During the oxidation of hydrocarbons using hydrogen peroxide solutions, the evolution of gaseous oxygen is a side and undesirable process, in which the consumption of the oxidizer is not associated with the formation of target products. Therefore, no attention is paid to the systematic study of the chemical composition of the gas and the mechanisms of its formation. Filling this gap, the authors discovered a number of new, previously unidentified, interesting facts concerning both gas evolution and the oxidation of hydrocarbons. In a 33% H2O2/Cu2Cl4·2DMG/CH3CN system, where DMG is dimethylglyoxime (Butane-2,3-dione dioxime), and is at 50 °C, evidence of significant evolution of gaseous hydrogen, along with the evolution of gaseous oxygen was found. In the authors’ opinion, which requires additional verification, the ratio of gaseous hydrogen and oxygen in the discussed catalytic system can reach up to 1:1. The conditions in which only gaseous oxygen is formed are selected. Using a number of oxidizable hydrocarbons with the first adiabatic ionization potentials (AIPs) of a wide range of values, it was found that the first stage of such a process of evolving only gaseous oxygen was the single electron transfer from hydrogen peroxide molecules to trinuclear copper clusters with the formation, respectively, of hydrogen peroxide radical cations H2O2•+ and radical anions Cu3Cl5•− (AIP = 5 eV). When the conditions for the implementation of such a single electron transfer mechanism are exhausted, the channel of decomposition of hydrogen peroxide molecules into gaseous hydrogen and oxygen is switched on, which is accompanied by the transition of the system to an oscillatory mode of gas evolution. In some cases, the formation of additional amounts of gaseous products is provided by the catalytically activated decomposition of water molecules into hydrogen and oxygen after the complete consumption of hydrogen peroxide molecules in the reaction of gaseous oxygen evolution. The adiabatic electron affinity of various forms of copper molecules involved in chemical processes is calculated by the density functional theory method.