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.

Question

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 &#34;degree of freedom&#34; 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 &#34;random&#34; 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.

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