A water molecule can vibrate in various ways, but the easiest type of vibration to excite is the "flexing" mode in which the hydrogen atoms move toward and away from each other but the HO bonds do not stretch. The oscillations of this mode are approximately harmonic, with a frequency of 4.8 x 1013 Hz. As for any quantum harmonic oscillator, the energy levels are hf, hf, hf, and so on. None of these levels are degenerate. (a) Calculate the probability of a water molecule being in its flexing ground state and in each of the first two excited states, assuming that it is in equi- librium with a reservoir (say the atmosphere) at 300 K. (Hint: Calculate Z by adding up the first few Boltzmann factors, until the rest are negligible.) (b) Repeat the calculation for a water molecule in equilibrium with a reservoir at 700 K (perhaps in a steam turbine).
A water molecule can vibrate in various ways, but the easiest type of vibration to excite is the "flexing" mode in which the hydrogen atoms move toward and away from each other but the HO bonds do not stretch. The oscillations of this mode are approximately harmonic, with a frequency of 4.8 x 1013 Hz. As for any quantum harmonic oscillator, the energy levels are hf, hf, hf, and so on. None of these levels are degenerate. (a) Calculate the probability of a water molecule being in its flexing ground state and in each of the first two excited states, assuming that it is in equi- librium with a reservoir (say the atmosphere) at 300 K. (Hint: Calculate Z by adding up the first few Boltzmann factors, until the rest are negligible.) (b) Repeat the calculation for a water molecule in equilibrium with a reservoir at 700 K (perhaps in a steam turbine).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:A water molecule can vibrate in various ways, but the easiest
type of vibration to excite is the "flexing" mode in which the hydrogen atoms move
toward and away from each other but the HO bonds do not stretch. The oscillations
of this mode are approximately harmonic, with a frequency of 4.8 x 1013 Hz. As
for any quantum harmonic oscillator, the energy levels are hf, hf, hf, and so
on. None of these levels are degenerate.
(a) Calculate the probability of a water molecule being in its flexing ground
state and in each of the first two excited states, assuming that it is in equi-
librium with a reservoir (say the atmosphere) at 300 K. (Hint: Calculate Z
by adding up the first few Boltzmann factors, until the rest are negligible.)
(b) Repeat the calculation for a water molecule in equilibrium with a reservoir
at 700 K (perhaps in a steam turbine).
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