Temperature Gradients in Reacting Systems

A study of the heat conduction equation for reactions going on in a spherical flask leads to a calculation of three parameters, τ the mean lifetime for the establishment of stationary states, ΔTm the maximum temperature difference between the walls of the vessel and the reacting gas, and the average temperature difference 〈ΔTm〉Av. These are given by τ=ρcvr02π2K, ΔTm=RHr026K, 〈ΔTm〉Av=RHr0215K in which ρ is the density, cv the specific heat at constant volume, r0 the radius of the vessel, K the coefficient of thermal conductivity, R the specific rate of reaction, and H the heat of reaction. In liquid systems under usual conditions τ is of the order of r02 minutes (r0 in cm) while 〈ΔTm〉Av is about ±0.2°C (for H = ±10 kcal, half-life = 1 hour, r0 = 5 cm) so that such effects are usually negligible. In gases on the other hand 〈ΔTm〉Av can be of the order of magnitude of 2°C or higher depending on the various conditions, resulting in significant errors in observed rate constants and activation energies. The effects of added inert gases, pressure, vessel packing, and other parameters are discussed. A very crude treatment of convection shows that it is not expected to alter appreciably the magnitude of 〈ΔTm〉Av.