Ch.7 - Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 4th Edition

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Tro 4th Edition

Ch.7 - Quantum-Mechanical Model of the Atom

Problem 62

Chapter 7, Problem 62

Which combinations of n and l represent real orbitals, and which do not exist? a. 1s b. 2p c. 4s d. 2d

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Identify the principal quantum number (n) and the azimuthal quantum number (l) for each orbital. The principal quantum number, n, determines the energy level of the electron in the atom, and l determines the shape of the orbital. For s orbitals, l=0; for p orbitals, l=1; for d orbitals, l=2; and for f orbitals, l=3.

Check the validity of each combination by ensuring that the value of l is less than n. This is because the azimuthal quantum number l can range from 0 to n-1 for a given principal quantum number n.

Analyze each option: a. 1s has n=1 and l=0, which is valid as l < n. b. 2p has n=2 and l=1, which is valid as l < n. c. 4s has n=4 and l=0, which is valid as l < n.

Examine option d: 2d has n=2 and l=2. Check if this is valid by comparing l and n. Since l is not less than n, this combination does not represent a real orbital.

Conclude which combinations represent real orbitals and which do not. In this case, 1s, 2p, and 4s are valid combinations representing real orbitals, while 2d does not exist as a real orbital.

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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 numerical values that describe the unique quantum state of an electron in an atom. The principal quantum number (n) indicates the energy level, while the azimuthal quantum number (l) defines the shape of the orbital. Valid combinations of n and l must satisfy the condition that l can take on integer values from 0 to n-1.

Orbital Types

Orbitals are regions in an atom where there is a high probability of finding electrons. Each type of orbital is designated by a letter: s (l=0), p (l=1), d (l=2), and f (l=3). The type of orbital is determined by the value of the azimuthal quantum number (l), which must be less than the principal quantum number (n) for the orbital to exist.

Existence of Orbitals

Not all combinations of n and l correspond to real orbitals. For example, while 1s, 2p, and 4s are valid orbitals, 2d is not valid because the azimuthal quantum number l cannot equal 2 when n is 2; it must be less than n. Therefore, understanding the restrictions on quantum numbers is essential for determining which orbitals exist.

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