Part AConsider a monatomic gas of particles each with mass m. What is vzrmsExpress your answer in terms of T, kB, m, and other given quantities.▸ View Available Hint(s)Vz,rmsSubmitPart BUrms = √(1²) =Submit/ (v²) =Part CV-ΑΣΦNow consider the same system: a monatomic gas of particles of mass M, except in three dimensions. Find Urms, the rms speed if the gas is at an absolute temperature T.Express your answer in terms of T, kB, m, and other given quantities.▸ View Available Hint(s)20 =n|VL]ΑΣΦ[www]Submit Previous Answers Request Answer[www]What is the rms speed of molecules in air at 0°C? Air is composed mostly of N₂ molecules, so you may assume that it has molecules of average mass 28.0 x 1.661 × 10-27 kgExpress your answer in meters per second, to the nearest integer.=[5] ΑΣΦ??m/s(v2), the root mean square (rms) of the x component of velocity of the gas particles if the gas is at an absolute temperature T??4.65 x 10-26 kg.ActivaGo to Se In statistical mechanics, thermal energy is the random motion of the microscopic world. The average kinetic orpotential energy of each degree of freedom of the microscopic world therefore depends on the temperature. Ifheat is added, molecules increase their translational and rotational speeds, and the atoms constituting themolecules vibrate with larger amplitude about their equilibrium positions. It is a fact of nature that the energy ofeach degree of freedom is determined solely by the temperature. The Equipartition Theorem states thisquantitatively:The average energy associated with each degree of freedom in a system at absolute temperature Tis(1/2)kßT, where kß = 1.38 x 10-23 J/K is Boltzmann's constant.A "degree of freedom" corresponds to any dynamical variable that appears quadratically in the energy. Forinstance, (1/2)mv² is the kinetic energy of a gas particle of mass m with velocity component ₂ along the xaxis.The Equipartition Theorem follows from the fundamental postulate of statistical mechanics--that everyenergetically accessible quantum state of a system has equal probability of being populated, which in turn leadsto the Boltzmann distribution for a system in thermal equilibrium. From the standpoint of an introductory physicscourse, equipartition is best regarded as a principle that is justified by observation.In this problem we first investigate the particle model of an ideal gas. An ideal gas has no interactions among itsparticles, and so its internal energy is entirely "random" kinetic energy. If we consider the gas as a system, itsinternal energy is analogous to the energy stored in a spring. If one end of the gas container is fitted with a slidingpiston, the pressure of the gas on the piston can do useful work. In fact, the empirically discovered ideal gas law,pV = NkBT, enables us to calculate this pressure. This rule of nature is remarkable in that the value of themass does not affect the energy (or the pressure) of the gas particles' motion, only the temperature. It providesstrong evidence for the validity of the Equipartition Theorem as applied to a particle gas:\m(v²) = m(v}) = {m(v²) = {kBT.where the angle brackets represent the average value of that quantity.

Answered: Image /qna-images/answer/15ddd995-df71-40a5-affc-adc410e4323a.jpg

Consider a monatomic gas of particles each with mass m. What is Vz,rms Express your answer in terms of T, KB, m, and other given quantities. ► View Available Hint(s) Vz.rms (v²) = Submit Part B Urms = √(1²) = Submit |VL] ΑΣΦ Part C Now consider the same system: a monatomic gas of particles of mass m, except in three dimensions. Find Urms, the rms speed if the gas is at an absolute temperature T Express your answer in terms of T, KB, m, and other given quantities. ▸ View Available Hint(s) 9 [IV] ΑΣΦ [w] C (v2), the root mean square (rms) of the x component of velocity of the gas particles if the gas is at an absolute temperature T? ? What is the rms speed of molecules in air at 0°C? Air is composed mostly of N₂ molecules, so you may assume that it has molecules of average mass 28.0 x 1.661 x 10-27 kg 4.65 x 10-26 kg. Express your answer in meters per second, to the nearest integer.

