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Chapter 6: Problem 35

Does the hydrogen atom "expand" or "contract" when an electron is excited from the \(n=1\) state to the \(n=3\) state?

Short Answer

Expert verified
When an electron is excited from the \(n=1\) state to the \(n=3\) state, the average radius of the electron orbit increases from \(a_0\) to \(9a_0\). Therefore, the hydrogen atom "expands" during this transition.

Step by step solution

01

Write down the formula for the average radius of an electron orbit in a hydrogen atom

The formula for the average radius of an electron orbit in a hydrogen atom is given by: \[ r_n = a_0 n^2 \] where \(r_n\) is the average radius of the orbit for an electron in the principal quantum number (n), \(a_0\) is the Bohr radius (approximately \(5.29 \times 10^{-11}\) meters), and \(n\) is the principal quantum number.
02

Calculate the average radius of the electron orbit for n=1

Now, we will calculate the average radius of the electron orbit for \(n=1\). Substitute \(n=1\) into the formula: \[ r_1 = a_0 (1)^2 = a_0 \] So, the average radius of the electron orbit for \(n=1\) is equal to the Bohr radius, \(a_0\).
03

Calculate the average radius of the electron orbit for n=3

Next, we calculate the average radius of the electron orbit for \(n=3\). Substitute \(n=3\) into the formula: \[ r_3 = a_0 (3)^2 = 9a_0 \] The average radius of the electron orbit for \(n=3\) is 9 times the Bohr radius, \(9a_0\).
04

Compare the radii for n=1 and n=3

Now, we compare the radii for \(n=1\) and \(n=3\): - For \(n=1\), the average radius is \(a_0\). - For \(n=3\), the average radius is \(9a_0\). Since \(9a_0 > a_0\), the average radius of the electron orbit when the electron is excited from the \(n=1\) state to the \(n=3\) state increases.
05

Conclude whether the hydrogen atom expands or contracts

Because the average radius of the electron orbit increases when the electron is excited from the \(n=1\) state to the \(n=3\) state, we can conclude that the hydrogen atom "expands" during this transition.

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Most popular questions from this chapter

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+} .\) (a) Why is the Bohr model applicable to He \(^{+}\) ions but not to neutral He atoms? (b) The ground-state energies of \(\mathrm{H}, \mathrm{He}^{+},\) and \(\mathrm{Li}^{2+}\) are tabulated as follows: $$ \begin{array}{l}{\text { Atom or ion } \quad \quad\quad\quad\quad\quad \mathrm{H} \quad\quad\quad\quad\quad\quad \text { He }^{+} \quad\quad\quad\quad\quad\quad\quad \mathrm{Li}^{2+}} \\ {\text { Ground- state }\quad-2.18 \times 10^{-18} \mathrm{J}\quad-8.72 \times 10^{-18} \mathrm{J}\quad-1.96 \times 10^{-17} \mathrm{J}} \\ {\text { energy }}\end{array} $$ By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z .(\mathbf{c})\) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{C}^{5+}\) ion.

A stellar object is emitting radiation at 3.55 \(\mathrm{mm}\) . (a) What type of electromagnetic spectrum is this radiation? (b) If a detector is capturing \(3.2 \times 10^{8}\) photons per second at this wavelength, what is the total energy of the photons detected in 1.0 hour?

Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at \(8.85 \times 10^{5} \mathrm{cm} / \mathrm{s}\) .

A diode laser emits at a wavelength of 987 \(\mathrm{nm}\) . (a) In what portion of the electromagnetic spectrum is this radiation found? (b) All of its output energy is absorbed in a detector that measures a total energy of 0.52 \(\mathrm{J}\) over a period of 32 s. How many photons per second are being emitted by the laser?

The speed of sound in dry air at \(20^{\circ} \mathrm{C}\) is 343 \(\mathrm{m} / \mathrm{s}\) and the lowest frequency sound wave that the human ear can detect is approximately 20 \(\mathrm{Hz}\) (a) What is the wavelength of such a sound wave? (b) What would be the frequency of electromagnetic radiation with the same wavelength? (c) What type of electromagnetic radiation would that correspond to? [Section 6.1]
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