molecule has states with the following energies: 0, 1ε, 2ε, 3ε, and 4ε, where ε = 1.0 x 10-20 J. Calculate the average number of molecules in the first excited state (1ε) for a collection of 1000 molecules in thermal equilibrium at T = 300 K. Note that the average number of molecules in a state is just the probability that a molecule is in the state times the number of molecules. Provide your answer as a number in normal form.

molecule has states with the following energies: 0, 1ε, 2ε, 3ε, and 4ε, where ε = 1.0 x 10-20 J. Calculate the average number of molecules in the first excited state (1ε) for a collection of 1000 molecules in thermal equilibrium at T = 300 K. Note that the average number of molecules in a state is just the probability that a molecule is in the state times the number of molecules. Provide your answer as a number in normal form.

University Physics Volume 2

18th Edition

ISBN:9781938168161

Author:OpenStax

Publisher:OpenStax

Chapter2: The Kinetic Theory Of Gases

Section: Chapter Questions

Problem 68P: Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a...

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Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between 3.451021 J and 3.501021 J. `
(a) Repeat Exercise 31.2, and convert the energy to joules or calories. (b) If all of this energy is converted to thermal energy in the gas, what is its temperature increase, assuming 50.0 cm3 of ideal gas at 0.250atm pressure? (The small answer is consistent with the fact that the energy is large on a quantum mechanical scale but small on a macroscopic scale.)
A molecule has states with the following energies: 0, 1ε, 2ε, where ε = 1.0 x 10-20 J. Calculate the average energy of this molecule in thermal equilibrium at T = 300 K. Provide your answer in units of ε. In other words, your solution should have the form of (number)ε, such as 2.137ε, but the answer that you enter is just 2.137, a number in normal form with 3 decimal places (X.XXX). It is a good idea to keep 4 decimal places during your calculation, then round to 3 decimal places for your submitted answer.
A molecule has states with the following energies: 0, 1ε, 2ε, 3ε, and 4ε, where ε = 1.0 x 10-20 J. Calculate the probability that a molecule is in the ground state (with zero energy) for a collection of molecules in thermal equilibrium at T = 300 K. Provide your answer as a number in normal form to 3 decimal places (in the form X.XXX). It is a good idea to keep 4 decimal places during your calculation, then round to 3 decimal places for your submitted answer. Hint: note that this molecule has a finite number of states so you must take a finite sum, do not use expressions for infinite sums. Also note that your calculations for this problem will be useful for the next two problems, so keep them.
What is the ratio of the probability of finding a molecule moving with the average speed to the probability of finding a molecule moving with 3 times the average speed? How does this ratio depend on temperature?
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A triatomic molecule can have a linear configuration, as does CO2 (as shown), or it can be nonlinear, like H2O (as shown). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible. What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant, (a) if the molecules are linear and (b) if the molecules are nonlinear? At high temperatures, a triatomic molecule has two modes of vibration, and each contributes 1/2 R to the molar specific heat for its kinetic energy and another 1/2 R for its potential energy. Identify the high-temperature molarspecific heat at constant volume for a triatomic ideal gas of (c) linear molecules and (d) nonlinear molecules. (e) Explain how specific heat data can be used to determine whether a triatomic molecule is linear or nonlinear. Are the data as shown sufficient to make this determination?
You measure the average free path λ and the average collision time τ of the molecules of a diatomic gas of molecular mass 6.00 × 10-²⁵ kg and radius r = 1.0 x 10-¹⁰ m. From these microscopic data can we obtain macroscopic properties such as temperature T and pressure P? If so, consider λ = 4.32 x 10-⁸ m and τ = 3.00 x 10-¹⁰ s and calculate T and P. indicate the correct answer: 1- Not possible2- Yes, T =150 K and P ~ 2.04 atm.3- Yes, T = 150 K and P ~ 4.08 atm.4- Yes, T = 300 K and P ~ 4.08 atm.5- Yes, T = 300 K and P ~ 5.32 atm6- Yes, T = 400 K and P ~ 4.08 atm.7- Yes, T = 400 K and P ~ 5.32 atm. obs.: If necessary, consider: R = 8.314 J/mol∙K1 cal = 4.19 Jkb =1,38 x 10⁻²³ m² kg s⁻² K⁻¹
A gas is cooled and trapped in a two-dimensional region by laser beams and magnets. The speed distribution function of the gas molecules obeys the two-dimensional Maxwell-Boltzmann distribution law, f(v)=2π(m2πkB T)ve−mv22kB T where m is the mass of a gas molecule, T is the temperature, and v is the speed. What is the average speed of the gas molecule?
I'm hoping for a good explanation of how to do this. I'm also wondering why it matters if the configuration is linear or nonlinear? A triatomic molecule can have a linear configuration, as does CO2 (Figure a), or it can be nonlinear, like H2O (Figure b). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible. (a) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are linear?Eint/nT = ? (b) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are nonlinear?Eint/nT = ? At high temperatures, a triatomic molecule has two modes of vibration, and each contributes 0.5R to the molar specific heat for its kinetic energy and another 0.5R for its potential energy. (c) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the linear molecules. (Use…
A thermocouple, with a spherical junction diameter of 0.5 mm, is used for measuring the temperature of hot air flow in a circular duct. The convection heat transfer coefficient of the air flow can be related with the diameter (D) of the spherical junction and the average air flow velocity (V) as h = 2.2(V/D)0.5, where D, h, and V are in m, W/m2?K and m/s, respectively. The properties of the thermocouple junction are k = 35 W/m?K, r = 8500 kg/m3, and cp = 320 J/kg?K. Determine the minimum air flow velocity that the thermocouple can be used, if the maximum response time of the thermocouple to register 99 percent of the initial temperature difference is 5 s.