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Boltzmann constant The fundamental Bose-Einstein statistics In quantum statis-
physical constant k = R/L = 1.380 = 658 × tics of the distribution of particles among vari-
−1
10 −23 JK , where R is the gas constant and ous possible energy values there are two types of
L the Avogadro constant. In the ideal gas particles, fermions and bosons, which obey the
law PV = NkT , where P is the pressure, V Fermi-Dirac statistics and Bose-Einstein statis-
the volume, T the absolute temperature, N tics, respectively. In the Fermi-Dirac statis-
the number of moles, and k is the Boltzmann tics, no more than one set of identical particles
constant. may occupy a particular quantum state (i.e., the
Pauli exclusion principle applies), whereas in the
Bose-Einstein statistics the occupation number is
Boltzmann equation Boltzmann’s equation
not limited in any way.
for a density function f(x, v,t) is the equation
of continuity (mass conservation)
Boson A particle described by Bose-
Einstein statistics.
∂f ∂f ∂f
(x,v,t) +˙x (x,v,t) +˙v (x,v,t) = 0.
∂t ∂x ∂v
boundary Let A ⊂ S be topological spaces.
Theboundary ofAistheset∂A = A−A , where
◦
¯
Bolzano-Weierstrass theorem (for the real line)
◦
A is the closure and A is the interior of A in S.
¯
If A ⊂ R is infinite and bounded, then there
exists at least one point x ∈ R that is an accu-
boundary layer The motion of a fluid of low
mulation point of A; equivalently every bounded
viscosity (e.g., air, water) around (or through)
sequence in R has a convergent subsequence.
a stationary body possesses the free velocity of
In metric spaces: compactness and sequential
an ideal fluid everywhere except in an extremely
compactness are equivalent.
thin layer immediately next to the body, called
the boundary layer.
bond There is a chemical bond between
two atoms or groups of atoms in the case that boundary value problem The problem of
the forces acting between them are such as to finding a solution to a given differential equation
lead to the formation of an aggregate with suf- in a given set A with the solution required to meet
ficient stability to make it convenient for the certain specified requirements on the boundary
chemisttoconsideritasanindependent“molecu- ∂A of that set.
lar species.”
See also coordination. bounded linear operator A bounded linear
operator from a normed linear space (X , . )
1
1
to another normed linear space (X , . ) is a
bond order, p p p The theoretical index of the 2 2
rs rs rs
1
degree of bonding between two atoms relative to map T : X → X which satisfies
2
that of a single bond, i.e., the bond provided by (i.) T(αx + βy) = αT (x) + βT (y) for all
one localized electron pair. In molecular orbital x, y ∈ X ,α,β ∈ R ; linearity
1
theory it is the sum of the products of the cor-
(ii.) Tx 2 ≤ C x , for some constant
1
responding atomic orbital coefficients (weights)
C ≥ 0, all x ∈ X ; boundedness.
1
over all the occupied molecular spin-orbitals.
Boundedness is equivalent to continuity.
Borel sets The sigma-algebra of Borel sets Bourbaki, N. A pseudonym of a changing
n
n
of R is generated by the open sets of R .
group of leading French mathematicians. The
An element of this algebra is called Borel
Association of Collaborators of Nicolas Bour-
measurable.
baki was created in 1935. With the series of
monographs El´ements de Math´ematique they
Bose-Einstein gas A gas composed of par- tried to write a foundation of mathematics based
ticles with integral spin. on simple structures.
© 2003 by CRC Press LLC