Ch.6 - Electronic Structure of Atoms

Brown - Chemistry: The Central Science 14th Edition

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Brown 14th Edition

Ch.6 - Electronic Structure of Atoms

Problem 93

Chapter 6, Problem 93

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: (a) n = 3, l = 3, m_l = 2, m_s = +1/2 (b) n = 4, l = 3, m_l = -3, m_s = +1/2 (c) n = 3, l = 1, m_l = 2, m_s = +1/2 (d) n = 5, l = 0, m_l = 0, m_s = 0 (e) n = 2, l = 1, m_l = 1, m_s = -1/2

Verified step by step guidance

Step 1: The principal quantum number (n) is a positive integer (n = 1, 2, 3, ...). In this case, n = 2, which is a valid value.

Step 2: The azimuthal quantum number (l) can have any integer value from 0 to n-1. For n = 2, the possible values of l are 0 and 1. Here, l = 1, which is a valid value.

Step 3: The magnetic quantum number (ml) can have any integer value from -l to +l. For l = 1, the possible values of ml are -1, 0, and 1. Here, ml = 1, which is a valid value.

Step 4: The spin quantum number (ms) can only have two possible values: -1/2 or +1/2. Here, ms = -12, which is not a valid value.

Step 5: Therefore, the set of quantum numbers is not valid because the spin quantum number (ms) has an invalid value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Numbers

Quantum numbers are a set of four numbers that describe the unique quantum state of an electron in an atom. They include the principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (ml), and spin quantum number (ms). Each quantum number has specific rules governing its possible values, which are essential for determining the electron's energy level and orbital shape.

Principal Quantum Number (n)

The principal quantum number (n) indicates the main energy level of an electron in an atom and can take positive integer values (1, 2, 3, ...). Higher values of n correspond to electrons that are further from the nucleus and have higher energy. In the given set, n = 2 is valid as it is a positive integer.

Spin Quantum Number (ms)

The spin quantum number (ms) describes the intrinsic spin of an electron and can have values of +1/2 or -1/2. This quantum number reflects the two possible orientations of an electron's spin within an orbital. In the provided set, ms = -12 is invalid because it does not conform to the allowed values for spin.

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Related Practice

Textbook Question

The human retina has three types of receptor cones, each sensitive to a different range of wavelengths of visible light, as shown in this figure (the colors are merely to differentiate the three curves from one another; they do not indicate the actual colors represented by each curve):

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Textbook Question

The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed.

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Textbook Question

The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (b) Calculate the wavelengths of the first three lines in the Paschen series—those for which ni = 4, 5, and 6.

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Textbook Question

Bohr's model can be used for hydrogen-like ions—ions that have only one electron, such as He⁺ and Li²⁺. (a) Why is the Bohr model applicable to He⁺ ions but not to neutral He atoms?

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