Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 48a

Chapter 20, Problem 48a

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H₂). The number density is about 1 atom/cm³ and the temperature is about 3 K. Estimate the pressure in interstellar space. Give your answer in Pa and in atm.

Verified step by step guidance

Start by recalling the ideal gas law, which is given by the equation:

P = n k T

, where

P

is the pressure,

n

is the number density (in particles per cubic meter),

k

is the Boltzmann constant (

1.38 \times 10 ⁻ 23 J / K

), and

T

is the temperature in kelvins.

Convert the number density from atoms per cubic centimeter to atoms per cubic meter. Since there are

10 ⁶

cubic centimeters in a cubic meter, multiply the given number density (

1 atom / cm ³

) by

10 ⁶

to get the number density in

atoms / m ³

Substitute the values into the ideal gas law. Use

n

(number density in

atoms / m ³

k

(Boltzmann constant), and

T

(temperature in kelvins) into the equation

P = n k T

to calculate the pressure in pascals (Pa).

Convert the pressure from pascals to atmospheres. Use the conversion factor

1 atm = 1.013 \times 10 ⁵ Pa

. Divide the pressure in pascals by this value to get the pressure in atmospheres.

Summarize the results: The pressure in interstellar space is calculated in both pascals (Pa) and atmospheres (atm) using the steps above. Ensure the units are consistent and the calculations are accurate.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. In this context, it can be simplified to P = nkT, where P is pressure, n is the number density of particles, k is the Boltzmann constant, and T is the temperature in Kelvin. This law is essential for estimating the pressure of gases in low-density environments like interstellar space.

Number Density

Number density refers to the number of particles per unit volume, typically expressed in particles per cubic centimeter (atoms/cm³). In the given scenario, the number density of hydrogen atoms is approximately 1 atom/cm³. This concept is crucial for calculating the pressure in interstellar space using the Ideal Gas Law, as it directly influences the resulting pressure value.

Boltzmann Constant

The Boltzmann constant (k) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. Its value is approximately 1.38 x 10^-23 J/K. In the context of interstellar space, it is used in the Ideal Gas Law to convert temperature into energy, allowing for the calculation of pressure based on the thermal motion of hydrogen atoms at low temperatures.

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The pressure inside a tank of neon is 150 atm. The temperature is 25℃. On average, how many atomic diameters does a neon atom move between collisions?

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Dust particles are ≈ 10 μm in diameter. They are pulverized rock, with ρ ≈ 2500 kg/m³. If you treat dust as an ideal gas, what is the rms speed of a dust particle at 20℃?

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Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H₂). The number density is about 1 atom/cm³ and the temperature is about 3 K. What is the edge length L of an L ✕ L ✕ L cube of gas with 1.0 J of thermal energy?

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A mad engineer builds a cube, 2.5 m on a side, in which 6.2-cm-diameter rubber balls are constantly sent flying in random directions by vibrating walls. He will award a prize to anyone who can figure out how many balls are in the cube without entering it or taking out any of the balls. You decide to shoot 6.2-cm-diameter plastic balls into the cube, through a small hole, to see how far they get before colliding with a rubber ball. After many shots, you find they travel an average distance of 1.8 m. How many rubber balls do you think are in the cube?

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Textbook Question

Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away. How far would you be able to see if all the molecules around you suddenly doubled in volume?

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Textbook Question

Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away. How far would you be able to see if the temperature suddenly rose from 20°C to a blazing hot 1500°C with the pressure unchanged?

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