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Chapter 6: Problem 56

How many unique combinations of the quantum numbers \(l\) and \(m_{l}\) are there when (a) \(n=3,\) (b) \(n=4 ?\)

Short Answer

Expert verified
For \(n=3\), there are 9 unique combinations of \(l\) and \(m_l\), and for \(n=4\), there are 16 unique combinations of \(l\) and \(m_l\).

Step by step solution

01

Finding unique combinations for n = 3

1. Identify the possible values of l: For n = 3, the possible values for l = 0, 1, 2. 2. Identify the possible values of m_l for each value of l: - For l = 0: m_l can only be 0 (1 value). - For l = 1: m_l = -1, 0, 1 (3 values). - For l = 2: m_l = -2, -1, 0, 1, 2 (5 values). 3. Calculate the total number of unique combinations: Sum of possible m_l values: 1 + 3 + 5 = 9 unique combinations
02

Finding unique combinations for n = 4

1. Identify the possible values of l: For n = 4, the possible values for l = 0, 1, 2, 3. 2. Identify the possible values of m_l for each value of l: - For l = 0: m_l can only be 0 (1 value). - For l = 1: m_l = -1, 0, 1 (3 values). - For l = 2: m_l = -2, -1, 0, 1, 2 (5 values). - For l = 3: m_l = -3, -2, -1, 0, 1, 2, 3 (7 values). 3. Calculate the total number of unique combinations: Sum of possible m_l values: 1 + 3 + 5 + 7 = 16 unique combinations Thus, for n = 3, there are 9 unique combinations of l and m_l, and for n = 4, there are 16 unique combinations of l and m_l.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Number n
Quantum number n, also known as the principal quantum number, plays a pivotal role in determining the energy level of an electron in an atom. It is the first of a set of quantum numbers which describe the unique state of an electron and can take on any positive integer value. The higher the value of n, the greater the energy and the farther away the electron is from the nucleus, residing in a higher energy level or shell. For instance, when n equals 1, the electron is in the closest orbit to the nucleus, which is the lowest energy state.
Quantum Number l
The quantum number l, known as the azimuthal or angular momentum quantum number, defines the shape of an electron's orbital. It can take on any integer value from 0 to n-1, where n is the principal quantum number. The values of l are often associated with specific orbital shapes; for l=0, we have s-orbitals which are spherical, p-orbitals with l=1 are dumbbell-shaped, d-orbitals for l=2 are more complex, and f-orbitals for l=3 are even more intricate. Thus, the possible values of l provide a means of identifying the type of orbital an electron occupies.
Magnetic Quantum Number ml
The magnetic quantum number ml further refines the description of an electron's position within an orbital by indicating its orientation in space. This quantum number can take on any integer value between -l and +l, including zero. Each value of ml corresponds to a specific orientation of an orbital, and thus a different magnetic behavior under an external magnetic field. For each value of l, there are 2l +1 possible ml values, which means the number of allowed orientations exponentially increases with higher l values.
Unique Combinations of Quantum Numbers
The unique combinations of quantum numbers define the unique state an electron can occupy within an atom. These combinations arise from the possible values of n, l, and ml for each electron. An electron is described by its set of quantum numbers, with no two electrons in the same atom having the same set (in accordance with Pauli's exclusion principle). The step-by-step solution to the given problem illustrates how, for a given value of n, the possible combinations of l and ml can be determined and summed up to find the total number of unique combinations available for electrons to occupy.
Atomic Orbitals
Atomic orbitals are regions in space around an atom's nucleus where the probability of finding an electron is highest. Orbitals are not fixed paths, but rather fuzzy clouds where electrons are likely to be found. They are defined by the quantum numbers n, l, and ml, with each set corresponding to a particular orbital shape and orientation. The principal quantum number n determines the shell of the orbital, l its shape, and ml its orientation. For example, an electron in a p-orbital (l=1) could be found in one of three p-orbitals, each oriented differently in space, as defined by the three possible ml values (-1, 0, and +1) at this level.

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Most popular questions from this chapter

(a) For \(n=4,\) what are the possible values of \(l ?(\mathbf{b})\) For \(l=2\) what are the possible values of \(m_{l} ?(\mathbf{c})\) If \(m_{l}\) is \(2,\) what are the possible values for \(l\) ?

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=6,(\mathbf{b})\) from an orbit of radius 4.76\(\hat{\mathrm{A}}\) to one of radius \(0.529 \mathrm{A},(\mathbf{c})\) from the \(n=6\) to the \(n=9\) state.

The following electron configurations represent excited states. Identify the element and write its ground-state condensed electron configuration. (a) \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1},\) (b) \([\) Ar \(] 4 s^{1} 3 d^{10} 4 p^{2} 5 p^{1}\) \((\mathbf{c})[\mathrm{Kr}] 5 s^{2} 4 d^{2} 5 p^{1}\)

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths in the range of \(320-380 \mathrm{nm},\) whereas UV-B radiation has wavelengths in the range of \(290-320 \mathrm{nm} .\) (a) Calculate the frequency of light that has a wavelength of 320 \(\mathrm{nm}\) . (b) Calculate the energy of a mole of 320 -nm photons. (c) Which are more energetic, photons of UV-A radiation or photons of UV-B radiation? (d) The UV-B radiation from the Sun is considered a greater cause of sunburn in humans than is UV-A radiation. Is this observation consistent with your answer to part (c)?

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: $$ \begin{array}{l}{\text { (a) } n=4, l=1, m_{l}=2, m_{s}=-\frac{1}{2}} \\\ {\text { (b) } n=4, l=3, m_{l}=-3, m_{s}=+\frac{1}{2}}\\\\{\text { (c) } n=3, l=2, m_{l}=-1, m_{s}=+\frac{1}{2}} \\ {\text { (d) } n=5, l=0, m_{l}=0, m_{s}=0} \\ {\text { (e) } n=2, l=2, m_{l}=1, m_{s}=+\frac{1}{2}}\end{array} $$
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