The adsorption of a gas on a solid surface can be described mathematically using adsorption isotherms. Adsorption isotherms are equations that relate the amount of gas adsorbed on a solid surface to the pressure or concentration of the gas at a constant temperature. Two of the most common adsorption isotherms are the Langmuir isotherm and the Freundlich isotherm.1. Langmuir Isotherm: The Langmuir isotherm is based on the assumption that adsorption occurs on a homogeneous surface with a finite number of adsorption sites, and that there is no interaction between adsorbed molecules. The Langmuir isotherm equation is given by: = Kp / 1 + Kp where is the fractional coverage of the surface the ratio of occupied sites to total sites , K is the Langmuir adsorption constant, and p is the partial pressure of the gas.2. Freundlich Isotherm: The Freundlich isotherm is an empirical equation that describes adsorption on heterogeneous surfaces. The Freundlich isotherm equation is given by:x/m = KF p ^1/nwhere x is the amount of gas adsorbed, m is the mass of the adsorbent, KF is the Freundlich constant, p is the partial pressure of the gas, and 1/n is the heterogeneity factor.Factors that influence the extent of adsorption include:1. Nature of the adsorbent: The surface area, porosity, and chemical nature of the adsorbent can affect the extent of adsorption. A larger surface area and higher porosity generally lead to greater adsorption.2. Nature of the adsorbate: The size, shape, and polarity of the adsorbate molecules can influence their interaction with the adsorbent surface and thus affect the extent of adsorption.3. Temperature: Adsorption is generally an exothermic process, so increasing the temperature usually decreases the extent of adsorption. However, in some cases, increasing the temperature can increase the rate of adsorption, leading to a higher extent of adsorption.4. Pressure: Increasing the pressure of the gas generally increases the extent of adsorption, as it increases the concentration of the gas molecules near the adsorbent surface.5. Presence of other gases or impurities: The presence of other gases or impurities can compete with the target gas for adsorption sites, potentially decreasing the extent of adsorption of the target gas.