The following table gives the volume of nitrogen (reduced to 0^∘ C and 1 bar) adsorbed per gram

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The following table gives the volume of nitrogen (reduced to...

The following table gives the volume of nitrogen (reduced to $0^{\circ} \mathrm{C}$ and 1 bar) adsorbed per gram of active carbon at $0^{\circ} \mathrm{C}$ at a series of pressures: \[ \begin{array}{lccccc} P / \mathrm{Pa} & 524 & 1731 & 3058 & 4534 & 7497 \\ v / \mathrm{cm}^{3} \mathrm{g}^{-1} & 0.987 & 3.04 & 5.08 & 7.04 & 10.31 \end{array} \] Plot the data according to the Langmuir isotherm, and determine the constants.

The following table gives the volume of nitrogen (reduced to $0^{\circ} \mathrm{C}$ and 1 bar) adsorbed per gram of active carbon at $0^{\circ} \mathrm{C}$ at a series of pressures:
\[
\begin{array}{lccccc}
P / \mathrm{Pa} & 524 & 1731 & 3058 & 4534 & 7497 \\
v / \mathrm{cm}^{3} \mathrm{g}^{-1} & 0.987 & 3.04 & 5.08 & 7.04 & 10.31
\end{array}
\]
Plot the data according to the Langmuir isotherm, and determine the constants.

Key Concepts

Langmuir Isotherm Model

The Langmuir isotherm is a model that describes the adsorption of molecules onto a surface forming a monolayer. It is based on the assumptions that the surface has a finite number of identical sites, each site holds at most one molecule, and there is no interaction between adsorbed molecules. The isotherm is typically expressed by the equation v = (v_max * K * P) / (1 + K * P), where v is the adsorbed amount, v_max is the maximum adsorption capacity, K is the adsorption equilibrium constant, and P is the pressure. This model is fundamental in surface chemistry and catalysis for describing adsorption phenomena.

Linearization of Isotherm Equations

In order to determine the constants v_max and K from experimental data, the Langmuir isotherm equation is often linearized. A common linearization involves rearranging the original equation into the form P/v = (1/(K*v_max)) + (1/v_max) * P, which produces a straight line when P/v is plotted against P. The slope and intercept of this line can then be used to extract the values of v_max and K, making data analysis more straightforward.

Adsorption Equilibrium

Adsorption equilibrium refers to the state where the rate at which molecules adsorb onto a surface is equal to the rate at which they desorb from that surface. This equilibrium is reached under constant temperature and pressure conditions, and the resulting isotherm reflects how the amount of adsorbed substance varies with pressure. This concept is essential in understanding and modeling systems where surface interactions play a critical role.

Graphical Analysis and Regression Techniques

Graphical analysis and regression techniques are employed to fit experimental data to theoretical models, such as the Langmuir isotherm. By plotting data in a linearized form, one can use linear regression to accurately estimate model parameters like the maximum adsorption capacity and equilibrium constant. This approach not only simplifies the extraction of these parameters but also helps in assessing the validity of the underlying assumptions of the model.

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