Laplace's formula
Spherical drop.
For arbitrary smooth surface:
where
and
-
are the main radii of curvature of the surface.
Examples.
1. Drop. ( = )
2. Soap bubble. ( = )
3. Bubble inside the liquid. ( = )
4. Liquid
jet. (
=
;
)
Home work.
W
hat
is happened with two bubbles at different sides of the tube?
Phenomena on the border of surfaces
T
he
conditions of equilibrium for tangential forces:
Contact
angle
.
The formula is valid only if:
For
we have
1.
and the
equilibrium is impossible – liquid will spread on all solid surface
(
).
2.
absolutely
unloosening
Home work.
How much water one can bring with sieve?
Capillarity
Water – glass Mercury – glass
If
then
;
If
then
.
If
then
.
The problem with water jet
Conservation laws:
1. The mass conservation:
2. The energy conservation:
Crystals
The main property of crystals - long range ordering. As usual the crystals are anisotropic. It means that the physical properties of crystal depend on direction.
There are polycrystals that haven't anisotropy in large scales. Monocrystals have ideal long range ordering inside all the volume.
Symmetry of lattice – it is major property to describe the crystals.
General definition of symmetry – "you are doing something but the result is the same".
Forms of lattice symmetries
1. Translation.
P
eriod
of identity: a,
b,
c,
so
,
,
.
Low-level cell. Primitive cell – it is minimal low-level cell.
2. Rotation.
n-fold
axis. The figure is coincided after turning on angle
.
There are only n-fold axes in crystals with n = 2, 3, 4 and 6. There are only these turning angels respectively: 180o, 120 o, 90 o, 60 o. It is a property of 3-dimentional space.
3. Reflection in plane.
Plane of symmetry. The figure is coincided after reflection in plane of symmetry.
4. Reflection about a point.
C
– it is the center
of symmetry.
The figure is coincided after reflection about of center of symmetry.
5. Rotation-reflection symmetry.
n-fold rotation-reflection axis. The figure is coincided after rotation about this axis on some angle and reflection in plane that hold this axis.
Group theory is the base for classification the crystals.
Fedorov's group contains 230 types of crystals. There were discovered 177 types of them in Nature up to this time.
There are seven crystal's systems – syngonies:
Triclinic lattice -
;
;Monoclinic lattice - ;
;
;Rhombic lattice - ;
;Tetragonal lattice -
;
;Rhombohedral lattice -
;
;Hexagonal lattice - ;
;
;Cubic lattice - ; .
Physical types of crystals
There are four types of crystals:
1. Ionic
crystals
(NaCl). Ionic
crystals consist of ions (
)
not of molecules. Ionic crystals have heteropolar (electrostatic)
bond of its elements.
2. Atomic crystals (Diamond, graphite). Atomic crystals have homeopolar (covalent) bond of its elements.
3. Metallic crystals (Cu). Lattice formed by the ions of metal and gas of free electrons.
4. Molecular crystals (H2O, CO2, O2, N2). molecular crystals have Van-der-Waals's bond of its elements.
Defects in crystals
Defects – it is the disturbances in an ideal lattice of a crystal.
Point defect.
Linear defects (or dislocations): glide dislocation, screw dislocation, edge dislocation and so on.
Defects can change the physical properties of crystals terrifically. The strength of the crystal changes on powers!
Specific heat of crystals
Average energy for the molecule in crystal is a sum of kinetic and potential energies in one degree of freedom:
Therefore the classic theory gives for chemically simple matter:
,
and
.
Because of
insignificant thermal expansion for solids
and one can say about common heat capacity
:
It is the classical law for solid's heat capacity.
Experiment gives another result – the heat capacity of solids depends on its temperature (at low temperatures):
T
he
true result was obtained by Einstein and Debye. They used the quantum
theory and obtained the formula that was in a good convenience with
practice.
Thermal conductivity at steady-state conditions
The experimental law (Fourier's law):
- it is the density of heat flux ;
- it is the thermo conductivity .
- it is the gradient of the temperature.
Some problems.
One dimensional heat transfer.
Heat flux along x axis:
S – it is area of cross section.
When there are no sources in the media we conclude from the energy conservation law that the heat flux along x axis must be constant.
If the area S of cross section is constant (it is so in one dimensional case) then the density of heat flux must be constant too:
Let's solve this differential equation (we suppose that thermoconductivity κ is constant).
And
So we have a linear function
The constants A and B one can receive from boundary conditions:
when
;
when
.
Finally we fined a linear function for temperature that satisfies the boundary conditions:
Heat flux along x axis:
Why the usual window-frame has two lists of glass?
Let's calculate the heat flux from that window.
Heat flux along x axis is constant:
Heat transfer from round pipe.
Heat flux along r axis:
– it is
area of cross section.
When there are no sources in the media we conclude from the energy conservation law that the heat flux along r axis must be constant.
To solve this equation let's divide the variables:
Then integrate this equality:
Let's take into account the boundary conditions:
when
;
when
.
and
Finally:
It is logarithmic dependence of temperature versus radius.
Heat flux along r axis:
The model of the atomic reactor.
Let us consider the spherically symmetric task.
Let
is the power of heat sources in the unit volume due to nuclear
fission reaction:
Heat flux through the closed surface along r axis:
The balance of energy:
It gives the equation:
After the division of variables:
Boundary conditions:
when
;
and
Finally we have:
The temperature inside the center of reactor is:
The energy flux from the reactor is:
