16118
J. Phys. Chem. B 2009, 113, 16118–16123
Application of the Compensated Arrhenius Formalism to Dielectric Relaxation Matt Petrowsky and Roger Frech* Department of Chemistry and Biochemistry, UniVersity of Oklahoma, Norman, Oklahoma 73019 ReceiVed: July 23, 2009; ReVised Manuscript ReceiVed: October 21, 2009
The temperature dependence of the dielectric rate constant, defined as the reciprocal of the dielectric relaxation time, is examined for several groups of organic solvents. Early studies of linear alcohols using a simple Arrhenius equation found that the activation energy was dependent on the chain length of the alcohol. This paper re-examines the earlier data using a compensated Arrhenius formalism that assumes the presence of a temperature-dependent static dielectric constant in the exponential prefactor. Scaling temperature-dependent rate constants to isothermal rate constants so that the dielectric constant dependence is removed results in calculated energies of activation Ea in which there is a small increase with chain length. These energies of activation are very similar to those calculated from ionic conductivity data using compensated Arrhenius formalism. This treatment is then extended to dielectic relaxation data for n-alkyl bromides, n-nitriles, and n-acetates. The exponential prefactor is determined by dividing the temperature-dependent rate constants by the Boltzmann term exp(-Ea/RT). Plotting the prefactors versus the static dielectric constant places the data on a single master curve for each group of solvents. Debye equation:
1. Introduction 1
Previous work has demonstrated that the temperaturedependent ionic conductivity of dilute, liquid electrolytes based on n-alcohols and 2-ketones as solvents can be expressed in the following form: -Ea/RT
σ(T,εs) ) σo(εs(T))e
(1)
where σ(T, εs) is the temperature-dependent conductivity, σo(εs(T)) is the exponential prefactor, T is the temperature, εs is the static (low frequency) dielectric constant, R is the gas constant, and Ea is the energy of activation. In these electrolyte solutions, cation-anion interactions have been shown to be minimal due to the nature of the salt.2 Temperature-dependent conductivities of these liquid electrolytes deviate from Arrhenius behavior due to the temperature dependence of the static dielectric constant contained in the exponential prefactor. Scaling the temperature-dependent conductivities to conductivities at a reference temperature removes the dielectric constant dependence and allows the energy of activation to be calculated. The exponential prefactor can then be determined by dividing the conductivity by the Boltzmann factor exp(-Ea/RT). When the temperature-dependent prefactors are plotted against the temperature-dependent static dielectric constants, the prefactors lie on a single master curve. Since the temperature dependence in the exponential prefactor must be scaled out to elucidate Arrhenius behavior, the temperature-dependent conductivity described by eq 1 is said to obey a “compensated” Arrhenius equation. In this paper, we extend the scope of our earlier work. Here, the compensated Arrhenius formalism will be applied to temperature-dependent dielectric relaxation times for the following solvent groups: n-alcohols, n-alkyl bromides, n-nitriles, and n-acetates. Comparisons will then be made between conductivity and dielectric relaxation data. The calculation of dielectric relaxation properties is somewhat arbitrary. A specific model must be chosen to determine the dielectric relaxation time from the experimental data. The data are usually fit to one of the following equations:3
ε ) ε∞ +
Cole-Cole equation: ε ) ε∞ +
(εs - ε∞) 1 + iωτ
(εs - ε∞)
0eR
