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The exponential atmosphere.(a) Consider a horizontal slab

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

Solution for problem 16P Chapter 1

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 16P

Problem 16P

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: 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:
Step 1 of 3

ANSWER:

a)

The basic formula for the vertical pressure variation is,

          --------------(1)

Where is the change in height.

In our question, the height is dZ.

So, ----------------(2)

Where is the density of air.

b)

The ideal gas equation is,

         -------------(3)

   [multiplied mass in both the sides]

----------------------(4)

If we put this result in equation (2),

. (proved)

The barometric equation.

c)

The pressure as a function of height we got as,

       

We also have equation (4) which gives the relation pressure and density. So, after putting the value we got in equation (4) in the above equation,

        

So, the density also obeys a similar type of equation apart from a constant multiplied.

Step 2 of 3

Chapter 1, Problem 16P is Solved
Step 3 of 3

Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

The full step-by-step solution to problem: 16P from chapter: 1 was answered by , our top Physics solution expert on 07/05/17, 04:29AM. Since the solution to 16P from 1 chapter was answered, more than 597 students have viewed the full step-by-step answer. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. This full solution covers the following key subjects: pressure, air, equation, height, slab. 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 answer to “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: 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 227 words.

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The exponential atmosphere.(a) Consider a horizontal slab