(^230) Aptitude Test Problems in Physics
2.24, The condition of equilibrium for the soap
bubble film consists in that the air pressure pbubi
is the sum of the external pressure pi and the excess
pressure 4a/r due to surface tension. It should be
noted that there are two air-soap film interfaces
in the soap bubble, each of which produces a pres-
sure 2a/r. For this reason, the excess pressure is
2 X 2a/r = 4a/r. Therefore, we can first write the
equilibrium condition in the form
40
Phut,' =-
,
r •After the radius of the bubble has been reduced
by half, the pressure produced by surface tension
becomes 8a/r. By hypothesis, the temperature is
maintained constant, and hence (according to
Boyle's law) a decrease in the volume of the bubble
by a factor of eight (its radius has decreased by
half) leads to an eight-fold increase in the air pres-
sure in the bubble (it becomes 8pbubi), so that we
can write
, 80
8 Pbubi = P2-1- —
Substituting Pbubi into this formula from the first
equation, we can finally writeP2=8P1+24a2.25. In the fireplace, large temperature gradients
may take place. If the bricks and the mortar are
made of different materials, i.e. materials with
different temperature expansion coefficients, the
fireplace can crack.
2.26. Let us suppose that the temperature of the
mixture of the liquids having the initial temper-
atures T 1 and T 2 has become T. Since the vessel
containing the mixture is thermally insulated
= 0), we can writec1m1 (T — T^1 ) c^2 m^2 (T — T2) = 0,