Consider a monatomic gas of particles each with mass m. What is Vz,rms Express your answer in terms of T, KB, m, and other given quantities. ► View Available Hint(s) Vz.rms (v²) = Submit Part B Urms = √(1²) = Submit |VL] ΑΣΦ Part C Now consider the same system: a monatomic gas of particles of mass m, except in three dimensions. Find Urms, the rms speed if the gas is at an absolute temperature T Express your answer in terms of T, KB, m, and other given quantities. ▸ View Available Hint(s) 9 [IV] ΑΣΦ [w] C (v2), the root mean square (rms) of the x component of velocity of the gas particles if the gas is at an absolute temperature T? ? What is the rms speed of molecules in air at 0°C? Air is composed mostly of N₂ molecules, so you may assume that it has molecules of average mass 28.0 x 1.661 x 10-27 kg 4.65 x 10-26 kg. Express your answer in meters per second, to the nearest integer.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Diatomic Gas

Molecule, is a collection of two or more atoms together (which constitute the smallest observable unit) into a pure material. There are some ways to represent the structure of a molecule. Factor symbols represent atoms in Lewis structures, and dots represent electrons that accompany them. A pair of mutual electrons (covalent bond) may be represented by a single dash. The ball-and-stick model depicts the spatial structure of the atoms better.

Question

Part A
Consider a monatomic gas of particles each with mass m. What is vzrms
Express your answer in terms of T, kB, m, and other given quantities.
▸ View Available Hint(s)
Vz,rms
Submit
Part B
Urms = √(1²) =
Submit
/ (v²) =
Part C
V-ΑΣΦ
Now consider the same system: a monatomic gas of particles of mass M, except in three dimensions. Find Urms, the rms speed if the gas is at an absolute temperature T.
Express your answer in terms of T, kB, m, and other given quantities.
▸ View Available Hint(s)
20 =
n
|VL]ΑΣΦ
[www]
Submit Previous Answers Request Answer
[www]
What is the rms speed of molecules in air at 0°C? Air is composed mostly of N₂ molecules, so you may assume that it has molecules of average mass 28.0 x 1.661 × 10-27 kg
Express your answer in meters per second, to the nearest integer.
=
[5] ΑΣΦ
?
?
m/s
(v2), the root mean square (rms) of the x component of velocity of the gas particles if the gas is at an absolute temperature T?
?
4.65 x 10-26 kg.
Activa
Go to Se

In statistical mechanics, thermal energy is the random motion of the microscopic world. The average kinetic or
potential energy of each degree of freedom of the microscopic world therefore depends on the temperature. If
heat is added, molecules increase their translational and rotational speeds, and the atoms constituting the
molecules vibrate with larger amplitude about their equilibrium positions. It is a fact of nature that the energy of
each degree of freedom is determined solely by the temperature. The Equipartition Theorem states this
quantitatively:
The average energy associated with each degree of freedom in a system at absolute temperature Tis
(1/2)kßT, where kß = 1.38 x 10-23 J/K is Boltzmann's constant.
A "degree of freedom" corresponds to any dynamical variable that appears quadratically in the energy. For
instance, (1/2)mv² is the kinetic energy of a gas particle of mass m with velocity component ₂ along the x
axis.
The Equipartition Theorem follows from the fundamental postulate of statistical mechanics--that every
energetically accessible quantum state of a system has equal probability of being populated, which in turn leads
to the Boltzmann distribution for a system in thermal equilibrium. From the standpoint of an introductory physics
course, equipartition is best regarded as a principle that is justified by observation.
In this problem we first investigate the particle model of an ideal gas. An ideal gas has no interactions among its
particles, and so its internal energy is entirely "random" kinetic energy. If we consider the gas as a system, its
internal energy is analogous to the energy stored in a spring. If one end of the gas container is fitted with a sliding
piston, the pressure of the gas on the piston can do useful work. In fact, the empirically discovered ideal gas law,
pV = NkBT, enables us to calculate this pressure. This rule of nature is remarkable in that the value of the
mass does not affect the energy (or the pressure) of the gas particles' motion, only the temperature. It provides
strong evidence for the validity of the Equipartition Theorem as applied to a particle gas:
\m(v²) = m(v}) = {m(v²) = {kBT.
where the angle brackets represent the average value of that quantity.

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