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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 double-slit interference experiment is performed with 2.0-ev photons. The same pair of slits is then used for an experiment with electrons. What is the kinetic energy of the electrons if the interference pattern is the same as for the photons (i.e., the spacing between maxima is the same)?
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.
A marble of mass \(10 \mathrm{g}\) is confined to a box \(10 \mathrm{cm}\) long and moves at a speed of \(2 \mathrm{cm} / \mathrm{s} .\) (a) What is the marble's quantum number \(n ?\) (b) Why can we not observe the quantization of the marble's energy? [Hint: Calculate the energy difference between states \(n\) and \(n+1 .\) How much does the marble's speed change?]
(a) Make a qualitative sketch of the wave function for the \(n=5\) state of an electron in a finite box \([U(x)=0\) for \(00\) elsewherel. (b) If \(L=1.0 \mathrm{nm}\) and \(U_{0}=1.0\) keV, estimate the number of bound states that exist.
How many electron states of the H atom have the quantum numbers \(n=3\) and \(\ell=1 ?\) Identify each state by listing its quantum numbers.
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