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Chapter 38: Q. 53 (page 1116)

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n=2 to n =1. What is the length of the box?

Short Answer

Expert verified

Tthe length of the box is $L = 426.4 pm$

Step by step solution

01

Given information

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n=2 to n =1.

02

Explanation

Energy levels of the particle in the box:

$E_{n} = \frac{h^{2} n^{2}}{8 m L^{2}}$

Energy of the photon:

$Δ E_{12} = \frac{h c}{λ}$

Substitute energy levels:

role="math" localid="1648514586435" $E_{2} - E_{1} = \frac{h^{2} \cdot 2^{2}}{8 m L^{2}} - \frac{h^{2} \cdot 1^{2}}{8 m L^{2}} E_{2} - E_{1} = \frac{3 h^{2}}{8 m L^{2}}$

Substitute energy of the photon:

$\frac{h c}{λ} = \frac{3 h^{2}}{8 m L^{2}}$

03

Calculations:

Express the above equation in terms of L:

$L = \sqrt{\frac{3 h^{2} λ}{8 m h c}}$

Substitute the values:

$L = \sqrt{\frac{3 {(6.63 \cdot 10^{- 34})}^{2} \cdot 200 \cdot 10^{- 9}}{8 \cdot 9.11 \cdot 10^{- 31} \cdot 6.63 \cdot 10^{- 34} \cdot 3 \cdot 10^{8}}} L = 426.4 pm$

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Most popular questions from this chapter

Photoelectrons are observed when a metal is illuminated by light with a wavelength less than 388 nm. What is the metal’s work function?

a. What quantum number of the hydrogen atom comes closest to giving a 100-nm-diameter electron orbit?

b. What are the electron’s speed and energy in this state?

An electron with 2.00 eV of kinetic energy collides with the atom shown in Figure Ex 38.24.

a. Is the electron able to excite the atom? Why or why not?

b. If your answer to part a was yes, what is the electron’s kinetic energy after the collision?

Electrons in a photoelectric-effect experiment emerge from an aluminum surface with a maximum kinetic energy of 1.30 eV. What is the wavelength of the light?

What is the third-longest wavelength in the absorption spectrum of hydrogen?

See all solutions

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