Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

An electron is trapped in a one-dimensional infinite well of width $250 pm$ and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a, single photon absorption?

The longest wavelength is 68.6 nm .
The second longest wavelength is 25.72 nm .
The third longest wavelength is 13.72 nm .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Introduction

An electron is a negatively charged subatomic particle. It can be either free (not attached to any atom), or bound to the nucleus of an atom. Electrons in atoms exist in spherical shells of various radii, representing energy levels. The larger the spherical shell, the higher the energy contained in the electron.

Step 2: Concept

An infinite potential well is a device for confining an electron. From the confinement principle we expect that the matter wave representing a trapped electron can exist only in a set of discrete states.

Energy of electron in an energy level in one-dimensional potential well is,

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

Here, h is the Planck’s constant, L is the length of the well, and m is the mass of the electron, and n is the integer.

Planck’s constant, $h = 6.626 \times 10^{- 34} Js$

Mass of the electron, $m = 9.109 \times 10^{- 31} kg$

Convert the length of the well from Pico meters to meters as follows:

$L = 250 pm = (250 pm) (10^{- 12} m / pm) = 250 \times 10^{12} m$

Step 3: Calculation

When electron jumps from $n = n_{1}$ state to $n = n_{f}$ state by absorbing a

Photon, then the frequency of the photon is given by

$f = \frac{∆ E}{h} = (\frac{h^{2}}{8 m L^{2}}) (\frac{1}{h}) (n_{f}^{2} - n_{i}^{2})$

Wavelength of the photon, $λ = \frac{c}{f}$

Substitute $(\frac{h^{2}}{8 m L^{2}}) (\frac{1}{h}) (n_{f}^{2} - n_{i}^{2})$ for f in the above equation.

$λ = \frac{c}{(\frac{h^{2}}{8 m L^{2}}) (n_{f}^{2} - n_{i}^{2})} = (\frac{8 m L^{2}}{h (n_{f}^{2} - n_{i}^{2})})$

Here, the electron is the ground state is $n_{i} = 1$

Hence, the wavelength is

$λ = \frac{8 m L^{2}}{h (n_{f}^{2} - n_{i}^{2})}$

Substitute $9.10 \times 10^{- 31} kg for m, 250 \times 10^{- 12} m for L, 2.998 \times 10^{8} m / s$ for c , $6.626 \times 10^{- 34} j . s$ for h , and 1 for $n_{i}$ in the above equation.

$λ = \frac{8 (9.10 \times 10^{- 31} kg) {(250 \times 10^{- 12} m)}^{2} (2.998 \times 10^{8} m / s)}{(6.626 \times 10^{- 34} j . s) (n_{f}^{2} - 1^{2})} = (\frac{2.058 \times 10^{7} m}{n_{f}^{2} - 1}) = (\frac{2.058 \times 10^{- 9} m (10^{9} nm / m)}{n_{f}^{2} - 1}) = \frac{205.8 nm}{n_{f}^{2} - 1}$

Step 4: (a) Determine the longest wavelength

For longest wavelength, $n_{f} = 2$

The longest wavelength is,

$λ = \frac{205.8 nm}{n_{f}^{2} - 1}$

Substitute 2 for in the above equation.

role="math" localid="1661859531615" $λ = \frac{205.8 nm}{2^{2} - 1} = 68.6 nm$

Therefore, the longest wavelength is $68.6 nm$ .

Step 5: (b) Determine the second longest wavelength.

For second longest wavelength, $n_{f} = 3$

The second longest wavelength is,

$λ = \frac{205.8 nm}{n_{f}^{2} - 1}$

Substitute 3 for $n_{f}$ in the above equation.

$λ = \frac{205.8 nm}{3^{2} - 1} = 25.72 nm$

Therefore, the second longest wavelength is $25.72 nm$ .

Step 6: (c) Determine third longest wavelength

For third longest wavelength, $n_{f} = 4$

The third longest wavelength is,

$λ = \frac{205.8 nm}{n_{f}^{2} - 1}$

Substitute 4 for $n_{f}$ in the above equation.

$λ = \frac{205.8 nm}{4^{2} - 1} = 13.72 nm$

Therefore, the third longest wavelength is $13.72 nm$ .

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Short Answer

An electron is trapped in a one-dimensional infinite well of width $250 pm$ and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a, single photon absorption?

Step by Step Solution

Step 1: Introduction

Step 2: Concept

Step 3: Calculation

Step 4: (a) Determine the longest wavelength

Step 5: (b) Determine the second longest wavelength.

Step 6: (c) Determine third longest wavelength

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Fundamentals Of Physics

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Short Answer

An electron is trapped in a one-dimensional infinite well of width250pm and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a, single photon absorption?

Step by Step Solution

Step 1: Introduction

Step 2: Concept

Step 3: Calculation

Step 4: (a) Determine the longest wavelength

Step 5: (b) Determine the second longest wavelength.

Step 6: (c) Determine third longest wavelength

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An electron is trapped in a one-dimensional infinite well of width $250 pm$ and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a, single photon absorption?