posited from solutions containing a
sufÜcient excess of sulphuric acid.irreducible representation A
representation of a symmetry opera-
tion of a group, which cannot be ex-
pressed in terms of a representation
of lower dimension. When the repre-
sentation of the group is in matrix
form (i.e. a set of matrices that multi-
ply in the same way as the elements
of the group), the matrix represen-
tation cannot be put into block-
diagonal form by constructing a lin-
ear combination of the basis func-
tions. The importance of irreducible
representations in *quantum me-
chanics is that the energy levels of
the system are labelled by the irre-
ducible representations of the sym-
metry group of the system, thus
enabling *selection rules to be de-
duced. In contrast to an irreducible
representation, a reducible represen-
tationcan be expressed in terms of a
representation of lower dimension,
with a reducible matrix representa-
tion that can be put into block diago-
nal form by constructing a linear
combination of the basis functions.irreversibility The property of a
system that precludes a change to
the system from being a *reversible
process. The paradox that although
the equations describing the bodies
in a system, such as Newton’s laws of
motion, Maxwell’s equation, or
Schrödinger’s equation are invariant
under time reversal, events involving
systems made up from large num-
bers of these bodies are not re-
versible. The process of scrambling
an egg is an example. The resolution
of this paradox requires the concept
of *entropy using *statistical me-
chanics. Irreversibility occurs in the
transition from an ordered arrange-
ment to a disordered arrangement,
which is a natural trend, sincechanges in a closed system occur in
the direction of increasing entropy.irreversible process See irre-
versibility; reversible process.
irreversible reaction See chemi-
cal reaction.IR spectroscopy See infrared
spectroscopy.
isentropic process Any process
that takes place without a change of
*entropy. The quantity of heat trans-
ferred, δQ, in a reversible process is
proportional to the change in en-
tropy, δS, i.e. δQ = TδS, where T is the
thermodynamic temperature. There-
fore, a reversible *adiabatic process
is isentropic, i.e. when δQ equals
zero, δS also equals zero.Ising modelA model for magnetic
systems in which atomic *spins have
to be aligned either parallel or an-
tiparallel to a given direction. The
Ising model was introduced, and
solved in the case of one dimension,
by E. Ising in 1925. Ising found that
in one dimension, in the absence of
an external magneticÜeld, there is
no spontaneous magnetization at any
temperature above absolute zero.
The study of *phase transitions in
the Ising model in dimensions
greater than one has been very im-
portant to the general understanding
of phase transitions. In two dimen-
sions, the Ising model wasÜrst solved
exactly by L. Onsager in 1944. In
three dimensions, approximation
techniques, frequently involving
renormalization have to be used.ISIS/DrawA commonly used chem-
ical drawing program for 2D and 3D
structures, copyright of MDL Infor-
mation Systems, Inc. The program
has certain additional features in-
cluding calculation of molecular
weight, calculation of percentages ofirreducible representation 294i