Question

What values of ${m_l}$ are permitted for an electron with $l = 3$?

Answer 1

There are four types of quantum numbers in atomic physics namely, principal quantum number $(n)$, Azimuthal quantum number $(l)$, magnetic quantum number $({m_l})$ and electron spin quantum number $({m_s})$. Magnetic orbital quantum number, ${m_l}$ tells us about the orientation of the orbital with respect to the standard set of coordinate axes.

Complete answer:
The value of ${m_l}$ tells us about the number of ways in which the orbitals can be oriented. For a given value of $l$, ${m_l}$ has the same number of orbitals per subshell. Hence, we can say that the number of orbitals is equal to the number of ways in which they are oriented.
The values that the magnetic quantum number shows depend upon the value of angular quantum number and it is described by the following relation:
${m_l} = (2l + 1)$
In the question, the value of $l$ is equal to $3$. Hence, the total magnetic quantum number value will be,
${m_l} = (2 \times 3 + 1)$
${m_l} = 7$
Hence, there are $7$ values of magnetic quantum number. The permitted value ${m_l}$ is dependent on the value of angular momentum number in the following manner:
${m_l} = \\{ - l, - (l - 1), - (l - 2),...., - 1,0,1,....,(l - 2),(l - 1),l\\}$
Since, $l = 3$, therefore ${m_l}$ will have the following values:
${m_l} = \\{ - 3, - 2, - 1,0,1,2,3\\}$
Here, the value of $l$ describes the $f$ subshell. The value of ${m_l}$ tells us that this $f$ subshell can hold a total of $7$ orbitals.

Note:
The Azimuthal quantum number $(l)$ is also called orbital angular quantum number or subsidiary quantum number. The value of $l$ also identifies the subshell and also determines its shape. For $l = 0$, there will be one $s$orbital. For $l = 1$, there will be three $p$orbitals. For $l = 2$, there will be five $d$orbitals and for $l = 3$, there will be seven $f$orbitals and so on.