Solution Found!
A water molecule can vibrate in various ways, but the
Chapter 6, Problem 10P(choose chapter or problem)
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 \(4.8\times10^{13}\mathrm{\ Hz}\). As for any quantum harmonic oscillator, the energy levels are \(\frac{1}{2}hf,\ \frac{3}{2}hf,\ \frac{5}{2}hf\) 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 equilibrium 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 700K (perhaps in a steam turbine).
Questions & Answers
(3 Reviews)
QUESTION:
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 \(4.8\times10^{13}\mathrm{\ Hz}\). As for any quantum harmonic oscillator, the energy levels are \(\frac{1}{2}hf,\ \frac{3}{2}hf,\ \frac{5}{2}hf\) 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 equilibrium 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 700K (perhaps in a steam turbine).
ANSWER:Step 1 of 8
(a)
The expression for the partition function for the vibrating atom is,
\(Z=e^{-h f / 2 k T}+e^{-3 h f / 2 k T}+e^{-5 h f / 2 k T}+\ldots\)
Here, h is Planck's constant, f is the frequency of the vibrating atom, k is the Boltzmann constant, and T is the temperature.
Use the numerical data of the above variables to calculate the value of \(\frac{h f}{2 k T}\).
Reviews
Review this written solution for 22787) viewed: 299 isbn: 9780201380279 | An Introduction To Thermal Physics - 1 Edition - Chapter 6 - Problem 10p
Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.
No thanks, I don't want to help other students
Review this written solution for 22787) viewed: 299 isbn: 9780201380279 | An Introduction To Thermal Physics - 1 Edition - Chapter 6 - Problem 10p
Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.
No thanks, I don't want to help other students
Review this written solution for 22787) viewed: 299 isbn: 9780201380279 | An Introduction To Thermal Physics - 1 Edition - Chapter 6 - Problem 10p
Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.
No thanks, I don't want to help other students