In 1885 J. J. Hood published a paper On the Influence of Heat on the Rate of Chemical Change in Philosophical Magazine in which he gave the empirical fit by an exponential function for some data on the reaction rate for the oxidation of ferrous sulfate (FeSO4) by potassic chlorate (KClO3) at various temperatures. The justification for the use of an exponential function is that the data appears to be almost linear in a semi-log plot.
Later in 1889 Svante Arrhenius published Über die Reaktionsgeschwindigkeit bei der Inversion of Rohrzucker durch Säuren (On the Reaction Rate in the Inversion of Cane Sugar by Acids) in Zeitschrift für Physicalische Chemie which gave a better fit for Hood's data based on some thermodynamics considerations. In the table on the page of the link "beob." is the observed rate of the reaction, "ber.1" is the calculated rate using formula (1) in the paper and "ber.2" gives the calculated rates for Hood's empirical formula for comparison.
Formula (1) in this paper is equivalent to the statement that the reaction rate ρ divided by e-A/T is equal to a constant which is essentially the Arrhenius equation. This is the exponential that is found in Schrödinger's equation for the 'time of expectation'. The Arrhenius constant A is now written as activation energy divided by the gas constant R, the constant from the ideal gas law, PV = nRT. The k in Schrödinger's equation is the corresponding constant for a molecule. His 'time of expectation' is inversely proportional to the reaction rate.
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