The visible flame is a gas composed of several types of particles. There are whole molecules, parts of molecules (many of these are the free radicals), maybe a few isolated atoms. There are molecules (and fragments thereof) of the original fuel, of oxygen, of breakdown products from the oxidation reactions that may well go on to further reactions, etc. In fact, the flame cannot continue without all of these things in the mix.
Every chemical reaction, including oxidations and open flames, requires the input of activation energy to begin. From where? We are all accustomed to using a small flame as the source of activation energy to light something else on fire. We also use electrical sparks to do this - inside an automobile engine, or using the igniter in a backyard gas barbeque, for examples. That's because the AVERAGE energy content of a fuel molecule and an oxygen molecule is not high enough to initiate the first step of the oxidation reaction sequence. But in any collection of molecules there is a broad range of energies among the molecules. In a few cases where the activation energy required is not very high there may be a few molecules that already have enough, and we get spontaneous combustion. Or, in many cases although the original collection of molecules does not have enough, there is a small input from outside that does provide the kick necessary. For example, the infamous cases of rags soaked in very volatile petroleum materials like gasoline and stored in a place exposed to heat or sunlight - enough heat may come in to start the fire, even though we might consider that amount of heat to be too small to care about. Another example: striking a match on a rough surface - we input enough heat to start the match head chemicals to oxidize rapidly. So this still follows the principle that activation energy is needed - the questions are, how much? and from where?
I severely doubt you could create a flame from a rapid oxidation reaction of fuel with liquid oxygen if the system were so compressed that it was kept in the liquid state. That goes back to what I said in a previous post - in liquid of solid states the chain reactions necessary could not propogate because the free radicals formed would give up their excess energy too fast to their neighboring molecules, and there would be different reactions forming different products.
A flame is complex mixture of many components (as above) involved in a series of several chemical reactions. Some of these reactions produce "intermediate products" such as fragements of the original molecules which may go on to participate in further reactions as the "fuel". And some of these are free radicals which are particularly important in initiating other reactions to keep the whole system going. Eventually at the end of the chain of reactions you end up with final products like water and carbon dioxide (both gases) at high tempertures, and these hot gases are part of the heat source we try to use. But in the meantime the reactions also manage to produce heat and light by radiating them out as electromagnetic radiation of different energies; some are in a frequency (energy) range that we humans sense as light, and some that we sense as heat.
In most cases of burning fuels we see that some of the intermediate products manage to escape the flame zone without participating in subsequent oxidation reactions. So they never get all the way to water and carbon dioxide. These unburned products are a major source of air pollution. Things like PAH (polycyclic aromatic hydrocarbons), dioxins, and even the polychlorinated dioxins that are known carcinogens can be produced in small quantities in combustion reactions. But if there is a lot of this going on, the total amount can be significant. Do you know that the two largest sources, BY FAR, of dioxins in our world are from forest fires and internal combustion engines? Moreover, this type of problem is more likely to occur if the input mix contains things that will slow down the rapid reactions; that is why incineration of domestic garbage is such a problem - it contains many things that don't burn well, and lots of water to grab and steal the heat energy produced.
A flame contains temporarily what we call a dynamic equilibrium. Fuel and oxygen molecules begin a reaction sequence and produce intermediate products, some of which will be free radicals (and some won't). These intermediates are consumed in subsequent reactions in the sequence, and their supply is replaced with the fresh fuel at the start of the sequence. Heat and light are radiated out, and product molecules also contain energy which they keep (for example, the water molecules stay as gas because they have not given up enough energy to condense into liquid water). In the steady state the rate of energy loss from the flame matches the rate of energy release from the oxidation reactions. But eventualy the source of at least one reactant (fuel or oxygen) is exhausted and the flame will go out. Or, if somehow the balance of energy release and energy loss is altered (e.g. by spraying liquid water into the flame) we may see the flame extinguished because the new conditions rob the flame of the necessary concentration of high-energy intermediates to keep it going.
By the way, all this has concentrated on a fuel and oxygen as the two main reactants. There are other types of reactions that can proceed rapidly enough to look like a flame without oxygen. For example, chlorine gas will react with simple hydrocarbons (for example, petroleum products) pretty quickly; you could actually "burn" kerosene in a chamber filled with chlorine and no oxygen. The flame produces dense clouds of black smoke, too, because the chlorine reacts almost exclusively with the hydrogen atoms from the kerosene, leaving unreacted carbon. And since there is no oxygen this time, the carbon cannot burn up to carbon dioxide gas - it simply remains as tiny black carbon particles.
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