33. The barometric height formula. Boltzmann distribution
The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude.
There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). The first equation is used when the value of standard temperature lapse rate is not equal to zero; the second equation is used when standard temperature lapse rate equals zero.
Equation 1:
Equation 2:
where
=
static pressure (pascals)
=
standard temperature (K)
=
standard temperature lapse rate (K/m) in ISA
=
height above sea level (meters)
=
height at bottom of layer b (meters;
e.g., h1 =
11,000 meters)
= universal
gas constant for air: 8.31432 N·m /(mol·K)
= gravitational
acceleration (9.80665 m/s2)
=
molar mass of Earth's air (0.0289644 kg/mol)
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution[1]) is aprobability distribution, probability measure, or frequency distribution of particles in a system over various possiblestates. The distribution is expressed in the form
where 
is
state energy (which varies from state to state), and 
(a
constant of the distribution) is the product of Boltzmann's
constant and thermodynamic
temperature.
The ratio of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference.
34.Speed distribution of molecules. Maxwell distribution. Root-mean-square, average and the most probable velocities of molecules.
In physics, particularly statistical mechanics, the Maxwell–Boltzmann distribution or Maxwell speed distribution describes particle speeds in idealized gases where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. Particle in this context refers to gaseous atoms or molecules, and the system of particles is assumed to have reached thermodynamic equilibrium.
The distribution is a probability distribution for the speed of a particle within the gas - the magnitude of its velocity. This probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The distribution depends on the temperature of the system and the mass of the particle.
The mean speed, most probable speed (mode), and root-mean-square can be obtained from properties of the Maxwell distribution.
The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the same mass m) in the system and corresponds to the maximum value or mode of f(v). To find it, we calculate df/dv, set it to zero and solve for v:
which yields:
where R is the gas constant and M = NA m is the molar mass of the substance.
For
diatomic nitrogen (N2,
the primary component of air)
at room
temperature (300 K),
this gives 
m/s
The mean speed is the expected value of the speed distribution
The root mean square speed is the second-order moment of speed:
The typical speeds are related as follows:
