Which of the following statements is (are) correct for an electron with \(n=4\) and \(m_{\ell}=2 ?\) Explain. (a) The electron is in the fourth principal shell. (b) The electron may be in a \(d\) orbital. (c) The electron may be in a \(p\) orbital. (d) The electron must have \(m_{s}=+\frac{1}{2}\)

Quantum numbers are numbers assigned to atomic orbitals that describe various properties of the orbitals and the electrons they contain. Here in this exercise, three quantum numbers are mentioned: n, m_l, and m_s.\nThe principal quantum number n describes the energy level of an electron, the apparent 'shell' the electron is in.\nThe magnetic quantum number m_l can take on any integer values between -n and +n, and it tells us about the orientation of an orbital.\nThe spin quantum number m_s, however, only has two possible values: +1/2 or -1/2, representing the two possible spin states of an electron: spin up and spin down.

(a) The electron is in the fourth principal shell. This statement is correct. Here, n=4, which means that the electron is located in the fourth principal shell (or energy level).\n(b) The electron may be in a d orbital. This is correct as well. m_l could take values from -n to +n. For n=4, m_l could be from -4 to +4. The value given here, m_l=2, indicates that the electron could be in a d orbital because d orbitals correspond to m_l values from -2 through +2.\n(c) The electron may be in a p orbital. This statement is incorrect. p orbitals correspond to m_l values of -1, 0, and +1. Since m_l=2, it means the electron cannot be in a p orbital.\n(d) The electron must have m_s=+1/2. This statement is incorrect. The spin quantum number m_s is independent of the values of n and m_l. Regardless of its principal quantum number or its magnetic quantum number, an electron may have a spin quantum number of either +1/2 (spin up) or -1/2 (spin down). These spin states are not determined by the value of n or m_l.

After analyzing each statement, it can be concluded that Statements A and B are correct, while Statements C and D are incorrect. These conclusions are based on the definition and implications of the principal quantum number (n), the magnetic quantum number (m_l), and the spin quantum number (m_s).