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An orbital is a one-electron wavefunction, usually derived by solving the Schrodinger equation. This tag applies to questions about all forms of orbitals; additionally, questions about the construction and properties of molecular orbitals should be tagged with [molecular-orbital-theory].

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What is the basis for naming the p-orbitals with reference to the Cartesian coordinate syste...

EDIT: Since you seem to have an issue here, I'll mention this as well: Quoting from chemguide.co.uk A) $\mathrm{d_{xy}}$, $\mathrm{d_{yz}}$, $\mathrm{d_{zx}}$ The names tell you that these orbitals … B) $\mathrm{d_{x^2 - y^2}}$, $\mathrm{d_{z^2}}$ Although these two orbitals look totally different, what they have in common is that their lobes point along the various axes. …
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Explain!"An orbital of smaller size means there is more chance of finding an electron near t...

Of course there are variations in shape [spherical(s-orbitals), dumbbells (p-orbitals), Double dumbbells (d-orbitals), etc] and size depending on which orbital, sub-shell and shell/energy-level you're … Chances of finding electrons outside of the orbitals is pretty slim. Have a look at this diagram on atomic orbitals: Source. I found this via a Google Images search. …
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