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Consider a monatomic ideal gas that lives at a height z

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 37P Chapter 3

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 37P

Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgz in addition to its kinetic energy.

(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz:

(You can derive this result from either the definition μ = –T(∂S/∂N)U,V or the formula μ = (∂U/∂N)S,V.)

(b) Suppose you have two chunks of helium gas, one at sea level and one at height z, each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is

N(z) = N(0)e‒mgz/kT,

in agreement with the result of below Problem 1.

Problem 1:

The exponential atmosphere.

(a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest , the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.

(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation

called the barometric equation.

(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e–mgz/kT. Show also that the density obeys a similar equation.

(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft , 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)

Problem 2:

Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%).

Step-by-Step Solution:
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Chapter 3, Problem 37P is Solved
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Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

This full solution covers the following key subjects: pressure, height, air, level, Sea. This expansive textbook survival guide covers 10 chapters, and 454 solutions. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1. The full step-by-step solution to problem: 37P from chapter: 3 was answered by , our top Physics solution expert on 07/05/17, 04:29AM. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. Since the solution to 37P from 3 chapter was answered, more than 326 students have viewed the full step-by-step answer. The answer to “Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgz in addition to its kinetic energy.(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz: (You can derive this result from either the definition ? = –T(?S/?N)U,V or the formula ? = (?U/?N)S,V.)________________(b) Suppose you have two chunks of helium gas, one at sea level and one at height z, each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk isN(z) = N(0)e?mgz/kT,in agreement with the result of below 1. 1:The exponential atmosphere.(a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest , the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.________________(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation called the barometric equation.________________(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e–mgz/kT. Show also that the density obeys a similar equation.________________(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft , 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.) 2:Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%).” is broken down into a number of easy to follow steps, and 346 words.

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Consider a monatomic ideal gas that lives at a height z