,N′‐tris(2‐pyridylmethyl)ethylenediamine) displays an S=1/2?S=3/2 spin crossover (SCO) behavior (T_1/2=370 K, ΔH=12.48 kJ mol^?1, ΔS=33 J K^?1 mol^?1) stemming from strong magnetic coupling between the NO radical (S=1/2) and thermally interconverted (S=0?S=2) ferrous spin states. The crystal structure of this robust complex has been investigated in the temperature range 120–420 K affording a detailed picture of how the electronic distribution of the t_2g–e_g orbitals modulates the structure of the {FeNO}⁷ bond, providing valuable magneto–structural and spectroscopic correlations and DFT analysis.     

Thermal Behavior and Non-isothermal Decomposition Reaction Kinetics of NEPE Propellant with Ammonium Dinitramide

Thermal decomposition behavior and non‐isothermal decomposition reaction kinetics of nitrate ester plasticized polyether NEPE propellant containing ammonium dinitramide (ADN), which is one of the most important high energetic materials, were investigated by DSC, TG and DTG at 0.1 MPa. The results show that there are four exothermic peaks on DTG curves and four mass loss stages on TG curves at a heating rate of 2.5 K·min^?1 under 0.1 MPa, and nitric ester evaporates and decomposes in the first stage, ADN decomposes in the second stage, nitrocellulose and cyclotrimethylenetrinitramine (RDX) decompose in the third stage, and ammonium perchlorate decomposes in the fourth stage. It was also found that the thermal decomposition processes of the NEPE propellant with ADN mainly have two mass loss stages with an increase in the heating rate, that is the result of the decomposition heats of the first two processes overlap each other and the mass content of ammonium perchlorate is very little which is not displayed in the fourth stage at the heating rate of 5, 10, and 20 K·min^?1 probably. It was to be found that the exothermal peak temperatures increased with an increase in the heating rate. The reaction mechanism was random nucleation and then growth, and the process can be classified as chemical reaction. The kinetic equations of the main exothermal decomposition reaction can be expressed as: dα/dt=10^12.77(3/2)(1?α)[?ln(1?α)]^1/3 e^{?1.723×104/T}. The critical temperatures of the thermal explosion (T_be and T_bp) obtained from the onset temperature (T_e) and the peak temperature (T_p) on the condition of β→0 are 461.41 and 458.02 K, respectively. Activation entropy (ΔS^≠), activation enthalpy (ΔH^≠), and Gibbs free energy (ΔG^≠) of the decomposition reaction are ?7.02 J·mol^?1·K^?1, 126.19 kJ·mol^?1, and 129.31 kJ·mol^?1, respectively.     

Syntheses and crystal structures of two Co(II) coordination polymers based on 4,6-bis(4-methylbenzoyl)isophthalic acid and 4,4′-bipyridine

Two coordination complexes, [Co₂L₂(4,4′-bpy)₂(H₂O)₄]?·?6H₂O (1) and [CoL(4,4′-bpy)] (2) (H₂L?=?4,6-bis(4-methylbenzoyl)isophthalic acid and 4,4′-bpy?=?4,4′-bipyridine), have been synthesized with the same starting materials under conventional and hydrothermal condition, respectively. Their structures have been characterized by X-ray diffraction, elemental analysis, IR spectra, and thermogravimetric analysis. Complex 1 features a 2-D sheet structure (space group C2/c) with (4,4) grid units. The non-covalent interactions (O–H?·?·?·?O, C–H?·?·?·?π, and weak π??·?·?·?π interactions) extend 1 into a 3-D supramolecular network. Complex 2 displays a (3,5)-connected network (space group P 1) with a (4²?·?6)(4²?·?6⁸) topology.     

Ultimate tensile properties of elastomers. VII. Effect of crosslink density on time–temperature dependence

Data on tensile strength and elongation at break for a series of Viton A-HV vulcanizates are discussed. The data were obtained at various extension rates at temperatures from ?5 to 230°C (25 ? T — T_g ? 260°C) on seven vulcanizates having crosslink densities v_e (estimated from C₁ in the Mooney-Rivlin equation) from 0.46 × 10^?5 to 24.4 × 10^?5 mole/cm³. At an extension rate of 1 min^?1, an increase in v_e affects the tensile strength σ_b (based on the undeformed cross-sectional area) and the true tensile strength σ_bσ_b (based on the cross-sectional area of a deformed specimen) as follows: σ_b is essentially constant at a low temperature; it passes through a decided maximum at intermediate temperatures; and it increases to a plateau at elevated temperatures. In contrast, λ_bσ_b decreases markedly at all temperatures, an exception being the most lightly crosslinked vulcanizate(s). Application of time—temperature superposition to the ultimate-property data gave log a_T; its temperature dependence is that typical of nonpolar rubbery polymers. Data on the vulcanizates were compared in corresponding temperature states by plotting log 273σ_b/T, log 273λ_bσ_b/T, and (λ_b — 1)/(λ_b — 1)_max against logt_b/(t_b)_max, where t_b is the temperature-reduced time to break and (t_b)_max is the value at which the ultimate extension ratio λ_b attains its maximum, (λ_b)_max. Except for the most lightly crosslink vulcanizate, the comparison shows that 273λ_bσ_b/T and (λ_b — 1)/(λ_b — 1)_max are substantially independent of (or only weakly dependent on) crosslink density, that 273λ_b/T increases with v_e, and that 273λ_b/T ∝? v_e^0.6 and λ_b ∝? v_e^?0.4 at a large value of t_b/(t_b)_max.