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Electrons get pulled in when there is a higher positive charge (there are more protons) which is why they get smaller. This explains why cations are smaller than anions. Cations have lost some amount of electrons but still have the same number of protons as the neutral atom (this is why it's still the same element as the neutral atom) so less electrons means that the nucleus has less electrons to hold on to, which causes the remaining electrons to be pulled in further by the protons and decreasing the atomic radius. Anions are bigger for the opposite reason, they gain electrons so the protons have more electrons to hold on to than they would in the neutral atom which causes the electrons to be more spread out creating a larger atomic radius.
Cations are smaller than anions (and the neutral atoms) because when they lose electrons, electron-electron repulsion decreases and, therefore, the nucleus' positive charge pulls the remaining electrons inwards, towards the nucleus, decreasing the radius of the atom. Vice versa happens with anions, because as they gain electrons, shielding and electron-electron repulsion increases, increasing the atom's radius.
Essentially the number of protons and electrons equal each other in a neutral atom. And after a neutral atom loses a certain amount of electrons, this new cation now has less electrons in comparison to protons. Now that the positive charge protons outnumber the e-, the e- are pulled in tighter to the nucleus.
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