(a) the probability of atoms gathering in one half of the vessel;
(b) the approximate numerical value of N ensuring the occurrence
of this event within the time interval t 101 ° years (the age of the
Universe).
2.155. Find the statistical weight of the most probable distribution
of N = 10 identical molecules over two halves of the cylinder's
volume. Find also the probability of such a distribution.
2.156. A vessel contains N molecules of an ideal gas. Dividing
mentally the vessel into two halves A and B, find the probability
that the half A contains n molecules. Consider the cases when N = 5
and n = 0, 1, 2, 3, 4, 5.
2.157. A vessel of volume V, contains N molecules of an ideal
gas. Find the probability of n molecules getting into a certain separat-
ed part of the vessel of volume V. Examine, in particular, the case
V = V 0 /2.
2.158. An ideal gas is under standard conditions. Find the diame-
ter of the sphere within whose volume the relative fluctuation of the
number of molecules is equal to i = 1.0.10-3. What is the average
number of molecules inside such a sphere?
2.159. One mole of an ideal gas consisting of monatomic molecules
is enclosed in a vessel at a temperature T, = 300 K. How many
times and in what way will the statistical weight of this system
(gas) vary if it is heated isochorically by AT = 1.0 K?
2.5. Liquids. Capillary Effects
- Additional (capillary) pressure in a liquid under an arbitrary surface
(Laplace's formula):
1 1 \
" a
where a is the surface tension of a given liquid. - Free energy increment of the surface layer of a liquid:
dF = a dS, (2.5b)
where dS is the area increment of the surface layer. - Amount of heat required to form a unit area of the liquid surface layer
during the isothermal increase of its surface:
„, du (2.5c)
dT
2.160. Find the capillary pressure
(a) in mercury droplets of diameter d = 1.5 pm;
(b) inside a soap bubble of diameter d = 3.0 mm if the surface
tension of the soap water solution is a = 45 mN/m.
2.161. In the bottom of a vessel with mercury there is a round
hole of diameter d = 70 pm. At what maximum thickness of the
mercury layer will the liquid still not flow out through this hole?(2.5a)93