\sect{Chemical equilibrium A=A+e (Exam 2011)} $N_0$ atoms of type $A$ are placed in an empty box of volume $V$, such that their initial density is $n_0=N_0/V$. The ionization energy of the atoms is $\varepsilon_0$. The box is held in temperature $T$, and eventually a chemical equilibrium ${A \rightleftarrows A^{+} + e^{-}}$ is reached. The fraction of ionized atoms is ${x=N^{+}/N_0}$. The masses of the particles are $m_e$ for the electron, and $m_{A^+} \approx m_A$ for the atoms and the ions. \Dn (1) Define temperature $T_0$ such that ${T \gg T_0}$ is a sufficient condition for treating the gas of atoms in the Boltzmann approximation. \Dn (2) Assuming the Boltzmann approximation for both the atoms and the electrons, write an equation for $x$. Write its {\em approximate} solution assuming ${x\ll1}$. Write the condition for the validity of the latter assumption. \Dn (3) Assuming that ${x\ll1}$, write a condition on the density $n_0$, that above $T_0$ it was legitimate to treat the electrons in the Boltzmann approximation. Note: the condition is a simple inequality and should be expressed using $(m_e,m_A,\varepsilon_0)$. \Dn Assume that the condition in (3) breaks down. It follows that there is a regimes ${T_0 \ll T \ll T_1}$ where the atoms can be treated in the Boltzmann approximation, while the electrons can be treated as a low temperature quantum gas. \Dn (4) Write an equation for $x$ assuming that the electrons can be treated approximately as a zero temperature Fermi gas. Exotic functions should not appear. You are not expected to solve this transcendental equation. \Dn (5) What would be the equation for $x$ if the electrons were Bosons instead of Fermions. \Dn {\em Note:} Express all the final answers using $(m_e,m_A,n_0,\varepsilon_0,T)$, and {\em elementary} functions. \\ Exotic functions should not appear. It is allowed to use the notation $\lambda_e(T)=(2\pi/\mass_e T)^{1/2}$.