\sect{Chemical potential, reaction in ideal quantum gases} ${N}$ fermions of type ${F}$ that have spin $1/2$ are placed in a box of volume ${V}$. Each fermion can decay into a boson of type ${B}$ that have spin ${0}$, and a fermion of type ${A}$ that has spin $1/2$. The reaction is ${F\rightarrow A+B}$, and it has an energy gain ${\epsilon_0}$. This means that ${A+B}$ has a lower binding energy than ${F}$. The masses of the particles are $m_{F},m_{A},m_{B}$ respectively. \Dn (1) Assuming ideal gases at temperature ${T}$, write the chemical equilibrium condition that determine the densities ${n_{F}, n_{A}, n_{B}}$ at equilibrium. \Dn (2) Write the chemical equilibrium condition at ${T=0}$. Describe the dependece of the densities on ${\epsilon_{0}}$. Find $\epsilon_c$ such that for $\epsilon_0>\epsilon_c$ the number of F fermions vanishes. \Dn (3) Assume that the condensation of bosons B occurs at~$T_c$ such that ${T_c \ll p_F^2/(2m_A) < \epsilon_0}$, where $p_F$ is the Fermi momentum of fermions A. Evaluate $T_c$ and rewrite the condition on $T_c$ in terms of the given parameters. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%