Book edition 2nd Edition

Author(s) Randy Harris

Pages 633 pages

ISBN 9780805303087

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Short Answer

Calculate the average speed of a gas molecule in a classical ideal gas.
What is the average velocity of a gas molecule?

The average speed of gas molecule in a classical ideal gas is $\sqrt{\frac{8 k_{B} T}{πm}}$ .
The average velocity of gas molecule is zero.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Maxwell Probability Distribution.

$P (v) = {(\frac{m}{2 {πk}_{B} T})}^{\frac{3}{2}} 4 {πv}^{2} e^{- \frac{mv}{2 k_{B} T}}$ …..(1)

Where,

m is the mass of the particle.

v is velocity of particle.

T is temperature.

k_B is Boltzmann constant.

Average speed

$v_{avg} = \int_{0}^{\infty} vP (v) dv$

Substituting expression (1) in (2).

$v_{avg} = \int_{0}^{\infty} v {(\frac{m}{2 {πk}_{B} T})}^{\frac{3}{2}} 4 {πv}^{2} e^{- \frac{{mv}^{2}}{2 k_{B} T}} dv$

Let $b = \frac{1}{2 a^{2}} = \frac{m}{2 k_{B} T}$

$\begin{array}{rcl} v_{avg} & = & 4 π {(\frac{b}{π})}^{\frac{3}{2}} \int_{0}^{\infty} v^{3} e^{- {bv}^{2}} dv \\ = & 4 π {(\frac{b}{π})}^{\frac{3}{2}} \frac{1}{2 b^{2}} \\ = & \sqrt{\frac{4}{πb}} \\ = & \sqrt{\frac{8 k_{B} T}{πm}} \end{array}$

Step 2: Average Velocity.

The average velocity of a body is the pace at which it changes position from one location to another. It's a quantity with a vector. The fact that gas molecules move in random directions is well known, and so the gas molecules have velocity in all possible directions. As a result, the vector sum of all velocities equals zero. As a result, a gas molecule's average velocity is zero.

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Modern Physics

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Short Answer

Calculate the average speed of a gas molecule in a classical ideal gas.
What is the average velocity of a gas molecule?

Step by Step Solution

Step 1: Maxwell Probability Distribution.

Step 2: Average Velocity.

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Modern Physics

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Short Answer

Calculate the average speed of a gas molecule in a classical ideal gas. What is the average velocity of a gas molecule?

Step by Step Solution

Step 1: Maxwell Probability Distribution.

Step 2: Average Velocity.

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