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Chapter 8: Problem 71

Identify the orbital that has (a) one radial node and one angular node; (b) no radial nodes and two angular nodes; (c) two radial nodes and three angular nodes.

Short Answer

Expert verified
The orbitals are (a) \(3p\), (b) \(3d\), and (c) \(6f\).

Step by step solution

01

Understand the concepts

Orbital is a specific region within an atom where there is a high probability of finding an electron. Nodes are areas where the probability of finding an electron is zero. There are two types of nodes: radial and angular. A radial node occurs when the electron is never found at a certain distance from the nucleus. An angular node (also known as a nodal plane) is a plane where the electron cannot be found. The number of radial nodes is given by \( n - l - 1 \) and the number of angular nodes is equal to \( l \). where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.
02

Solve for (a) with one radial node and one angular node

From the formula for the number of radial nodes, we have \( n - l - 1 = 1 \), and from the formula for the number of angular nodes, we just get \( l = 1 \). From the principle quantum number and azimuthal quantum number, it's found that \( n = 3 \) and \( l = 1 \), corresponding to the \( 3p \) orbitals.
03

Solve for (b) with no radial nodes and two angular nodes

Following a similar process, for no radial nodes, we get \( n = l + 1 \) and for the two angular nodes, we just get \( l = 2 \). From these equations, we get \( n = 3 \) and \( l = 2 \), corresponding to the \( 3d \) orbitals.
04

Solve for (c) with two radial nodes and three angular nodes

Again, for two radial nodes we get \( n = l +3 \) and for three angular nodes, we just get \( l = 3 \). From these equations, we get \( n = 6 \) and \( l = 3 \), corresponding to the \( 6f \) orbitals.

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Most popular questions from this chapter

In what region of the electromagnetic spectrum would you expect to find radiation having an energy per photon 100 times that associated with 988 nm radiation?

An atom in which just one of the outer-shell electrons is excited to a very high quantum level \(n\) is called a "high Rydberg" atom. In some ways, all these atoms resemble a Bohr hydrogen atom with its electron in a high-numbered orbit. Explain why you might expect this to be the case.

Draw an energy-level diagram that represents all the possible lines in the emission spectrum of hydrogen atoms produced by electron transitions, in one or more steps, from \(n=5\) to \(n=1\).

What electron transition in a hydrogen atom, ending in the orbit \(n=3,\) will produce light of wavelength \(1090 \mathrm{nm} ?\)

On the basis of the periodic table and rules for electron configurations, indicate the number of (a) \(2 p\) electrons in \(\mathrm{N} ;\) (b) \(4 \mathrm{s}\) electrons in \(\mathrm{Rb} ;\) (c) \(4 \mathrm{d}\) electrons in As; (d) \(4 f\) electrons in \(\mathrm{Au} ;\) (e) unpaired electrons in \(\mathrm{Pb} ;\) (f) elements in group 14 of the periodic table; (g) elements in the sixth period of the periodic table.
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