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

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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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