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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

A scanning electron microscope is used to look at cell structure with 10 -nm resolution. A beam of electrons from a hot filament is accelerated with a voltage of \(12 \mathrm{kV}\) and then focused to a small spot on the specimen. (a) What is the wavelength in nanometers of the beam of incoming electrons? (b) If the size of the focal spot were determined only by diffraction, and if the diameter of the electron lens is one fifth the distance from the lens to the specimen, what would be the minimum separation resolvable on the specimen? (In practice, the resolution is limited much more by aberrations in the magnetic lenses and other factors.)
What are the possible values of \(L_{z}\) (the component of angular momentum along the \(z\) -axis) for the electron in the second excited state \((n=3)\) of the hydrogen atom?
(a) What are the electron configurations of the ground states of fluorine \((Z=9)\) and chlorine \((Z=17) ?\) (b) Why are these elements placed in the same column of the periodic table?
An image of a biological sample is to have a resolution of \(5 \mathrm{nm} .\) (a) What is the kinetic energy of a beam of electrons with a de Broglie wavelength of \(5.0 \mathrm{nm} ?\) (b) Through what potential difference should the electrons be accelerated to have this wavelength? (c) Why not just use a light microscope with a wavelength of 5 nm to image the sample?
(a) What are the electron configurations of the ground states of lithium \((Z=3),\) sodium \((Z=11),\) and potassium \((Z=19) ?\) (b) Why are these elements placed in the same column of the periodic table?
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