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 48c

Chapter 20, Problem 48c

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?

Verified step by step guidance

Start by recalling the formula for the thermal energy of an ideal gas:

E = rac{3}{2} Nk_B T

, where

E

is the thermal energy,

N

is the number of particles,

k_B

is the Boltzmann constant (

1.38 	imes 10^{-23} \(\text{ J/K}\)

), and

T

is the temperature in kelvins.

Rearrange the formula to solve for

N

, the number of particles:

N = rac{2E}{3k_B T}

. Substitute the given values:

E = 1.0 \(\text{ J}\)

T = 3 \(\text{ K}\)

, and

k_B = 1.38 	imes 10^{-23} \(\text{ J/K}\)

Once

N

is calculated, use the relationship between the number density

n

(particles per unit volume) and the total number of particles:

N = n 	imes V

, where

V

is the volume of the cube. Rearrange to find

V

V = rac{N}{n}

. The number density

n

is given as

1 \(\text{ atom/cm}\)^3

, which must be converted to SI units:

1 \(\text{ atom/cm}\)^3 = 10^6 \(\text{ atoms/m}\)^3

The volume

V

of the cube is related to its edge length

L

by the formula

V = L^3

. Solve for

L

L = V^{1/3}

. Substitute the value of

V

obtained in the previous step.

Finally, ensure all units are consistent (e.g., meters for

L

) and verify the calculation steps to confirm the edge length

L

of the cube.

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Key Concepts

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

Thermal Energy

Thermal energy is the total kinetic energy of the particles in a substance due to their motion. In the context of gases, it is related to temperature and can be quantified using the equation E = (3/2)nRT for ideal gases, where E is the thermal energy, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates pressure, volume, temperature, and the number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps in understanding the behavior of gases under various conditions.

Volume and Density

Volume is the amount of space occupied by a substance, and in this case, it is represented by the cube's edge length L, where the volume V = L³. Density, defined as mass per unit volume, is crucial for determining how many particles are present in a given volume, which directly affects the thermal energy and behavior of the gas in interstellar space.

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Related Practice

Textbook Question

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.

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Textbook Question

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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Textbook Question

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

You are watching a science fiction movie in which the hero shrinks down to the size of an atom and fights villains while jumping from air molecule to air molecule. In one scene, the hero's molecule is about to crash head-on into the molecule on which a villain is riding. The villain's molecule is initially 50 molecular radii away and, in the movie, it takes 3.5 s for the molecules to collide. Estimate the air temperature required for this to be possible. Assume the molecules are nitrogen molecules, each traveling at the rms speed. Is this a plausible temperature for air?

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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