Several literature relations to calculate the vapour pressures of pure compounds are re-examined to evaluate their reliability in predicting the experimental data in a very low pressure range, chosen between the triple point and 1 mmHg. Following a previous work of the author, the performances of the Riedel equation are greatly improved by allowing a temperature dependence of the Riedel parameter 0.0838, which is assumed constant in the unmodified Riedel equation. The resulting relations, which vary for each class of compounds, embody only pure component parameters which can be evaluated by knowing only the normal boiling temperature and the critical constants. With these modifications, the Riedel equation is significantly better than the Miller, Lee-Kesler and Ambrose-Walton equations. This superiority is particularly evident in the case of strongly associated compounds, like alcohols. The strong dependence of the calculated pressures at very low temperatures on the critical temperature value is also stressed. The proposed method is verified on the basis of nearly 90 compounds, by using as "true" experimental data those reported in three sources: the DIPPR Data Compilation of Pure Compounds Properties, Project 801, the NIST program and a literature paper. (c) 2005 Elsevier B.V. All rights reserved.