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Magnetic Quantum Number

27th Jan 2020 @ 3 min read

Physical Chemistry

The magnetic quantum number is a set of integers that determine the spatial orientation of an orbital. It defines the orbital and is unique to each orbital for a given value of the azimuthal quantum number. It is symbolized as ml. m stands for magnetic and the subscript l for azimuthal. ml = … −2, −1, 0, 1, 2…

The number along with the principal quantum number, azimuthal quantum number, and spin quantum number is used to identify each and every electron in an atom. It was introduced by Arnold Sommerfeld, a German theoretical physicist, who also proposed the azimuthal quantum number.

It determines the direction of the orbital angular momentum. The magnetic quantum number does not decide the energy of an orbital. However, it affects the energy of an orbital in the presence of an external static magnetic field. The magnetic quantum number together with the spin quantum number contributes to the magnetic moment of an electron.

The values of the magnetic quantum number are restricted by the azimuthal quantum number l. For a given value of l, we can have 2l + 1 possible values of ml, ranging from −l, −(l − 1) … −2, −1, 0, 1, 2 … (l − 1), l.

For l = 0, ml = 0. This represents an s orbital, which is spherical in shape. For an s orbital, the orbital angular momentum is zero and it does not point in any specific directions.

s orbital
s orbital

For l = 1, ml = {−1, 0, 1}. Each value represents a p orbital. When ml = 0, it is the pz orbital. It is aligned along the z-direction having each lobe on either of the origin. ml = ±1 are the px and py orbitals. px and py rest on x- and y-axes.

p orbitals
p orbitals

l = 2 is the d subshell and it has five orbitals. The magnetic quantum numbers for d subshell are −2, −1, 0, 1, and 2. ml = 0 is the dz2 orbital, which is oriented along the z-axis. The orbitals dxz and dyz have ml = ±1 and lie in the xz and yz planes. ml = ±2 corresponds to dxy and dx2 − y2; both lie in the xy plane.

d orbital
d orbitals

l for f subshell is 3 and its has seven orbitals: fz3, fxz2, fyz2, fxyz, fz(x2 − y2), fx(x2 − 3y2), and fy(3x2 − y2). And the magnetic quantum number associated with these orbitals are −3, −2, −1, 0, 1, 2, and 3.

f orbital
f orbitals

Similarly for g orbital, ml = −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, and 5.

Each orbital can hold utmost two electrons. Thus, every ml can have two electrons; however, it cannot distinguish those two electrons. They can only be identified by the spin quantum number.

The magnetic quantum number ml depends on the azimuthal quantum number l, which relies on the principal quantum number n. We can establish the relation between ml and n. For every value of n, there can be n values of l ranging from 0, 1, 2, 3 … n − 2, n − 1.

n^2

Therefore, n2 values of ml are possible for every n.

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Quantum Number Atomic Structure

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Kalyani
27th Mar 2020
For px and py, how do we decide which one will have 1 or −1 value? Is there any order? Reply: No, there is no order. For pz, it is clear: ml = 0. But for px and py, it is wrong to assign 1 or −1 to anyone.


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