Question

For the azimuthal quantum number $ l = 3 $ , the total number of the different possible values of the magnetic azimuthal quantum number $ {m_l} $ is:
(A) 3
(B) 4
(C) 5
(D) 7

Answer 1

Hint: We know that the atoms of the periodic table possess unique atomic arrangement and these arrangements are the basis of which they are classified. The arrangements are denoted by unique quantum numbers of each atom. Each of the four quantum numbers are interdependent and denote unique specifications of the atoms.

Formulas used: $ {m_l} = + l{\text{ to }} - l $

Complete Step by Step answer
We know that, in chemistry at the atomic level, quantum numbers are values of quantities that describe the quantum system and its arrangement. They are basically used to find the energy levels of electrons in a specific atom and its stability in atomic level. There are many different quantum states and hence quantum numbers. However, to describe an atom or specifically an electron in an atom, 4 quantum numbers can be used: Principal quantum number $ \left( n \right) $ ,Azimuthal quantum number $ \left( l \right) $ ,Magnetic quantum number $ \left( {{m_l}} \right) $ ,Spin quantum number $ \left( {{m_s}} \right) $ .
Each of these quantum numbers are interdependent. For example, consider an element with electronic configuration $ 1{s^2}2{s^2}2{p^6}3{s^2}3{p^6}4{s^2}3{d^5} $ which is the electronic configuration of Manganese. In this each aspect describes a quantum number. The $ s,p,d $ denotes subshells of an atom which is characterized by the azimuthal quantum number. The $ s $ subshell has azimuthal quantum number $ l = 0 $ , the p subshell has $ l = 1 $ and so on.
This azimuthal quantum number ranges from, $ l = 0 $ to $ l = n - 1 $ . Similarly, the magnetic quantum number values range from $ {m_l} = - l $ to $ {m_l} = + l $ .
Now we have an electron with azimuthal quantum number $ l = 3 $ we are supposed to find the maximum number of magnetic quantum number values possible. We know, the magnetic quantum number values range from, $ {m_l} = - l $ to $ {m_l} = + l $ . Thus for $ l = 3 $ we have the values of $ {m_l} = - 3, - 2, - 1,0, + 1, + 2, + 3 $ which makes a total of 7 values.
Thus, for azimuthal quantum number $ l = 3 $ there can be up to a maximum of $ 7 $ magnetic quantum number values.

Note
Alternatively, we can also see that the magnetic quantum number specifies the orientation of an orbital in space. The $ {m_l} $ divides the subshells of an electron into individual orbitals that can hold the electrons. We know that there are $ 2l + 1 $ orbitals in each subshell which denotes the magnetic quantum number. Thus, there will be $ 2l + 1 $ values of magnetic quantum number for a given value of $ l $ .
Thus for $ l = 3 $ , the number of $ {m_l} $ will be $ 2\left( 3 \right) + 1 = 7 $ .