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Chapter 6: Problem 38

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=3,(\mathbf{b})\) from an orbit of radius 0.529 to one of radius \(0.476 \mathrm{nm},(\mathbf{c})\) from the \(n=9\) to the \(n=6\) state.

Short Answer

Expert verified
For the electronic transitions in hydrogen: (a) From n = 2 to n = 3, energy is absorbed. (b) From an orbit of radius 0.529 nm to one of radius 0.476 nm, energy is emitted. (c) From the n = 9 to the n = 6 state, energy is emitted.

Step by step solution

01

Transition (a): n = 2 to n = 3

The electron is transitioning from n = 2 to higher energy level n = 3. In this case, the electron is jumping to a higher energy state, and hence the atom needs to absorb energy for this transition to occur. Therefore, energy is absorbed in this case.
02

Transition (b): Orbit of radius 0.529 nm to orbit of radius 0.476 nm

In this case, we need to use the relation between the orbital radius and principal quantum number in a hydrogen atom: \(r_n = a₀n^2\), where \(a₀ \approx 0.529\) nm is the Bohr radius. First, let's find the initial principal quantum number, \(n_i\): \[ n_i^2 = \frac{r_i}{a₀} = \frac{0.529\, \text{nm}}{0.529\, \text{nm}} = 1. \] So, \(n_i = 1\); Next, let's find the final principal quantum number, \(n_f\): \[ n_f^2 = \frac{r_f}{a₀} = \frac{0.476\, \text{nm}}{0.529\, \text{nm}} \approx 0.81. \] So, \(n_f \approx 0.9\); Since \(n_f < n_i\), the electron is transitioning to lower energy level. Thus, the atom releases energy for this transition to occur, and energy is emitted.
03

Transition (c): n = 9 to n = 6

The electron is transitioning from higher energy level n = 9 to lower energy level n = 6. In this case, the electron is falling to a lower energy state, and hence the atom needs to release energy for this transition to occur. Therefore, energy is emitted in this case. To summarize: (a) Energy is absorbed. (b) Energy is emitted. (c) Energy is emitted.

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Most popular questions from this chapter

As discussed in the A Closer Look box on "Measurement and the Uncertainty Principle," the essence of the uncertainty principle is that we can't make a measurement without disturbing the system that we are measuring. (a) Why can't we measure the position of a subatomic particle without disturbing it? (b) How is this concept related to the paradox discussed in the Closer Look box on "Thought Experiments and Schrödinger's Cat"?

Sketch the shape and orientation of the following types of orbitals: \((\mathbf{a}) s,(\mathbf{b}) p_{z},(\mathbf{c}) d_{x y}\).

(a) A green laser pointer emits light with a wavelength of \(532 \mathrm{nm}\). What is the frequency of this light? (b) What is the energy of one of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of \(532-\mathrm{nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Give the values for \(n, l,\) and \(m_{l}\) for \((\mathbf{a})\) each orbital in the \(3 p\) subshell, (b) each orbital in the \(4 f\) subshell.

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.23 and 6.24 ) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?
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