External chemical potential. Consider a system at temperature τ, with N atoms of mass M in volume V. Let μ(0) denote the value of the chemical potential at the surface of the earth. (a) Prove carefully and honestly that the value of the total chemical potential for the identical system when translated to altitude h is
μ(h)=μ(0)+Mgh,
where g is the acceleration of gravity. (b) Why is this result different from that applicable to the barometric equation of an isothermal atmosphere?