Physical chemistry is the part of chemistry that seeks to account for the properties and transformations of matter in terms of concepts, principles, and laws drawn from physics. This glossary is a compilation of definitions, descriptions, formulae, and illustrations of concepts that are encountered throughout the subject. This section describes the concepts that begin with the letter K; where appropriate, the entries also describe subsidiary but related concepts. Refer to the Directory for a full list of all the concepts treated.
The
Karplus equation is an empirical expression for the variation with torsional angle
ϕ of the spin–spin coupling constant
3J between protons in a fragment HCCH. It comes in a variety of forms, two of which are
The parameters in these two expressions are related (by using the relation
) as follows:
A wide range of the parameters are reported in the literature. One set is (Figure K.1)
On the
Kelvin scale of temperature, the lowest point, ‘absolute zero’, lies at
T = 0 and higher temperatures are expressed as multiples of 1 kelvin (1 K). The scale was originally defined in terms of the triple point of water, taken to lie at 273.16 K exactly, but it is now defined in terms of fundamental constants. The conversions between the Celsius, Fahrenheit, and Kelvin scales are
The numbers are exact. The degree Celsius has the same size as 1 K.
The
Kelvin equation expresses the vapour pressure of a liquid that is dispersed as droplets of radius
r:
where
p* is the vapour pressure of the bulk liquid of molar volume
Vm and
γ is the surface tension of the liquid. For the vapour pressure inside a spherical cavity, change the sign of
χ.
The
kinetic chain length,
v, of a polymerization chain reaction is defined as
It is therefore a measure of the efficiency of the chain propagation mechanism: the faster the propagation of a chain, the greater is the kinetic chain length.
The
kinetic energy,
Ek and sometimes (in mechanics, when there is no chance of confusing it with temperature)
T, is the energy due to the motion of a body. For a body of mass
m travelling at a nonrelativistic speed
,
where
is the magnitude of the linear momentum of the body. In quantum mechanics, the operators in the position representation for kinetic energy in one and three dimensions are
It follows that the greater the local curvature of a wavefunction, then the greater is the contribution to the expectation value of the kinetic energy to the total energy. The kinetic energy of a body with moment of inertia
I rotating with an angular velocity
ω is
where
J is the magnitude of the angular momentum of the body.
The primary kinetic isotope effect is the reduction of the rate of a reaction by the replacement of an atom by a heavier isotope. The effect is most pronounced when hydrogen (strictly, protium, 1H) is replaced by deuterium (2H). It can be traced to the lowering of the zero-point energy of the X–H bond that is undergoing cleavage in the rate-determining step (Figure K.2). This lowering increases the activation energy of the reaction and hence reduces the rate of the reaction. The secondary kinetic isotope effect is the effect on rate caused by isotopic substitution elsewhere in the molecule.
The kinetic salt effect is the modification of the rate of reaction between ions in solution caused by a change in the ionic strength, I, of the solution. If the charges of the ions have the same sign, then the rate is increased when the ionic strength is increased; if they have opposite sign, then the rate is reduced.
The effect is ascribed to the modification of the ionic atmosphere when ions come together to form the activated complex. Ions of the same charge increase the charge density, strengthen the ionic atmosphere, lower the energy of the activated complex, so reduce the activation energy of the reaction, and thereby increase its rate. The opposite is true when ions of opposite charge come together and form an activated complex of lower total charge. The limiting (as
I → 0) rate constant can be expected to follow the relation
where
zA and
zB are the (signed) charge numbers of the ions and
A is the parameter that occurs in the Debye–Hückel limiting law (
Figure K.3).
The kinetic model of gases (KMT, or kinetic theory of gases) treats a gas as a collection of point molecules in ceaseless random motion and which have no interactions except when they undergo elastic collisions. An elastic collision is a collision in which the translational energy is unchanged.
The pressure,
p, of the gas is derived by assessing the force the molecules exert on the walls of the container of volume
V containing
N molecules as they change direction on impact:
where
is the root-mean-square speed of the molecules and
n is the amount of molecules of molar mass
M. This expression and the Maxwell distribution of speeds can be used to infer the following features of the speeds of the molecules:
where
is their
mean speed and
is their
relative mean speed.
Two further parameters are used to characterize the spatial and time scales of events in the gas. Both require the addition to the model of the notion of a
collision cross-section,
σ, an area around a molecule which, if the centre of a second molecule enters as it moves through space, is counted as a ‘collision’. If the diameter of the molecules are both taken to be
d, then
σ = π
d 2 (
Figure K.4). The
mean free path,
λ, is the average distance that a molecule travels before it collides with another molecule. The
collision frequency,
z, is the average number of collisions a molecule makes in a given time interval divided by the duration of the interval (colloquially, the ‘number of collisions per unit time’).
Figure K.4
The relation of the collision cross-section to molecular diameter.
Figure K.4
The relation of the collision cross-section to molecular diameter.
In a container of constant volume,
λ is independent of temperature (because then
p ∝
T). Typical collision frequencies in air at sea level are about 5 ns
−1. The kinetic model is used in to establish approximate expressions for transport properties.
Koopmans’ theorem states that the ionization energy of an electron from a specified orbital is equal to the negative of the one-electron energy of that orbital:
The theorem is only an approximation because it ignores the adjustment in electron density that occurs when one electron is removed.
