Enthalpies of melting and boiling for pure elements versus temperatures of transition

thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

, Trouton's rule states that the (molar)

entropy of vaporization In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. This is always positive, since the degree of disorder increases in the transition from a liquid in a relatively small volume to a vapor or ...

has almost the same value, about 85–88 J/(K·mol), for various kinds of

liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...

s at their

boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding envi ...

s. The entropy of

vaporization Vaporization (or vapo(u)risation) of an element or compound is a phase transition from the liquid phase to vapor. There are two types of vaporization: evaporation and boiling. Evaporation is a surface phenomenon, whereas boiling is a bulk phenome ...

is defined as the ratio between the

enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...

of vaporization and the

boiling Boiling or ebullition is the rapid phase transition from liquid to gas or vapor, vapour; the reverse of boiling is condensation. Boiling occurs when a liquid is heated to its boiling point, so that the vapour pressure of the liquid is equal to ...

temperature. It is named after Frederick Thomas Trouton. It is expressed as a function of the

gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...

: :

\Delta \bar S_\text \approx 10.5 R.

A similar way of stating this (Trouton's ratio) is that the

latent heat Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process—usually a first-order phase transition, like melting or condensation. ...

is connected to boiling point roughly as :

\frac \approx 8588\ \frac.

Trouton’s rule can be explained by using Boltzmann's definition of entropy to the relative change in free volume (that is, space available for movement) between the liquid and vapour phases. It is valid for many liquids; for instance, the

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

of vaporization of

toluene Toluene (), also known as toluol (), is a substituted aromatic hydrocarbon with the chemical formula , often abbreviated as , where Ph stands for the phenyl group. It is a colorless, water Water is an inorganic compound with the c ...

is 87.30 J/(K·mol), that of

benzene Benzene is an Organic compound, organic chemical compound with the Chemical formula#Molecular formula, molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar hexagonal Ring (chemistry), ring with one hyd ...

is 89.45 J/(K·mol), and that of

chloroform Chloroform, or trichloromethane (often abbreviated as TCM), is an organochloride with the formula and a common solvent. It is a volatile, colorless, sweet-smelling, dense liquid produced on a large scale as a precursor to refrigerants and po ...

is 87.92 J/(K·mol). Because of its convenience, the rule is used to estimate the enthalpy of vaporization of liquids whose boiling points are known. The rule, however, has some exceptions. For example, the entropies of vaporization of

water Water is an inorganic compound with the chemical formula . It is a transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance. It is the main constituent of Earth's hydrosphere and the fluids of all known liv ...

ethanol Ethanol (also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound with the chemical formula . It is an Alcohol (chemistry), alcohol, with its formula also written as , or EtOH, where Et is the ps ...

formic acid Formic acid (), systematically named methanoic acid, is the simplest carboxylic acid. It has the chemical formula HCOOH and structure . This acid is an important intermediate in chemical synthesis and occurs naturally, most notably in some an ...

and

hydrogen fluoride Hydrogen fluoride (fluorane) is an Inorganic chemistry, inorganic compound with chemical formula . It is a very poisonous, colorless gas or liquid that dissolves in water to yield hydrofluoric acid. It is the principal industrial source of fluori ...

are far from the predicted values. The entropy of vaporization of at its boiling point has the extraordinarily high value of 136.9 J/(K·mol), or 16.5''R''. The characteristic of those liquids to which Trouton’s rule cannot be applied is their special interaction between molecules, such as

hydrogen bonding In chemistry, a hydrogen bond (H-bond) is a specific type of molecular interaction that exhibits partial covalent character and cannot be described as a purely electrostatic force. It occurs when a hydrogen (H) atom, Covalent bond, covalently b ...

. The entropy of vaporization of water and ethanol shows positive deviance from the rule; this is because the hydrogen bonding in the liquid phase lessens the entropy of the phase. In contrast, the entropy of vaporization of formic acid has negative deviance. This fact indicates the existence of an orderly structure in the gas phase; it is known that formic acid forms a dimer structure even in the gas phase. Negative deviance can also occur as a result of a small gas-phase entropy owing to a low population of excited rotational states in the gas phase, particularly in small molecules such as methane a small

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...

giving rise to a large

rotational constant In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special r ...

, with correspondingly widely separated rotational energy levels and, according to

Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...

, a small population of excited rotational states, and hence a low rotational entropy. The validity of Trouton's rule can be increased by considering :

\Delta \bar S_\text \approx 4.5R + R \ln T.

Here, if , the right-hand side of the equation equals , and we find the original formulation for Trouton's rule.

References

Further reading