Chapter 7, Quantum Statistics Video Solutions, An Introduction to Thermal Physics

Problem 1

Near the cells where oxygen is used, its chemical potential is significantly lower than near the lungs. Even though there is no gaseous oxygen near these cells, it is customary to express the abundance of oxygen in terms of the partial pressure of gaseous oxygen that would be in equilibrium with the blood. Using the independent-site model just presented, with only oxygen present, calculate and plot the fraction of occupied heme sites as a function of the partial pressure of oxygen. This curve is called the Langmuir adsorption isotherm ("isotherm" because it's for a fixed temperature). Experiments show that adsorption by myosin follows the shape of this curve quite accurately.

Khoobchandra Agrawal

Numerade Educator

01:32

Problem 2

In a real hemoglobin molecule, the tendency of oxygen to bind to a heme site increases as the other three heme sites become occupied. To model this effect in a simple way, imagine that a hemoglobin molecule has just two sites, either or both of which can be occupied. This system has four possible states (with only oxygen present). Take the energy of the unoccupied state to be zero, the energies of the two singly occupied states to be $-0.55 \mathrm{eV}$, and the energy of the doubly occupied state to be $-1.3 \mathrm{eV}$ (so the change in energy upon binding the second oxygen is $-0.75 \mathrm{eV}$ ). As in the previous problem, calculate and plot the fraction of occupied sites as a function of the effective partial pressure of oxygen. Compare to the graph from the previous problem (for independent sites). Can you think of why this behavior is preferable for the function of hemoglobin?

Dominador Tan

Numerade Educator

03:56

Problem 3

Consider a system consisting of a single hydrogen atom/ion, which has two possible states: unoccupied (i.e., no electron present) and occupied (i.e., one electron present, in the ground state). Calculate the ratio of the probabilities of these two states, to obtain the Saha equation, already derived in Section 5.6 Treat the electrons as a monatomic ideal gas, for the purpose of determining $\mu$. Neglect the fact that an electron has two independent spin states.

Prashant Bana

Numerade Educator

03:56

Problem 4

Repeat the previous problem, taking into account the two independent spin states of the electron. Now the system has two "occupied" states, one with the electron in each spin configuration. However, the chemical potential of the electron gas is also slightly different. Show that the ratio of probabilities is the same as before: The spin degeneracy cancels out of the Saha equation.

Christopher Marshall

Numerade Educator

02:34

Problem 5

Consider a system consisting of a single impurity atom/ion in a semiconductor. Suppose that the impurity atom has one "extra" electron compared to the neighboring atoms, as would a phosphorus atom occupying a lattice site in
a silicon crystal. The extra electron is then easily removed, leaving behind a positively charged ion. The ionized electron is called a conduction electron, because it is free to move through the material; the impurity atom is called a donor, because it can "donate" a conduction electron. This system is analogous to the hydrogen atom considered in the previous two problems except that the ionization energy is much less, mainly due to the screening of the ionic charge by the dielectric behavior of the medium.
(a) Write down a formula for the probability of a single donor atom being ionized. Do not neglect the fact that the electron, if present, can have two independent spin states. Express your formula in terms of the temperature, the ionization energy $I$, and the chemical potential of the "gas" of ionized electrons.
(b) Assuming that the conduction electrons behave like an ordinary ideal gas (with two spin states per particle), write their chemical potential in terms of the number of conduction electrons per unit volume, $N_{c} / V$
(c) Now assume that every conduction electron comes from an ionized donor atom. In this case the number of conduction electrons is equal to the number of donors that are ionized. Use this condition to derive a quadratic equation for $N_{c}$ in terms of the number of donor atoms $\left(N_{d}\right),$ eliminating $\mu$. Solve for $N_{c}$ using the quadratic formula. (Hint: It's helpful to introduce some abbreviations for dimensionless quantities. 'Iry $x=N_{c} / N_{d}, t=k T / I$ and so on.)
(d) For phosphorus in silicon, the ionization energy is 0.044 eV. Suppose that there are $10^{17} \mathrm{P}$ atoms per cubic centimeter. Using these numbers, calculate and plot the fraction of ionized donors as a function of temperature. Discuss the results.

Dominador Tan

Numerade Educator

13:45

Problem 6

Show that when a system is in thermal and diffusive equilibrium with a reservoir, the average number of particles in the system is
$$\bar{N}=\frac{k T}{\mathcal{Z}} \frac{\partial \mathcal{Z}}{\partial \mu}$$
where the partial derivative is taken at fixed temperature and volume. Show also that the mean square number of particles is
$$
\overline{N^{2}}=\frac{(k T)^{2}}{Z} \frac{\partial^{2} \mathcal{Z}}{\partial \mu^{2}}
$$
Use these results to show that the standard deviation of $N$ is
$$
\sigma_{N}=\sqrt{k T(\partial \bar{N} / \partial \mu)}
$$
in analogy with Problem 6.18 . Finally, apply this formula to an ideal gas, to obtain a simple expression for $\sigma_{N}$ in terms of $\bar{N}$. Discuss your result briefly.

Jack Hou

Numerade Educator

13:12

Problem 7

In Section 6.5 I derived the useful relation $F=-k T \ln Z$ between the Helmholtz free energy and the ordinary partition function. Use an analogous argument to prove that $$\Phi=-k T \ln \mathcal{Z}$$
where $\mathcal{Z}$ is the grand partition function and $\Phi$ is the grand free energy introduced in Problem 5.23.

Jack Hou

Numerade Educator