Chapter 12, Introducing chemical potential Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

A gas is contained in a harmonic potential well $V(r)=\alpha r^2$, where $\alpha$ is a constant and $r$ is the distance from the centre of the well. Show that under isothermal conditions, the number density profile is $n=n_0$ $\exp \left(-\alpha r^2 / k_{\mathrm{B}} T\right)$

Supratim Pal

Numerade Educator

02:04

Problem 2

The chemical potentials for free electrons in metals A and B differ by $1.1 \mathrm{eV}$ when the metals are not in contact. When cubic blocks of these metals are brought into contact, electrons move from A to B until the chemical potential is uniform. How many electrons are transferred, if the contact area is $1 \mathrm{~cm}^2$ and the junction can be modelled as a parallel plate capacitor with gap $10^{-10} \mathrm{~m}$ ? Is this a significant fraction of the total number of free electrons in the metal blocks?

Narayan Hari

Numerade Educator

03:48

Problem 3

(i) Use the Saha equation to find the degree of ionization of the photosphere of a hot star having $T=12000 \mathrm{~K}, n \simeq 10^{23} \mathrm{~m}^{-3}$. (ii) In the limit $n \rightarrow 0$ the Saha equation predicts that a hydrogen cloud in thermal equilibrium would be completely ionized even at $T \ll E_{\mathrm{R}} / k_{\mathrm{B}}$. This seems to go against our ordinary intuition that an excited atom will quickly decay to its ground state and stay there. What is going on?

Keshav Singh

Numerade Educator

12:34

Problem 4

Generalize the argument for the Saha equation, so as to obtain the following equation for the ratio of densities of ions in a plasma:
$$
\frac{n_{i+1}}{n_i}=\frac{2 g_{i+1}}{g_i n_{\mathrm{e}}}\left(\frac{m_{\mathrm{e}} k_{\mathrm{B}} T}{2 \pi \hbar^2}\right)^{3 / 2} e^{-\Delta E / k_{\mathrm{B}} T},
$$
where $i$ is the degree of ionization and $g_i=Z_{\text {int, }, i}$.

Linda Winkler

Numerade Educator