Explanation Given information: The distance r between the electron and the nucleus is a random variable having probability density p ( r ) = 4 a − 3 0 r 2 e − 2 r a 0 for r ≥ 0 , where a 0 is a Bohr radius. The probability density function p ( r ) = 4 a − 3 0 r 2 e − 2 r a 0 is given by, Formula used : ∫ u d v = u v − ∫ v d u Calculation: p ( r ) = 4 a − 3 0 r 2 e − 2 r a 0 (Given) The probability P that the electron is within one Bohr radius of the nucleus is given by P = ∫ 0 a 0 p ( r ) d r = ∫ 0 a 0 4 a 0 − 3 r 2 e − 2 r a 0 d r = 4 a 0 − 3 ∫ 0 a 0 r 2 e − 2 r a 0 d r Substituting u = 2 r a 0 , then d u = 2 a 0 d r ⇒ r = a 0 u 2 . Using these in the above equation, we get P = 4 a 0 − 3 ∫ 0 a 0 ( a 0 u 2 ) 2 e − u a 0 2 d u = 4 a 0 − 3 a 0 3 8 ∫ 0 a 0 ( u ) 2 e − u d u = 1 2 ∫ 0 a 0 ( u ) 2 e − u d u By integration by parts, we get P = 1 2 ∫ 0 a<

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Chapter 9.1, Problem 15E

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To find: The probability P that the electron is within one Bohr radius of the nucleus.

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Chapter 9.1, Problem 14E

Chapter 9.1, Problem 16E

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Chapter 9 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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The probability P that the electron is within one Bohr radius of the nucleus.

Solution Summary: The author calculates the probability P that the electron is within one Bohr radius of the nucleus.

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To find: The probability P that the electron is within one Bohr radius of the nucleus.

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Chapter 9 Solutions