Equivalent Exchange

Humankind cannot gain anything without first giving something in return. To obtain, something of equal value must be lost. That is alchemy's first law of Equivalent Exchange. In those days, we really believed that to be the world's one, and only, truth.

— Alphonse Elric, Fullmetal Alchemist

Equivalent Exchange is a concept quoted and believed by alchemists from the anime Fullmetal Alchemist, notably the Elric Brothers. Equivalent Exchange is used in alchemy, and most closely resembles the law of conservation of matter of Antoine Lavoisier, one of the founders of modern chemistry. In Fullmetal Alchemist, conservation of energy seems to be an unknown concept thus other scientific laws such as Albert Einstein's E=mc² and the laws of thermodynamics do not directly apply.

The Alchemists treat Equivalent Exchange as both a physical law placing limitations on Transmutation and a philosophical law meaning that for every action or gain in life, an equal price must be paid.

Final Conclusion about the law from the Anime

Humankind cannot gain anything without first giving something in return. To obtain, something of equal value must be lost. That is alchemy's first law of Equivalent Exchange. In those days, we really believed that to be the world's one, and only, truth. But the real world is imperfect and there is not a single law that can explain everything. Even the law of Equivalent Exchange. In spite of that we still believe that people cannot gain anything without offering something in return and that everyone can gain something by putting forth the effort. Equivalent Exchange is not the world's one, and only truth.

— Alphonse Elric, Fullmetal Alchemist

The anime Fullmetal Alchemist examines the truth of Equivalent Exchange. The series concludes that the world is too complex and imperfect for any one law to explain everything. It is also said that trading a life for a life is never an Equivalent Exchange.

Comparisons to Contemporary Science

Thermodynamics

The scientific laws of thermodynamics draws a similarity to the alchemist law of equivalent exchange. The first law of thermodynamics states that energy can be converted from one form to another but cannot be created or destroyed. This shows the example of how obtaining something, or transmuting something through alchemy, requires material of equal value in order to operate.

The second law of thermodynamics continues on with the context of entropy, where there is a certain amount of energy that cannot be used for any work or process. Even alchemical processes are not immune to entropy, as most alchemists are required to create a transmutation circle. Though the circle can be repeatedly used (as it is simply a gateway equation to allow alchemical energy to be channeled), the power that is drawn from the transmutation circle equation cannot be apparently used to "create" or "destroy" matter. For alchemists that have seen the Gate of Alchemy, the knowledge of transmutation without a circle is gained and can "channel" alchemical energy directly. However this energy, with or without a circle, still comes from the same source: the lives that die on the other side of the gate. These lives power the energy required for alchemical reactions and completes the alchemist law of Equivalent Exchange.

Modern Chemistry

Much of today's modern chemistry does evolve from the study of alchemy. As a result, when alchemical processes are explained in the anime (such as Roy Mustang's ability to conjure up flames on his finger tips), they are explained through modern chemistry and terminology. This results in a hybrid system that integrates alchemy and chemistry more tightly than usual. Chemistry is used to explain the process and result of the transmutation while Alchemy is used to effect the transmutation. Left unexplained by chemistry is the nature of the Alchemic power that allows the reactions to take place.

Exceptions

Human Transmutation

Complete human transmutation using alchemy is not possible under the law of Equivalent Exchange. Even though the knowledge of what elements makes up a human body, one's soul could not be transmuted along with the artificial body. There is no Equivalent Exchange for a life, including another life. Failed attempts at human transmutation create homunculi, which are artificial humans with no souls. They sometimes have some memories of the ones that they were meant to be. When born they do not have a human shape, but when they devour certain red stones (incomplete or impure Philosopher's Stones), their body gains a human form (namely, the one they were supposed to be when the alchemist attempted human transmutation)

Philosopher's Stone

The Philosopher's Stone is thought by Alchemists to provide a way to bypass the law of Equivalent Exchange. But rather than bypass the law, the stones creation only provides a way to transform human souls into an energy source for transmutation and the creation of matter. When the Philosopher's stone is used, a certain number of souls are used up and taken away, lessening the stone's power; eventually the number of souls deplete, and the stone erodes away. The Philosopher's Stone can take on multiple forms. The material can take on solid or liquid form. It is also possible through transmutation to change an object or even a human body into the Philosopher's Stone. But when an object or body is transformed, it will decay and disappear in proportion to the erosion of the stone. In the TV series the Elric Brothers' Philosopher's Stone took the shape of Alphonse's armor. It was also suggested in the series that if one of the Homunculi consumed an object or human body made into the stone, it would revert to its crystal form.

In the manga version, a Philosopher's's Stone is instrumental in creating Homunculi. Human components are gathered around the Philosopher's stone and then transmuted into the Homunculus.