face Lecture
120 min.
chemical potential Gibbs distribution grand canonical ensemble statistical mechanics
These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.computer Computer Simulation
30 min.
face Lecture
5 min.
assignment Homework
group Small Group Activity
120 min.
assignment Homework
At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}
What is the energy of the ground state and the first and second excited states of the \(H_2\) molecule? i.e. the lowest three distinct energy eigenvalues.
At room temperature, what is the relative probability of
finding a hydrogen molecule in the \(\ell=0\) state versus finding it
in any one of the \(\ell=1\) states?
i.e. what is
\(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)
At what temperature is the value of this ratio 1?
assignment Homework
The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework
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 \(\textsf{O}_{2}\).