Component (thermodynamics)

Calculating the number of components in a system is necessary when applying Gibbs' phase rule in determination of the number of degrees of freedom of a system.

The number of components is equal to the number of distinct chemical species (constituents), minus the number of chemical reactions between them, minus the number of any constraints (like charge neutrality or balance of molar quantities).

Calculation

Suppose that a chemical system has M elements and N chemical species (elements or compounds). The latter are combinations of the former, and each species A_i can be represented as a sum of elements:

${\displaystyle A_i=\sum _j{a}_{ij}{E}_j,}$

where a_ij are the integers denoting number of atoms of element E_j in molecule A_i. Each species is determined by a vector (a row of this matrix), but the rows are not necessarily linearly independent. If the rank of the matrix is C, then there are C linearly independent vectors, and the remaining N-C vectors can be obtained by adding up multiples of those vectors. The chemical species represented by those C vectors are components of the system.

If, for example, the species are C (in the form of graphite), CO₂ and CO, then

${\displaystyle \left[\begin{array}{cc}1& 0\\ 1& 2\\ 1& 1\end{array}\right]\left[\begin{array}{c}C\\ \\ O\end{array}\right]=\left[\begin{array}{c}C\\ C{O}_2\\ CO\end{array}\right].}$

Since CO can be expressed as CO = (1/2)C + (1/2)CO₂, it is not independent and C and CO can be chosen as the components of the system.

There are two ways that the vectors can be dependent. One is that some pairs of elements always appear in the same ratio in each species. An example is a series of polymers that are composed of different numbers of identical units. The number of such constraints is given by Z. In addition, some combinations of elements may be forbidden by chemical kinetics. If the number of such constraints is R', then

${\displaystyle C=M-Z+R^{\prime }.}$

Equivalently, if R is the number of independent reactions that can take place, then

${\displaystyle C=N-Z-R.}$

The constants are related by N - M = R + R'.