## Particle Tunnelling

We now consider a situation where a particle is confined by walls of potential energy V, the potential being zero within the walls; the potential does not rise to infinity at the walls. The total energy E of the particle, which arises solely from the motion of the particle within the walls, is such that E < V. Classically, … Read more

## The Particle in A Two-Dimensional Box

In this model, we consider a particle that is confined to a rectangular plane, of length Lx in the x direction and Ly in the y direction. The potential energy is zero everywhere in this plane, and infinite at its walls and beyond. It should be clear that this is an extension of the particle in a one-dimensional box … Read more

## The Particle in a One-Dimensional Box

The heading of this page refers to the hypothetical system that will be considered on it. Though in itself this problem may appear quite a trivial one, it introduces various important concepts, and paves the way for exploration of some slightly more complex and physically relevant systems. It may also itself be used as a first approximation to some … Read more

## Uses of the Clapeyron Equation

We derived, on the previous page, the Clapeyron Equation: It now remains to show how it may be applied to the most commonly encountered phase boundaries: For the boundary between solid and liquid phases,  the entropy of transition, ΔStrs, may be replaced by ΔHtrs/T (This is because on the phase boundary the two phases are in equilibrium, so the heat transfer, which is … Read more

## Phase Boundaries

The phase boundaries are defined by the fact that they represent the precise conditions of temperature and pressure under which the chemical potentials of the two phases on either side of the boundary are equal. Therefore it is possible to construct equations for the phase boundaries by setting the chemical potentials of the two phases (which are … Read more

## Phase Stability and Chemical Potential

It is a consequence of the Second Law that at equilibrium, the chemical potential of a substance must be the same throughout, regardless of the number of phases present. (If chemical potentials were not equal throughout, it would be possible for the system to reduce its Gibbs energy by transfer of substance from the area of higher chemical potential to … Read more

## The Phase Diagram of Water

This diagram is a highly simplified representation of the phase diagram of water. At high pressures (greater than 2000 atmospheres) various allotropes of solid ice have been observed – these are omitted for clarity. The liquid-vapour phase boundary in the diagram summarises the variation in the vapour pressure of liquid water with temperature. Conversely, we can look at it … Read more

## Interpretation of Phase Diagrams

When a liquid is heated in an open vessel, its temperature and vapour pressure will both increase. At the temperature at which the liquid’s vapour pressure (the pressure at any given temperature at which both liquid and vapour are in equilibrium) would be equal to the external pressure, vaporisation can occur throughout the bulk of the liquid, and … Read more

## Introduction to Phase Diagrams

A phase of a substance is a form of matter that is uniform throughout in chemical composition and physical state. Thus we will typically encounter the solid, liquid and gaseous phases of a substance. Allotropes of the solid state, such as graphite and diamond, are also different phases. Processes such as vaporisation, melting and interconversion of allotropes … Read more

## Rotational Raman Spectra

The gross selection rule for rotational Raman spectroscopy is that the molecule must be anisotropically polarisable, which means that the distortion induced in the electron distribution in the molecule by an electric field must be dependent upon the orientation of the molecule in the field. i.e. An atom has a spherical electron distribution, and the dipole induced by an electric field … Read more