This can be easily separated using polar GC capillary cyano bonded column . I use this column for all my routine analysis and it very efficient column. This column can separate up to 52 fatty acids mixture including C12, C14,and C18 series. This column is also good for separating C18:1, C18:2 and its cis and trans isomers. You need to use at least 75-100 mt column for this purpose.
The method used is GC-FID.All you need is FAME of these acids. Hope this helps you
The best way to purify the palmitic acid is double crystallization or recrytalization from its amino salts. You can use primary amines such as cylcohexyl amine and solvents like acetone.First react palmitc acid with slight excess of amine.Heat if necessary .Cool it to 10-15 degree and filter. The cake should be recrystallized from acetone again .You can release amino salts after second crystallization with usual acidification followed by extraction in to a non polar solvent.