##### How to determine quantum numbers

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how to determine quantum numbers

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The three coordinates that come from Schrodinger's wave equations are the principal (*n*),
angular (*l*), and magnetic (*m*) quantum numbers. These quantum numbers
describe the size, shape, and orientation in space of the orbitals on an atom.

The **principal quantum number** (*n*) describes the size of the orbital.
Orbitals for which *n* = 2 are larger than those for which *n* = 1, for example.
Because they have opposite electrical charges, electrons are attracted to the nucleus of
the atom. Energy must therefore be absorbed to excite an electron from an orbital in which
the electron is close to the nucleus (*n* = 1) into an orbital in which it is further
from the nucleus (*n* = 2). The principal quantum number therefore indirectly
describes the energy of an orbital.

The **angular quantum number** (*l*) describes the shape of the orbital.
Orbitals have shapes that are best described as spherical (*l* = 0), polar (*l*
= 1), or cloverleaf (*l* = 2). They can even take on more complex shapes as the value
of the angular quantum number becomes larger.

There is only one way in which a sphere (*l* = 0) can be oriented in space.
Orbitals that have polar (*l* = 1) or cloverleaf (*l* = 2) shapes, however, can
point in different directions. We therefore need a third quantum number, known as the **magnetic
quantum number** (*m*), to describe the orientation in space of a particular
orbital. (It is called the *magnetic* quantum number because the effect of different
orientations of orbitals was first observed in the presence of a magnetic field.)

**Rules Governing the Allowed Combinations of
Quantum Numbers**

- The three quantum numbers (
*n*,*l*, and*m*) that describe an orbital are integers: 0, 1, 2, 3, and so on. - The principal quantum number (
*n*) cannot be zero. The allowed values of*n*are therefore 1, 2, 3, 4, and so on. - The angular quantum number (
*l*) can be any integer between 0 and*n*- 1. If*n*= 3, for example,*l*can be either 0, 1, or 2. - The magnetic quantum number (
*m*) can be any integer between -*l*and +*l*. If*l*= 2,*m*can be either -2, -1, 0, +1, or +2.

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