In earlier work [J. Chem. Phys. 100, 2154 (1994)] we presented a new method of determining the chemical potential in Monte Carlo or molecular dynamics simulations which makes use of a detailed balance method (DBM). In the present paper we present results of a careful study of this method applied to liquid palladium down to and below the zero-pressure melting temperature. A new surface detailed balance method (SDBM) is introduced, which is much more efficient near and below the melting temperature where the original method becomes too inefficient to determine an accurate value of the chemical potential. We also present results where the new surface detailed balance method is used to determine the chemical potential of the solid phase at a number of different temperatures.