# Adsorption

"Consumption" of matter from a gas or from an opening of the interface between them (or from the surface of a solid body). In other words, adsorption is the "consumption" by an adsorbate from a volume of gas on the surface of the adsorbent. Adsorption is a particular case of sorption.

The molecules of the adsorbate falling on the surface of the adsorbent are retained by the surface force for a period of time, which depends on the natures of the adsorbent and adsorbate, on the temperature $T$ and on the pressure $p$, after which they leave the surface (are desorbed). Under conditions of thermodynamic and molecular equilibrium, the rates of adsorption and desorption are equal. The relation between the relative pressure $\phi=p/p_s$ of the adsorbent and the relative concentration $\theta=c/c_s$, where the index $s$ stands for the limit value at a constant temperature, is known as the adsorption isotherm.

Langmuir's equation of mono-molecular adsorption has the form

$$\phi=\frac{\theta}{k(1-\theta)},$$

where $k$ is the equilibrium constant which roughly describes the interaction between the adsorbent and the adsorbate.

Brunauer's equation [1] is commonly used in the case of a homogeneous surface of the adsorbent and poly-molecular adsorption.

Posnov's formula [2], which is empirical, is widely employed for capillary bodies:

$$\frac1\theta=A\ln\phi+1,$$

where $A$ is a coefficient which varies with the temperature and with the structure of the adsorbent.

#### References

[1] | S. Brunauer, "Adsorption of gases and vapors" , Princeton Univ. Press (1943) |

[2] | V.A. Posnov, Zh. Tekhn. Fiz. : 23 (1953) pp. 865 |

[3] | B.V. Il'in, "The nature of adsorption forces" , Moscow-Leningrad (1952) (In Russian) |

[4] | J.H. de Boer, "The dynamical character of adsorption" , Clarendon Press (1968) |

#### Comments

The Brunauer–Emmett–Teller equation, or BET-equation, [a1] generalizes the Langmuir equation. It assumes multi-molecular adsorption for which each layer obeys a Langmuir equation. There are several modifications, cf. [a2].

#### References

[a1] | S. Brunauer, P.H. Emmett, E. Teller, J. Amer. Chem. Soc. , 60 (1938) pp. 309 |

[a2] | S. Brunauer, L.E. Copeland, "Surface tension, adsorption" E.U. Condon (ed.) H. Odishaw (ed.) , Handbook of physics , 2 , McGraw-Hill (1967) pp. Chapt. 7 |

**How to Cite This Entry:**

Adsorption.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Adsorption&oldid=34287