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Chapter 7: Problem 58

Describe the shapes of \(s, p,\) and \(d\) orbitals. How are these orbitals related to the quantum numbers \(n, \ell\), and \(m_{\ell} ?\)

Short Answer

Expert verified
The s, p, and d orbitals have spherical, dumbbell, and complex shapes respectively. The shape and number of each type of orbital is related to the quantum numbers. The azimuthal quantum number (l) categorizes the orbitals as s (for l=0), p (for l=1), and d (for l=2). The magnetic quantum number (m_l) denotes different orientations of the orbitals, with ranges from -l to +l, which is how there are multiple p and d orbitals per energy level.

Step by step solution

01

Title: Description of the shapes of s, p, and d orbitals

s, p, and d refer to the shapes of the electron cloud, often referred to as atomic orbitals. The s orbital is spherical and it is the simplest type of orbital. p orbitals are denser on two sides of a central line, forming a dumbbell shape. There are three different orientations of p-orbitals. d orbitals are complex, consisting of four cloverleaf-shaped orbitals and a dumbbell within a torus.
02

Title: Understanding Quantum Numbers

Quantum numbers describe the unique quantum state of an electron. The three quantum numbers in question are the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m_l). The quantum number 'n' is a positive integer that denotes the energy level or shell of an electron in an atom. The quantum number 'l' describes the shape of the orbital and has values ranging from 0 to (n-1), where l=0 corresponds to s orbitals, l=1 to p orbitals, and l=2 to d orbitals. The quantum number 'm_l' describes the orientation of the orbital in space and ranges from -l to +l.
03

Title: Relating Orbitals and Quantum Numbers

For an s orbital, l=0 and therefore m_l can only be 0. Hence, there is only one s orbital per energy level. p orbitals have l=1, and therefore m_l can have the values -1, 0, +1. This means there are three p orbitals per energy level. For d orbitals, l=2, and m_l can have the values -2, -1, 0, +1, +2. This means there are five d orbitals per energy level.

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Most popular questions from this chapter

Consider the following energy levels of a hypothetical atom: \(E_{4}\) \(-1.0 \times 10^{-19} \mathrm{~J}\) \(E_{3}\) \(--5.0 \times 10^{-19} \mathrm{~J}\) \(E_{2}\) \(--10 \times 10^{-19} \mathrm{~J}\) \(E_{1}\) \(-15 \times 10^{-19} \mathrm{~J}\) (a) What is the wavelength of the photon needed to excite an electron from \(E_{1}\) to \(E_{4} ?\) (b) What is the energy (in joules) a photon must have in order to excite an electron from \(E_{2}\) to \(E_{3} ?\) (c) When an electron drops from the \(E_{3}\) level to the \(E_{1}\) level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

What is the maximum number of electrons in an atom that can have the following quantum numbers? Specify the orbitals in which the electrons would be found. (a) \(n=2, m_{\mathrm{s}}=+\frac{1}{2}\) (b) \(n=4, m_{e}=+1\) (c) \(n=3, \ell=2 ;\) (d) \(n=2, \ell=0, m_{\mathrm{s}}=-\frac{1}{2} ;\) (e) \(n=4\) \(\ell=3, m_{\ell}=-2\)

Alveoli are the tiny sacs of air in the lungs (see Problem 5.136 ) whose average diameter is \(5.0 \times\) \(10^{-5} \mathrm{~m} .\) Consider an oxygen molecule \(\left(5.3 \times 10^{-26} \mathrm{~kg}\right)\) trapped within a sac. Calculate the uncertainty in the velocity of the oxygen molecule. (Hint: The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)

Spectral lines of the Lyman and Balmer series do not overlap. Verify this statement by calculating the longest wavelength associated with the Lyman series and the shortest wavelength associated with the Balmer series (in \(\mathrm{nm}\) ).

Calculate the total number of electrons that can occupy (a) one \(s\) orbital, (b) three \(p\) orbitals, (c) five \(d\) orbitals, (d) seven \(f\) orbitals.
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