The young student found some literary occupation and a little teaching in Edinburgh, and sometimes he was asked to make investigations in subjects connected with applied chemistry. Thus he struggled on for four or five years, during which time he began to publish papers on chemico-physical subjects. In the year 1829 he was appointed Lecturer on Chemistry at the Mechanics' Institution in Glasgow, and next year he was removed to the more important position of lecturer on the same science at the Andersonian Institution in that city. This position he occupied for seven years, when he was elected Professor of Chemistry in the University of London (now University College): he had been elected to the Fellowship of the Royal Society in the preceding year. During his stay at the Andersonian Institution Graham had established his fame as a physical chemist; he had begun his work on acids and salts, and had established the fundamental facts concerning gaseous diffusion. These researches he continued in London, and from 1837 to 1854 he enriched chemical science with a series of papers concerned for the most part with attempts to trace the movements of the atoms of matter.
In 1854 Graham succeeded Sir John Herschel in the important and honourable position of Master of the Mint. For some years after his appointment he was much engaged with the duties of his office, but about 1860 he again returned to his atomic studies, and in his papers on "Transpiration of Liquids" and on "Dialysis" he did much in the application of physical methods to solve chemical problems, and opened up new paths, by travelling on which his successors greatly advanced the limits of the science of chemistry. Graham was almost always at work; his holidays were "few and far between." By the year 1868 or so his general health began to grow feeble; in the autumn of 1869, during a visit to Malvern where he sought repose and invigorating air, he caught cold, which developed into inflammation of the lungs. On his return to London the disease was overcome by medical remedies, but he continued very weak, and gradually sank, till the end came on the 16th of September 1869.
I have said that the seven years during which Graham held the lectureship on chemistry in the Andersonian Institution, Glasgow, witnessed the beginning alike of his work on salts and of that on gaseous diffusion. He showed that there exists a series of compounds of various salts, _e.g._ chloride of calcium, chloride of zinc, etc., with alcohol. He compared the alcohol in these salts, which he called _alcoates_, to the water in ordinary crystallized salts, and thus drew the attention of chemists to the important part played by water in determining the properties of many substances. Three years later (1833) appeared one of his most important papers, bearing on the general conception of acids: "Researches on the Arseniates, Phosphates, and Modifications of Phosphoric Acid." Chemists at this time knew that phosphoric acid--that is, the substance obtained by adding water to pentoxide of phosphorus--exhibited many peculiarities, but they were for the most part content to leave these unexplained. Graham, following up the analogy which he had already established between water and bases, prepared and carefully determined the composition of a series of phosphates, and concluded that pentoxide of phosphorus is able to combine with a base--say soda--in three different proportions, and thus to produce three different phosphates of soda. But as Graham accepted that view which regards a salt as a metallic derivative of an acid, he supposed that three different phosphoric acids ought to exist; these acids he found in the substances produced by the action of water on the oxide of phosphorus. He showed that just as the oxide combines with a base in three proportions, so does it combine with water in three proportions. This water he regarded as chemically analogous to the base in the three salts, one atom (we should now rather say molecule) of base could be replaced by one atom of water, two atoms of base by two atoms of water, or three atoms of base by three atoms of water. Phosphoric acid was therefore regarded by Graham as a compound of pentoxide of phosphorus and water, the latter being as essentially a part of the acid as the former. He distinguished between _monobasic_, _dibasic_, and _tribasic_ phosphoric acids: by the action of a base on the _monobasic acid_, one, and only one salt was produced; the _dibasic acid_ could furnish two salts, containing different proportions (or a different number of atoms) of the same base: and from the _tribasic acid_ three salts, containing the same base but in different proportions, could be obtained.