2.9 The Arrhenius Equation

A petrologic example of the effect of temperature on rates is the reaction

CaCO3 (Calcite) + SiO2 (Quartz) = CaSiO3 (Wollastonite) + CO2

which may occur in marbles during contact metamorphism. Stephen Kridelbaugh (1973) studied this reaction using natural quartz and calcite in "cold-seal" (Tuttle, 1949) pressure vessels at CO2 pressures (PCO2 = Ptotal) ranging from 5000 to 25,000 psi (345 bars to 1724 bars). Reaction rate data for his 5000 psi experiments are shown in Figure 15. The temperatures of the experiments range from 800°C to 950°C, well above the equilibrium temperature (600°C) for this reaction at 5000 psi. Notice that the rate constant value increases significantly over the temperature range of the experiments.

Laboratory experiments like those shown in Figure 15 have demonstrated that the rates of many petrologic processes, such as chemical reactions, nucleation, crystal growth, and diffusion, depend exponentially on temperature. The variation with temperature (T) of rate constants (k) for these processes are typically given in terms of the Arrhenius equation, developed by the Swedish chemist Svante Arrhenius in 1899:

k = k0 exp [-Ea / (RT)]

where k0 is a constant, R is the gas constant, and Ea is a parameter he called the activation energy. Notice that the exponential term is negative. As temperature rises, the rate k increases. By taking the logarithm of both sides, the Arrhenius equation can be arranged to read:

ln [k] = ln [k0] - [Ea / R] [1 / T]

which is the equation of a straight line on a graph of ln [k] vs. [1/T], with a slope of
[-Ea / R]. Figure 16 above shows the Kridelbaugh (1973) data of Figure 15 on a logarithmic (Arrhenius) diagram.

The denominator of the exponential term in the Arrhenius equation is RT, which is proportional to the average kinetic energy of a gas (see Figure 14b). The numerator is called the activation energy based on the idea that atoms must have a minimum kinetic energy to participate in the reaction. For the Maxwell-Boltzmann distribution (Figure 14b), the fraction of atoms above any selected minimum value increases exponentially with temperature. You can read more about the Arrhenius equation here.

The rates of many petrological processes are limited by atom movements called diffusion. Diffusion data are well described by the Arrhenius equation.