# Component (thermodynamics)

In thermodynamics, a component is one of a collection of chemically-independent constituents of a system. The number of components represents the minimum number of independent species necessary to define the composition of all phases of the system.[1]

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 Ai can be represented as a sum of elements:

${\displaystyle A_{i}=\sum _{j}a_{ij}E_{j},}$

where Ej is the symbol for element j and aij are the components of an N x M matrix. 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.[2]

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

${\displaystyle {\begin{bmatrix}1&0\\1&2\\1&1\end{bmatrix}}{\begin{bmatrix}C\\\\O\end{bmatrix}}={\begin{bmatrix}C\\CO_{2}\\CO\end{bmatrix}}.}$

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.[3]

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

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

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'.[2]

## Examples

### CaCO3 - CaO - CO2 system

This is an example of a system with several phases, which at ordinary temperatures are two solids and a gas. There are three chemical species (CaCO3, CaO and CO2) and one reaction:

CaCO3 ⇌ CaO + CO2.

The number of components is then 3 - 1 = 2.[1]

### Water - Hydrogen - Oxygen

The reactions included in the calculation are only those that actually occur under the given conditions, and not those that might occur under different conditions such as higher temperature or the presence of a catalyst. For example, the dissociation of water into its elements does not occur at ordinary temperature, so a system of water, hydrogen and oxygen at 25 °C has 3 independent components.[1][3]

## References

1. ^ a b c Peter Atkins and Julio de Paula, "Physical Chemistry" 8th Edition (W.H. Freeman 2006), p.175-176
2. ^ a b Zeggeren, F. van; Storey, S. H. (2011). The computation of chemical equilibria (1st pbk. ed.). Cambridge University Press. pp. 15–18. ISBN 9780521172257.
3. ^ a b Zhao, Muyu; Wang, Zichen; Xiao, Liangzhi (July 1992). "Determining the number of independent components by Brinkley's method". Journal of Chemical Education. 69 (7): 539. doi:10.1021/ed069p539.