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Chapter 7: Problem 50

Which of the four quantum numbers \(\left(n, \ell, m_{\ell}, m_{s}\right)\) determine (a) the energy of an electron in a hydrogen atom and in a many- electron atom, (b) the size of an orbital, (c) the shape of an orbital, (d) the orientation of an orbital in space?

Short Answer

Expert verified
(a) Energy: \(n\) for hydrogen atom and \(n,\ell\) for multi-electron atoms. (b) Size: \(n\). (c) Shape: \(\ell\). (d) Orientation: \(m_{\ell}\).

Step by step solution

01

Understanding the quantum numbers

First, a brief review of what each quantum number indicates. The principal quantum number (n) indicates the energy level the electron is in, and it has a positive value of integer type. The azimuthal quantum number \(\ell\) (also known as the orbital quantum number or angular momentum quantum number) describes the shape of the orbital. The magnetic quantum number \(m_{\ell}\) tells us the orientation of the orbital in space, and the spin quantum number \(m_s\) indicates the direction of the electron spin.
02

Identifying quantum numbers related to energy level

(a) In a hydrogen atom, the energy of an electron is determined solely by the principal quantum number \(n\). In a multi-electron atom, the energy of an electron is determined by both the principal quantum number \(n\) and azimuthal quantum number \(\ell\). The energy increases with an increase in \(n\) and \(\ell\).
03

Identifying quantum number related to the size of an orbital

(b) The size of an orbital is determined by the principal quantum number \(n\). As \(n\) increases, the electron is greater the distance from the nucleus, so the size of the orbital increases.
04

Identifying quantum number related to the shape of an orbital

(c) The shape of an orbital is determined by the azimuthal quantum number \(\ell\). Different values of \(\ell\) correspond to different shapes (s, p, d, f).
05

Identifying quantum number related to the orientation of an orbital in space

(d) The orientation of an orbital in space is determined by the magnetic quantum number \(m_{\ell}\). Different values of \(m_{\ell}\) correspond to different orientations of the same shape in space.

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