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I read about the periodic properties — ionisation energy and electron affinity — in my book. It was quite intuitive that energy is required to remove an electron from the atom but I wonder why energy is released when an electron is added to an atom? Is it actually released or is it because of our convention that energy of electron at infinite distance from nucleus is 0, that we consider it to be released?

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It may help to take a look at the two processes you are comparing in terms of reactions. Ionisation follows the following reaction:

$$\ce{A ->[][$\Delta E_\mathrm{i}$] A+ + e-}\tag{1}$$

This process involves tearing charges apart, separating a positive and a negative one. Even if the original atom was somewhat unstable, it will be unfavourable to remove that electron simply due to electrostatic interaction. You write in the question that you were able to deduce this already.

The electron affinity on the other hand describes the process in this following reaction:

$$\ce{A + e- ->[][EA] A-}\tag{2}$$

Here, a charged particle is approaching a neutral particle and we have the same number of charged particles (and the same charges) on both sides of the equation. This means that we don’t have a strong, obvious physical effect that will make all electron affinities sign-equal for very simple reasons. Instead, we need to consider the details more.

The process often involves a release of energy because most atomic configurations have space for extra electrons at little or no extra cost. Except for the noble gases and the second group, atoms don’t have completely populated sub-shells so there should always be a position available in a subshell at an acceptable exothermic energy level. While adding an electron will increase electron-electron repulsion and increase all energy levels by a certain amount, this increase is typically less than the first ionisation energy which corresponds to the energy level of the highest occupied atomic orbital. Thus in electron affinity, we in principle gain the entire ionisation enthalpy minus the destabilising contribution of the new electron. (Caution! This is a very simplified picture and not suitable for generalisation or even strictly for explanation. It gets the basic point across though.)

Therefore, most electron affinities are exothermic with the notable exceptions of group 2, group 18 and nitrogen.

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Energy has to be released when electrons are added to atoms in most cases.

Otherwise, why would they ever form covalent bonds?

The whole point of a covalent bond is that electrons are stabilized by the attraction to multiple nuclei, which helps to hold the system together. Therefore, the system must be more stable when you add an electron to an atom.

Note that in the case of something like a neon atom, where adding an electron is not favorable is not the case, we don't form covalent bonds.

From the atomic perspective, even for a neutral atom, the nuclear charge is not completely shielded by all of the electrons. Some electrons are going to be on the other side of the nucleus from the new electron you're adding.

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This is because potential energy is more in this case when electron is further from nucleus as the forces between them are attractive. As the distance is decreased between nucleus and electron, potential energy decreases. Since energy is always conserved, the net decrease in potential energy is released as any form of energy. However the energy is not released in all the cases. It depends on the net effect of the interactions of the electron with the nucleus and the electrons that are already present in the atom.

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