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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

List the following types of electromagnetic radiation in order of increasing wavelength: (a) the gamma rays produced by a radioactive nuclide used in medical imaging; (b) radiation from an FM radio station at 93.1 \(\mathrm{MHz}\) on the dial; (c) a radio signal from an AM radio station at 680 \(\mathrm{kHz}\) on the dial; ( d ) the yellow light from sodium vapor streetlights; (e) the red light of a light-emitting diode, such as in a calculator display.

How many unique combinations of the quantum numbers \(l\) and \(m_{l}\) are there when (a) \(n=3,\) (b) \(n=4 ?\)

State where in the periodic table these elements appear: $$ \begin{array}{l}{\text { (a) elements with the valence-shell electron configuration }} \\ {n s^{2} n p^{5}} \\ {\text { (b) elements that have three unpaired p electrons }} \\ {\text { (c) an element whose valence electrons are } 4 s^{2} 4 p^{1}} \\ {\text { (d) the } d \text { -block elements [ Section } 6.9 ]}\end{array} $$

Identify the specific element that corresponds to each of the following electron configurations and indicate the number of unpaired electrons for each: (a) \(1 s^{2} 2 s^{2},(\mathbf{b}) 1 s^{2} 2 s^{2} 2 p^{4}\) (c) \([\operatorname{Ar}] 4 s^{1} 3 d^{5},(\mathbf{d})[\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4}\)

It is possible to convert radiant energy into electrical energy using photovoltaic cells. Assuming equal efficiency of conversion, would infrared or ultraviolet radiation yield more electrical energy on a per-photon basis?
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