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Chapter 28: Problem 38

What is the largest number of electrons with the same pair of values for \(n\) and \(\ell\) that an atom can have?

Short Answer

Expert verified
Answer: The largest number of electrons with the same pair of values for n and l within an atom is given by the expression 4n - 2, where n represents the principal quantum number.

Step by step solution

01

Understanding the quantum numbers and the Pauli Exclusion Principle

The principal quantum number (n) determines the energy level of an electron, ranging from 1 to \(\infty\). The azimuthal quantum number (l) determines the shape of the electron orbital and ranges from 0 to n-1. For a given pair of values of n and l, the magnetic quantum number (m_l) ranges from -l to +l. Lastly, the electron spin quantum number (m_s) can have two possible values: +1/2 or -1/2. The Pauli Exclusion Principle states that no two electrons within an atom can have the same set of quantum numbers (n, l, m_l, and m_s).
02

Identify the maximum number of electrons with the same pair of values for n and l

Given any pair of values for n and l, there can be a range of values for the magnetic quantum number (m_l) from -l to +l, resulting in a total of 2l+1 possible different values of m_l. Since each of these m_l values can be associated with an electron having m_s either +1/2 or -1/2 (two possible values), the maximum number of electrons with the same pair of n and l values would be: Number of electrons = (2l + 1) x 2 Let's simplify the expression: Number of electrons = 4l + 2
03

Find the largest possible value for l within an atom

In an atom, the largest possible value for l arises when it equals n-1 (for any given n). For example, in the ground state (n=1), there could be one possible l value (l=0), but for higher n values, the number of possible l values increases. Therefore, the largest number of electrons with the same pair of n and l values can be obtained when l is maximized to (n-1).
04

Calculate the maximum number of electrons with the same pair of values for n and l

Using the equation derived in Step 2 and plugging in the maximum value of l (n-1): Number of electrons = 4(n-1) + 2 Simplify the expression: Number of electrons = 4n - 4 + 2 Number of electrons = 4n - 2 Thus, the maximum possible number of electrons with the same pair of values for n and l within an atom is given by the expression 4n - 2, where n represents the principal quantum number.

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Most popular questions from this chapter

What are the de Broglie wavelengths of electrons with the following values of kinetic energy? (a) \(1.0 \mathrm{eV}\) (b) $1.0 \mathrm{keV} .
Before the discovery of the neutron, one theory of the nucleus proposed that the nucleus contains protons and electrons. For example, the helium-4 nucleus would contain 4 protons and 2 electrons instead of - as we now know to be true- 2 protons and 2 neutrons. (a) Assuming that the electron moves at nonrelativistic speeds, find the ground-state energy in mega-electron- volts of an electron confined to a one-dimensional box of length $5.0 \mathrm{fm}\( (the approximate diameter of the \)^{4} \mathrm{He}$ nucleus). (The electron actually does move at relativistic speeds. See Problem \(80 .)\) (b) What can you conclude about the electron-proton model of the nucleus? The binding energy of the \(^{4} \mathrm{He}\) nucleus - the energy that would have to be supplied to break the nucleus into its constituent particles-is about \(28 \mathrm{MeV} .\) (c) Repeat (a) for a neutron confined to the nucleus (instead of an electron). Compare your result with (a) and comment on the viability of the proton-neutron theory relative to the electron-proton theory.
What is the ground-state electron configuration of tellurium (Te, atomic number 52 )?
The particle in a box model is often used to make rough estimates of energy level spacings. Suppose that you have a proton confined to a one-dimensional box of length equal to a nuclear diameter (about \(10^{-14} \mathrm{m}\) ). (a) What is the energy difference between the first excited state and the ground state of this proton in the box? (b) If this energy is emitted as a photon as the excited proton falls back to the ground state, what is the wavelength and frequency of the electromagnetic wave emitted? In what part of the spectrum does it lie? (c) Sketch the wave function \(\psi\) as a function of position for the proton in this box for the ground state and each of the first three excited states.
If diffraction were the only limitation on resolution, what would be the smallest structure that could be resolved in an electron microscope using 10-keV electrons?
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