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Chapter 8: Problem 39

What is the maximum number of orbital angular momentum electron states in the \(n=2\) shell of a hydrogen atom? (Ignore electron spin.)

Short Answer

Expert verified
The maximum number of orbital angular momentum electron states in the \(n=2\) shell of a hydrogen atom, ignoring electron spin, is 4. This is calculated by finding the allowed values of the angular momentum quantum number (l) and the magnetic quantum number (m_l) for the given principal quantum number (n). In this case, there is 1 allowed state for \(l=0\) and 3 allowed states for \(l=1\), resulting in a total of 4 states.

Step by step solution

01

Find the allowed values of the angular momentum quantum number (l)

For a given principal quantum number n, the allowed values of the angular momentum (l) quantum number are \(0,1,2,...,n-1\). In this case, since the principal quantum number \(n=2\), the allowed values for angular momentum quantum number are \(l=0\) and \(l=1\).
02

Determine the allowed magnetic quantum numbers for each angular momentum value (l)

For each value of the angular momentum (l), the magnetic quantum number (m_l) ranges from \(-l\) to \(+l\), including 0. So, we will determine the allowed magnetic quantum numbers for \(l=0\) and \(l=1\): - For \(l=0\), the only possible magnetic quantum number is \(m_l=0\). - For \(l=1\), the possible magnetic quantum numbers are \(m_l=-1, 0, +1\).
03

Count the total number of orbital angular momentum electron states

Now that we have the allowed magnetic quantum numbers for each angular momentum value, we can count the total number of orbital angular momentum electron states. 1. For \(l=0\), there is 1 allowed state (\(m_l=0\)). 2. For \(l=1\), there are 3 allowed states (\(m_l=-1, 0, +1\)). Adding them together, we have a total of \(1+3=4\) orbital angular momentum electron states for a hydrogen atom in the \(n=2\) shell, ignoring electron spin.

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Most popular questions from this chapter

(a) How many angles can \(L\) make with the z-axis for an \(l=2\) electron? (b) Calculate the value of the smallest angle.

A student in a physics laboratory observes a hydrogen spectrum with a diffraction grating for the purpose of measuring the wavelengths of the emitted radiation. In the spectrum, she observes a yellow line and finds its wavelength to be \(589 \mathrm{nm}\). (a) Assuming that this is part of the Balmer series, determine \(n_{\mathrm{i}},\) the principal quantum number of the initial state. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

What is the probability that the 1 s electron of a hydrogen atom is found between \(r=0\) and \(r=\infty\) ?

What is the maximum number of orbital angular momentum electron states in the \(n=3\) shell of a hydrogen atom? (Ignore electron spin.)

An X-ray tube has an applied voltage of 100 kV. (a) What is the most energetic X-ray photon it can produce? Express your answer in electron volts and joules. (b) Find the wavelength of such an X-ray.
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