Explanation The expression for Boltzmann-Maxwell distribution of speeds is given by, f ( v ) = 4 π ( M 2 π R T ) 3 2 v 2 e − M v 2 2 R T Where, • R is the gas constant. • T is the temperature. • M is the molar mass of the gas. • v is the velocity. For a given gas at defined temperature, the molar mass is constant and temperature is constant. Therefore, the distribution of speeds can be re-written as, f ( v ) = K v 2 e − M v 2 2 R T Where, • K is a constant. Take the first derivative of the function, f ( v ) and equate it to zero to calculate the most probable speed. d f ( v ) d v = 0 d d v ( v 2 e − M v 2 / 2 R T )

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ISBN: 9780198769866

Author: ATKINS, P. W. (peter William), De Paula, Julio, Keeler, JAMES

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Chapter 1, Problem 1B.10P

Interpretation Introduction

Interpretation: The expression for the most probable speed of molecules by differentiation has to be derived.

Concept introduction: The expression for the most probable speed is derived from Maxwell Boltzmann distribution. The most probable speed for the gas particles is represented as,

vmps= (2RTM)1/2

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Chapter 1 Solutions

Atkins' Physical Chemistry

Ch. 1 - Prob. 1A.1ST Ch. 1 - Prob. 1A.2ST Ch. 1 - Prob. 1B.1ST Ch. 1 - Prob. 1C.1ST Ch. 1 - Prob. 1A.1DQ Ch. 1 - Prob. 1A.1AE Ch. 1 - Prob. 1A.1BE Ch. 1 - Prob. 1A.2AE Ch. 1 - Prob. 1A.2BE Ch. 1 - Prob. 1A.3AE

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The expression for the most probable speed of molecules by differentiation has to be derived. Concept introduction: The expression for the most probable speed is derived from Maxwell Boltzmann distribution. The most probable speed for the gas particles is represented as, v mps = ( 2 R T M ) 1 / 2

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Chapter 1 Solutions