How can a decomposition reaction be exothermic?

Decomposition reactions involve breaking of bonds which requires energy. Therefore, they are generally endothermic. 


But how can decomposition reactions release energy? Examples are the respiration reaction or the decomposition of vegetable matter.




It just can be so. Look for something that is favored by cooling instead of heating.


In the case of decomposition of something to pure elements that is exceedingly rare, but decomposition involving other compounds offers more possibilities. During hot processing of steel (in the hot rolled coil, actually) we can get this reaction which is favored by cooling below 570°C:


Wustite --> Magnetite + Iron


$4 \ce{Fe\_{1-x}O} \rightarrow \ce{Fe\_3O\_4}+(1-4x)\ce{Fe}$


$x$ typically 0.05-0.10.


See the phase diagram [here](https://www.researchgate.net/figure/Iron-oxygen-phase-diagram-17_fig6_303497669) noting what happens to the $\ce{Fe\_{1-x}O}$ phase on cooling.


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**You have to consider the whole reaction not just the initial step**


The problem with your first assumption is that, while often partially correct, it is an incomplete view of any chemical reaction. Yes, many–if not most–reactions start with some bonds breaking (which requires energy) but that is not the whole picture.


Consider the breakdown of nitrogen tri-iodide. This decomposition involves breaking a bunch of N-I bonds each of which requires some energy input. But, as many chemical students have witnessed, the input energy to kick of the explosive decomposition is low. A touch from a feather will trigger it. But the overall reaction releases a fair amount of energy, which is why it goes BANG and is an incredibly dangerous substance to work with.


Why this occurs is that the ultimate products that form (dinitrogen and molecular iodine as I2) *release* a lot of energy when they form, especially the N2. The reaction is not over when the N-I bond breaks but only when the intermediate products have reformed into stable compounds. And, if we tot up the *net* movement of energy we find that energy has been released.


The problem with looking at just the initial step is that this only tells us about the energy barrier to *starting* the reaction, not the overall amount of energy involved. In simple reactions this would be the *activation energy* barrier (the energy input required to kick things off) not the overall amount of energy involved once the reaction goes to completion.


Plenty of real world reactions need a lot of energy input to start but release even more once they get going. Burning dirty carbon in the form of coal, for example, takes a lot of effort to get going but was the primary way of providing domestic and industrial heat for many people for a century.


In short, for any reaction, you need to consider the *whole* reaction not just the first step to understand how much energy is involved.


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