2) (20 points) The energy of the vibrational modes of a molecule are the same as those of a (quantum) harmonic oscillator with frequency ??. There is a gas of nitrogen molecules in thermodynamic equilibrium for which ??/k_B = 3340 K . a) What fraction of the molecules are in the ground state, the 1st excited state, and the 2nd excited state of their vibrational modes at a temperature of 800 K? Use the exact partition function for this part. b) At what temperature will 4% of the molecules be in the 1st excited vibrational state? You may approximate the vibrational partition function with the largest two terms in it. 3) (25 points) Assume that a room at sea level was filled with a gas of oxygen molecules O? in thermal equilibrium at 0 Ā°C. There are 8 protons and 8 neutrons in the nucleus of an oxygen atom O. You may take the masses of the proton and the neutron to be the same, and ignore the mass of the electrons. 1 atm=1.01x10? N/mĀ² , ?=1.05x10?Ā³? JĀ·s , m_p=1.67x10?Ā²? kg. a) What would the (particle) number density be according to the ideal gas law? b) Compare the number density with the quantum concentration at the same temperature. c) Is the gas in the classical or quantum regime?
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The energy levels of a harmonic oscillator are given by: En = (n+1/2)āĻ where n is the quantum number. The probability of finding a molecule in a particular energy level is given by: Pn = e^(-En/(kB*T))/Z At a temperature of 800 K, we have Ī² = 1/(kB*800 K) = Show moreā¦
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Quantum mechanics is used to describe the vibrational motion of molecules, but analysis using classical physics gives some useful insight. In a classical model the vibrational motion can be treated as SHM of the atoms connected by a spring. The two atoms in a diatomic molecule vibrate about their center of mass, but in the molecule HI, where one atom is much more massive than the other, we can treat the hydrogen atom as oscillating in SHM while the iodine atom remains at rest. (a) A classical estimate of the vibrational frequency is $f=7 \times 10^{13} \mathrm{~Hz}$. The mass of a hydrogen atom differs little from the mass of a proton. If the HI molecule is modeled as two atoms connected by a spring, what is the force constant of the spring? (b) The vibrational energy of the molecule is measured to be about $5 \times 10^{-20} \mathrm{~J}$. In the classical model, what is the maximum speed of the H atom during its SHM? (c) What is the amplitude of the vibrational motion? How does your result compare to the equilibrium distance between the two atoms in the HI molecule, which is about $1.6 \times 10^{-10} \mathrm{~m} ?$
Consider the air in an oven at 500 K. The oven has a volume of 0.15 m^3 and contains 2.2 Ć 10^24 identical nitrogen molecules, each having five degrees of freedom and a mass of 4.8 Ć 10^ā26 kg. (a) What is the thermal energy of this system? (b) The magnitude of the momentum of any molecule can range from 0 to p0. Estimating p0 to be roughly twice the root mean square momentum, what is the volume in momentum space that is available to any particle? (c) We are going to calculate the number of accessible quantum states for the molecules' translational motions, ignoring the rotational states because the latter turn out to be relatively few in comparison. Considering the translational motion only, how many different quantum states would be accessible to any particle, if it were all by itself? (VrVp/h^3, where Vp = (4/3)Ļp^3/0.) (d) What is the number of states per particle Ļc corrected for the case of identical particles? (e) How many different quantum states are accessible to the entire system?
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