Explanation Given information: According to Maxwell’s Distribution law, in a gas molecular mass m, speed v of a molecule in a gas at temperature T (kelvins) is a random variable with density p ( v ) = 4 π ( m 2 π k T ) 3 2 v 2 e − m v 2 2 k T ( v ≥ 0 ) Where k is the Boltzmann’s constant. The average speed of oxygen molecule at room temperature isaround 450 m/s. Formula used: Mean, μ = ∫ − ∞ ∞ x p ( x ) d x Calculation: p ( v ) = 4 π ( m 2 π k T ) 3 2 v 2 e − m v 2 2 k T ( v ≥ 0 ) (Given) The average molecular speed formula is given by μ = ∫ − ∞ ∞ v p ( v ) d v = ∫ 0 450 v 4 π ( m 2 π k T ) 3 2 v 2 e − m v 2 2 k T d v = 4 π ( m 2 π k T ) 3 2 ∫ 0 450 v . v 2 e − m v 2 2 k T d v = 4 π ( m 2 π k T ) 3 2 ∫ 0 450 v 3 e − m v 2 2 k T d v = 4 π ( m 2 π k T ) 3 2 [ − ( m 2 k T v 2 + 1 ) e − m v 2 2 k T 2 ( m 2 k T ) 2 ]

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.8, Problem 22E

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To prove:The average molecular speed is equal to ( 8kT πm)12 .

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Chapter 8.8, Problem 21E

Chapter 8.8, Problem 23E

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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 average molecular speed is equal to ( 8 k T π m ) 1 2 .

Solution Summary: The author explains Maxwell's distribution law, where speed v is a random variable with density. The average speed of oxygen molecule at room temperature is around 450 m/s.

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To prove:The average molecular speed is equal to ( 8kT πm)12 .

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Chapter 8 Solutions