Book edition 1st

Author(s) Daniel V. Schroeder

Pages 356 pages

ISBN 9780201380279

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Short Answer

Use Boltzmann factors to derive the exponential formula for the density of an isothermal atmosphere, already derived in Problems 1.16 and 3.37. (Hint: Let the system be a single air molecule, let s1 be a state with the molecule at sea level, and let s2 be a state with the molecule at height z.)

Therefore, the exponential formula for the density of an isothermal atmosphere is: $ρ (z) = ρ (0) e^{- m g z / k T}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information

Let the system be a single air molecule, let S₁be a state with the molecule at sea level, and let S₂ be a state with the molecule at height z.

Step 2: Explanation

Consider a system with a single air molecule, where S₁ is the state when the molecule is at sea level and S₂is the state when the molecule is at a height of 2. Assume that the energy is only potential energy, so the difference in energy between the states S₁and S₂is the potential energy, which is $Δ E = m g z$ and the ratio of S₂state probability to state s1 probability is:

role="math" localid="1647369610229" $\frac{P (s_{2})}{P (s_{1})} = \frac{e^{- E_{2} / k T}}{e^{- E_{1} / k T}} = e^{- Δ E / k T} \frac{P (s_{2})}{P (s_{1})} = e^{- m g z / k T}$

This means that the air molecule is less likely to be at height of z than at the see level by a factor of $e^{- m g z / k T}$ , and that the number of molecules per unit volume at height of z is also smaller than the see level by the same ratio in the isothermal atmosphere, so:

$ρ (z) = ρ (0) e^{- m g z / k T}$

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An Introduction to Thermal Physics

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Short Answer

Use Boltzmann factors to derive the exponential formula for the density of an isothermal atmosphere, already derived in Problems 1.16 and 3.37. (Hint: Let the system be a single air molecule, let s1 be a state with the molecule at sea level, and let s2 be a state with the molecule at height z.)

Step by Step Solution

Step 1: Given information

Step 2: Explanation

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An Introduction to Thermal Physics

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Short Answer

Use Boltzmann factors to derive the exponential formula for the density of an isothermal atmosphere, already derived in Problems 1.16 and 3.37. (Hint: Let the system be a single air molecule, let s1 be a state with the molecule at sea level, and let s2 be a state with the molecule at height z.)

Step by Step Solution

Step 1: Given information

Step 2: Explanation

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