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Chapter 40: Q. 3 (page 1174)

| FIGURE EX 40.3shows the wave function of an electron in a rigid box. The electron energy is25eV. How long is the box?

Short Answer

Expert verified

The electron energy is 25eV so the box is0.736nm.

Step by step solution

01

Given Information 

We have to given The electron energy is25eV.

02

Simplify

The graph of ψn(x)has n−1nodes, excluding the ends, and nantinodes. In our case, the wave function shown in the graph has three maxima and three minima, meaning that it has six antinodes. Since the wave function is shown in the graph corresponds to the n=6state. The energy of levels of an electron in a rigid box is given by:

En=n2h28mL2

Then,

E6=36h28mL2

the equation to solve the numerical values of the different variables.

L=6h8mE6=66.626×10−34J⋅s89.11×10−31kg25eV×1.6×10−19J/eVL=7.36×10−10m=0.736nm

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Most popular questions from this chapter

Showed that a typical nuclear radius is 4fm As you’ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well isEn=-20MeV. To find out how “fuzzy” the edge of the nucleus is, what is the neutron’s penetration distance into the classically forbidden region as a fraction of the nuclear radius?

Show that the normalization constantAn for the wave functions of a particle in a rigid box has the value given in Equation 40.26.

Figure 40.27a modeled a hydrogen atom as a finite potential well with rectangular edges. A more realistic model of a hydrogen atom, although still a one-dimensional model, would be the electron + proton electrostatic potential energy in one dimension:

U(x)=-e24πε0x

a. Draw a graph of U(x) versus x. Center your graph at x=0.

b. Despite the divergence at x=0, the Schrödinger equation can be solved to find energy levels and wave functions for the electron in this potential. Draw a horizontal line across your graph of part a about one-third of the way from the bottom to the top. Label this line E2, then, on this line, sketch a plausible graph of the n=2wave function.

c. Redraw your graph of part a and add a horizontal line about two-thirds of the way from the bottom to the top. Label this line E3, then, on this line, sketch a plausible graph of the n=3 wave function.

Model an atom as an electron in a rigid box of length 0.100nm, roughly twice the Bohr radius.

a. What are the four lowest energy levels of the electron?

b. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label λn→mto indicate the transition.

c. Are these wavelengths in the infrared, visible, or ultraviolet portion of the spectrum?

d. The stationary states of the Bohr hydrogen atom have negative energies. The stationary states of this model of the atom have positive energies. Is this a physically significant difference? Explain.

e. Compare this model of an atom to the Bohr hydrogen atom. In what ways are the two models similar? Other than the signs of the energy levels, in what ways are they different?

Even the smoothest mirror finishes are “rough” when viewed at a scale of 100nm. When two very smooth metals are placed in contact with each other, the actual distance between the surfaces varies from0nmat a few points of real contact to ≈100nm. The average distance between the surfaces is≈50nm. The work function of aluminum is 4.3eV. What is the probability that an electron will tunnel between two pieces of aluminum that are 50nmapart? Give your answer as a power of10rather than a power ofe.
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