Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
For N_{2} gas molecules in a 10 cm cubic box, the rules of QM dictate the discrete allowed values for:Follows Quantum mechanical correction to the equipartition theorem.
In the prompt, explain that 50, 500 and 5000 should be equally spaced, for instance. 2, 5, and 10 are also approximately equally spaced.
As it turns out, the heat capacity is about \(\frac52\) for almost the entire temperature range.
The bond length of N_{2} is 1.1 Å. Its mass is \(m = 14\text{ amu}\approx 2\times 10^{-26}\text{ kg}\). The moment of inertia is thus \begin{align} I &= 2\cdot 2\times 10^{-26}\text{ kg} \left(0.55\times 10^{-10}\text{ m}\right)^2 \\ &\approx 1.2\times 10^{-46}\text{ kg}\cdot\text{m}^2 \end{align} From which we can find the energy eigenvalues: \begin{align} E_{\ell} &= \frac{\hbar^2}{2I}\ell(\ell+1) \\ &= \frac{\left(10^{-34} \text{ J}\cdot\text{s} \right)^2 }{2\cdot 1.2\times 10^{-46}\text{ kg}\cdot\text{m}^2} \ell(\ell+1) \\ &\approx \ell(\ell+1)\cdot 0.4\times 10^{-22}\text{ J} \end{align}
I looked up the experimental vibrational frequency, which is 2358 cm\(^{-1}\).
(use a logarithmic temperature axis)
group Small Group Activity
30 min.
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assignment Homework
assignment Homework
It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is \(22^\circ\)C at the ocean surface and \(4^{o}\)C at the ocean floor.
What is the maximum possible efficiency of an engine operating between these two temperatures?
assignment Homework
assignment Homework
group Small Group Activity
30 min.
assignment Homework
At a power plant that produces 1 GW (\(10^{9} \text{watts}\)) of electricity, the steam turbines take in steam at a temperature of \(500^{o}C\), and the waste energy is expelled into the environment at \(20^{o}C\).
What is the maximum possible efficiency of this plant?
Suppose you arrange the power plant to expel its waste energy into a chilly mountain river at \(15^oC\). Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 10 cents per kilowatt-hour?
At what rate will the plant expel waste energy into this river?
Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will the temperature of the river increase?
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In carbon monoxide poisoning the CO replaces the \(\textsf{O}_{2}\) adsorbed on hemoglobin (\(\text{Hb}\)) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or may be occupied either with energy \(\varepsilon_A\) by one molecule \(\textsf{O}_{2}\) or with energy \(\varepsilon_B\) by one molecule CO. Let \(N\) fixed heme sites be in equilibrium with \(\textsf{O}_{2}\) and CO in the gas phases at concentrations such that the activities are \(\lambda(\text{O}_2) = 1\times 10^{-5}\) and \(\lambda(\text{CO}) = 1\times 10^{-7}\), all at body temperature \(37^\circ\text{C}\). Neglect any spin multiplicity factors.
First consider the system in the absence of CO. Evaluate \(\varepsilon_A\) such that 90 percent of the \(\text{Hb}\) sites are occupied by \(\textsf{O}_{2}\). Express the answer in eV per \(\textsf{O}_{2}\).
Now admit the CO under the specified conditions. Fine \(\varepsilon_B\) such that only 10% of the Hb sites are occupied by \(\textsf{O}_{2}\).
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