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Bond formation is possible when the resulting entity has an energy less than the sum of individual energies of the constituents. That is, when a bond forms , we can immediately say that the particular system has become stabilized. For example, consider the reaction

A + B = AB

Here, E(AB) < E(A) + E(B).

If, the bond formed is covalent, it'll mean that now A and B are sharing electrons between them (1 pair or 2 pairs, etc).

If, the bond is ionic , it'll mean that one of A or B has transfered an electron to the other and now the duo is electrostatically bound.

With respect to chemical reactions, bond formation can produce a lot of changes :

-color change due to the property of the newly formed compound

-one may observe bubbles

-heating of the beaker/test tube (in case of exothermic reactions)

-pH change, etc

If you're interested in thermodynamics , it's worthwhile to know, that during the progress of a reaction (bond dissociation or formation, etc) the Gibb's free energy is bound to decrease. In fact, the direction of reaction is decided based on G (i.e, direction in which deltaG is -ve)

And, yes if a new bond forms in a chemical reaction, ne might even end up with an explosion .. (in case of highly exothermic reactions like combustion,etc)

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