Density and Phase Changes

Let us first consider how the density of the condensed phase changes with temperature. As our material in the vapour phase is cooled at constant pressure the density increases until the boiling point is reached. Further cooling then allows us to differentiate between the vapour and liquid states by the formation of a boundary. Further cooling increases the liquid density but at a much slower rate than that of the gaseous phase. The density of many liquids can be described by a simple linear equation over a wide temperature range 5 

Here T0 is a reference temperature and if this is 273K the coefficient A] 103. The temperature may then be lowered until the melting point, Tm, is reached. At this point a crystalline solid is formed, the density of which will usually be within 10% of the liquid. The density can be described by a similar equation to Equation (3.28) but the coefficient is quite different, i.e. As 1015 The origin of the thermal expansion in a 

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence  

Lindemann6 said melting would occur when jc became some significant fraction, a, of r, say of the order of 0.1. Now Equation (3.30) can be written in terms of the molar volume, Vm, and a  

This very approximate model gives reasonable agreement with such disparate materials as iron, quartz and sodium chloride with a 0.14. Alternatively, of course, if we know the melting point, we can obtain an estimate of the elastic modulus. 

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