Explanation Given information: The distance r between the electron and the nucleus in a hydrogen atom is a random variable with the probability density function p ( r ) = 4 a 0 − 3 r 2 e − 2 r / a 0 for r ≥ 0 . Here, a 0 is Bohr radius and given by a 0 = 5.29 × 10 − 11 m The given figure is Formula used: For a probability density function p ( x ) on ( − ∞ , ∞ ) , the mean of the random variable is given by μ = ∫ − ∞ ∞ x p ( x ) d x The following integration by parts can be calculated as shown below. ∫ x 3 e − x / a d x = − a x 2 e − x / a + 2 a ( − a x e − x / a − a 2 e − x / a ) + C Calculation: The mean of the distance r between the electron and the nucleus can be calculated as shown below. μ = ∫ 0 a 0 r p ( r ) d r = ∫ 0 a 0 4 a 0 − 3 r 3 e − 2 r / a 0 d r = 4 a 0 − 3 ∫ 0 a 0 r 3 e − 2 r / a 0 d r

Loose-leaf Version For Calculus: Early Transcendentals

4th Edition

ISBN: 9781319055912

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 8.1, Problem 16E

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Themean of the distance r between the electron and the nucleus is 3a02.

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Chapter 8.1, Problem 15E

Chapter 8.1, Problem 17E

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Chapter 8 Solutions

Loose-leaf Version For Calculus: Early Transcendentals

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Themean of the distance r between the electron and the nucleus is 3 a 0 2 .

Solution Summary: Themean of the distance r between the electron and the nucleus in the hydrogen atom is a random variable with the probability density function.

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Themean of the distance r between the electron and the nucleus is 3a02.

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Themean of the distance r between the electron and the nucleus is 3 a 0 2 .

Solution Summary: Themean of the distance r between the electron and the nucleus in the hydrogen atom is a random variable with the probability density function.

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To show: Themean of the distance r between the electron and the nucleus is 3a02.

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Chapter 8 Solutions

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Themean of the distance r between the electron and the nucleus is 3a02.