Explanation Given information: The probability density is p ( r ) = 4 a 0 − 3 r 2 e − 2 r a 0 . Formula Used: Integration by parts: ∫ u v d x = u ∫ v d x − ∫ u ′ ( ∫ v d x ) d x Substitution method: Consider the function ∫ f ( g ( x ) ) g ′ ( x ) d x . Let u = g ( x ) . d u = g ′ ( x ) d x . Now, the function becomes, ∫ f ( u ) d u L-Hospital rule: If the function has indeterminate form ( 0 0 or ∞ ∞ ) , then differentiate the numerator an denominator. After that take limit. Calculation: The mean of the distribution is μ = ∫ 0 ∞ r p ( r ) d r . Substitute 4 a 0 − 3 r 2 e − 2 r a 0 for p ( r ) in the above function. μ = ∫ 0 ∞ r ( 4 a 0 − 3 r 2 e − 2 r a 0 ) d r = ∫ 0 ∞ 4 a 0 − 3 r 3 e − 2 r a 0 d r Use substitution method to solve the above integral. Let u = 2 r a 0 . So, r = u a 0 2 . So, d u = 2 a 0 d r . Find the limits for the integral. When r = 0 then, u = 0 . When r = ∞ then u = ∞ . Thus, the integral becomes, μ = 4 a 0 − 3 ∫ 0 ∞ ( u a 0 2 ) 3 e − u ( a 0 2 d u ) . Simplify the integral. μ = 4 a 0 − 3 ∫ 0 ∞ ( u a 0 2 ) 3 e − u ( a 0 2 d u ) = 4 16 a 0 − 3 ⋅ a 0 3 ⋅ a 0 ∫ 0 ∞ u 3 e − u d u = a 0 4 ∫ 0 ∞ u 3 e − u d u Use integration by parts method, to solve the above integral. a 0 4 ∫ 0 ∞ u 3 e − u d u = a 0 4 [ ( − u 3 e − u ) 0 ∞ + 3 ∫ 0 ∞ u 2 e − u d u ] First, simplify the expression ( − u 3 e − u ) 0 ∞

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.8, Problem 14E

To determine

To prove:

The distance r between the electron and the nucleus has mean μ=3a02.

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Chapter 8.8, Problem 13E

Chapter 8.8, Problem 15E

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Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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To prove: The distance r between the electron and the nucleus has mean μ = 3 a 0 2 .

Solution Summary: The author explains that the distance between the electron and the nucleus has mean mu =3a_02

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To prove:

The distance r between the electron and the nucleus has mean μ=3a02.

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Chapter 8 Solutions

To prove: The distance r between the electron and the nucleus has mean μ = 3 a 0 2 .

Solution Summary: The author explains that the distance between the electron and the nucleus has mean mu =3a_02

Concept explainers

Videos

To prove: The distance r between the electron and the nucleus has mean μ=3a02.

Want to see the full answer?

Chapter 8 Solutions

To prove:

The distance r between the electron and the nucleus has mean μ=3a02.