Let and be the changes in the internal energies of the two systems, during the process. If in a change from an initial state to a final one, Q if joules of heat are transferred into a system from its environment , and its temperature goes from T i to T f , then we define its . Since is inexact, heat capacity is not yet fully defined. A more complete definition usually requires that at least one state variable other than T be held fixed. It is shown as a subscript.
Notation is c .
In particular, comparing the coefficient of dT in this and the previous equations, . If we now substitute this equation into the right side of the equation for , and collect the common factor, the result is . Here C P , C V and the expansivity are known for most substances, but clearly it would take rather special apparatus to change the volume of a solid or liquid at a fixed temperature.