Brewer spent the first ten years of his life with his family in Youngstown, Ohio, where his father worked as a shoe repairman.

In the shadow of the United States' entrance into World War II, Brewer pursued his Ph.D. with steady determination, and completed his dissertation on the effect of electrolytes upon the kinetics of aqueous reactions in November 1942, after only 28 months.

Following his doctoral work, Brewer was immediately recruited by UC Berkeley professor Wendell Mitchell Latimer to join the top-secret wartime research group that would become known as the Manhattan Engineering District Project.

Eastman (whose deteriorating health forced him to withdraw from the project soon after work had begun), Brewer headed a group composed of Leroy Bromley, Paul Gilles and Norman Lofgren, assigned with the threefold task of predicting the possible high-temperature properties of the newly discovered element plutonium, then available only in trace amounts; developing refractory materials capable of containing molten plutonium without excessive contamination, even if the worst predictions should be true; and developing a micro-analytical procedure for the determination of oxygen.

The immediate result of the research was the creation of the new material cerium sulfide (CeS), from which they made several hundred crucibles for use at Los Alamos National Laboratory.

In 1946, following his service as a member of the Manhattan Project, Brewer was appointed an assistant professor in the department of chemistry at the University of California.

Brewer served as a faculty member of the department of chemistry for over sixty years, during which time he directed 41 Ph.D. candidates, and nearly two-dozen post-doctoral research associates.

Brewer's dual appointment afforded him the opportunity to take an active role in all levels of academic instruction, both inside and outside of the laboratory.

In order to ensure a high standard of instruction at even the most basic levels, Brewer initiated a course for freshman-chemistry teaching assistants that reviewed principles and certified their ability to adequately fulfill their responsibilities.

In addition, Brewer single-handedly compiled and maintained Part II of the Bibliography on the High-Temperature Chemistry and Physics of Materials.

Besides his distinguished career as a chemist and educator, Brewer was also an avid gardener who held a keen interest in native California plant life.

In addition, in 1961, he and Kenneth Pitzer revised Gilbert N. Lewis and Merle Randall's classic 1923 text, Thermodynamics and the Free Energy of Chemical Substances.

Brewer's compilation of the thermodynamic properties and phase diagrams of 101 binary systems of molybdenum provides many examples of use of predictive models when no reliable experimental data are available.

The experience on the Manhattan Project on the use of platinum to reduce the volatility of lanthanides and actinides were extended to a wide range of transition metal intermetallic compounds through use of the Engel correlation of electronic and crystal structures that has led to the prediction of the structures and compositions of the phases of most of the two billion multi-component phase diagrams of the transition metals.

Brewer conducted a wide range of spectroscopic studies both at high temperatures and in matrices to fix the thermodynamic properties of high-temperature vapors.

A combination of high-temperature solid electrolyte cells, equilibration with oxides, carbides and nitrides, and vapor pressure measurements were used.

In 1984, a special festschrift in his honor was prepared by his former students and colleagues, published under the title Modern High Temperature Science.