or absorbed radiation of frequency ν,
the electron jumped from one orbit
to another; the energy emitted or ab-
sorbed by each jump is equal to hν.
This theory gave good results in pre-
dicting the lines observed in the
spectrum of hydrogen and simple
ions such as He+, Li2+, etc. The idea of
quantized values of angular momen-
tum was later explained by the wave
nature of the electron. Each orbit has
to have a whole number of wave-
lengths around it; i.e. nλ= 2πr, where
λis the wavelength and n a whole
number. The wavelength of a particle
is given by h/mv, so nh/mv = 2πr,
which leads to mvr= nh/2π. Modern
atomic theory does not allow sub-
atomic particles to be treated in the
same way as large objects, and Bohr’s
reasoning is somewhat discredited.
However, the idea of quantized angu-
lar momentum has been retained.boiling point (b.p.)The tempera-
ture at which the saturated vapour
pressure of a liquid equals the exter-
nal atmospheric pressure. As a conse-
quence, bubbles form in the liquid
and the temperature remains con-
stant until all the liquid has evapo-
rated. As the boiling point of a liquid
depends on the external atmospheric
pressure, boiling points are usually
quoted for standard atmospheric
pressure (760 mmHg = 101 325 Pa).
boiling-point–composition dia-
gramA graph showing how the
boiling point and vapour composi-
tion of a mixture of two liquids de-
pends on the composition of the
mixture. The abscissa shows the
range of compositions from 100% A
at one end to 100% B at the other.
The diagram has two curves: the
lower one gives the boiling points (at
aÜxed pressure) for the different
compositions. The upper one is plot-
ted by taking the composition of
vapour at each temperature on theboiling-point curve. The two curves
would coincide for an ideal mixture,
but generally they are different be-
cause of deviations from *Raoult’s
law. In some cases, they may show a
maximum or minimum and coincide
at some intermediate composition,
explaining the formation of
*azeotropes.boiling-point elevationSee
elevation of boiling point.
Boltzmann, Ludwig Eduard
(1844–1906) Austrian physicist. He
held professorships in Graz, Vienna,
Munich, and Leipzig, where he
worked on the kinetic theory of
gases (see maxwell–boltzmann dis-
tribution) and on thermodynamics
(see boltzmann equation). He suf-
fered from depression and commit-
ted suicide.
Boltzmann constantSymbol k.
The ratio of the universal gas con-
stant (R) to the Avogadro constant
(NA). It may be thought of therefore
as the gas constant per molecule:
k = R/NA= 1.380 658(12) × 10 –23JK–1
It is named after Ludwig *Boltzmann.Boltzmann equationAn equation
used in the study of a collection of
particles in *nonequilibrium statisti-
cal mechanics, particularly their
transport properties. The Boltzmann
equation describes a quantity called
the distribution function, f, which
gives a mathematical description of
the state and how it is changing. The
distribution function depends on a
position vector r, a velocity vector v,
and the time t; it thus provides a sta-
tistical statement about the positions
and velocities of the particles at any
time. In the case of one species of
particle being present, Boltzmann’s
equation can be written
∂f/∂t + a.(∂f/∂v) + v.(∂f/∂r) = (∂f/∂t)coll,
where ais the acceleration of bodiesboiling point 76b