Thermodynamics, Statistical Physics, and Quantum Mechanics

(^168) SOLUTIONS
The coefficient 2 in front of thesecond termresultsfrom thepresence of
two surfaces of thesoapfilmenclosing the air(comparewith Problem4.61).
From (S.4.62.2) and (S.4.62.3) we obtain
Substituting(S.4.62.4)into(S.4.62.1), weobtain
andso
Notethat if ais very small the volume of the new bubble isclose to the
sum of the original volumes, whereas if it is very large the surfacearea of
the new bubble is roughly the sum of theoriginalsurfaceareas.
4.63 Soap Bubbles in Equilibrium (Moscow
Phys-Tech)
a) The equilibrium is unstable. It isobvious from purelymechanicalconsid-
erationsthat if theradius ofone bubbledecreases and theotherincreases,
the pressure in thefirstbubble (which isinversely proportional to will
increase andthat in thesecondbubblewill decrease, leading tofurther
changes inrespectiveradii until thesystem becomes onebubble with ra-
dius (see FigureS.4.63). The sameresult can be obtained by considering
the freeenergy of the system.
b) The freeenergy of the bubble consists of twoparts: a volumepart,which
is just the freeenergy of a gas(seeProblem4.38), and a surfacepart,which