Question

According to Freundlich adsorption isotherm, which of the following is correct? (a) \(\frac{\mathrm{x}}{\mathrm{m}} \propto \mathrm{P}^{1 \mathrm{n}}\) (b) \(\frac{x}{m} \propto \mathrm{P}^{1}\) (c) \(\frac{x}{m} \propto P^{0}\) (d) All the above are correct for different ranges of pressure

Short answer

(d) All the above are correct for different ranges of pressure.

Step-by-step solution

Understand Freundlich Isotherm

The Freundlich adsorption isotherm is an empirical relation between the amount of gas adsorbed by a unit mass of solid adsorbent and the pressure of the gas at a fixed temperature. It is typically expressed as \( \frac{x}{m} = k P^{1/n} \), where \( x \) is the mass of the adsorbate, \( m \) is the mass of the adsorbent, \( k \) is a constant, \( P \) is the pressure, and \( \frac{1}{n} \) is a constant related to the adsorption intensity.

Analyze the Options

We need to compare each option with the standard form of the Freundlich isotherm: \( \frac{x}{m} = k P^{1/n} \).- Option (a) suggests \( \frac{x}{m} \propto P^{1/n} \), which matches the constant power relation in Freundlich's isotherm.- Option (b) proposes \( \frac{x}{m} \propto P^{1} \), implying a direct relationship as recognized when \( n = 1 \).- Option (c) states \( \frac{x}{m} \propto P^{0} \), which implies a constant adsorption irrespective of pressure changes.- Option (d) suggests that all above are correct for different pressure ranges, hinting at varying adsorption behavior with pressure changes.

Consider Pressure Ranges

In real situations, at low pressures, the gas often behaves according to option (a) with \( n > 1 \), which means the exponent \( \frac{1}{n} < 1 \).At moderate pressures, adsorption may behave as in option (b), which suggests a linear increase in adsorption with pressure.At high pressures, the adsorption often approaches saturation, acting closer to option (c) where the amount adsorbed becomes nearly constant.

Conclusion

Given that Freundlich adsorption can exhibit different characteristics depending on the pressure range, option (d) is indeed correct. It acknowledges the presence of all these behaviors under varying pressure situations.

Key concepts

Adsorption Intensity

When talking about adsorption intensity in the context of the Freundlich adsorption isotherm, it refers to how strongly a gas is adsorbed onto a solid surface. This intensity is described by the relationship between the amount of gas adsorbed (denoted as \(x\) in the equation), the mass of the solid adsorbent (\(m\)), and the pressure (\(P\)) at which adsorption occurs.
The Freundlich isotherm is expressed as:\[\frac{x}{m} = k P^{1/n}\]where \(k\) is a constant and \(\frac{1}{n}\) is directly related to the adsorption intensity.

• When \(1/n\) is closer to 1, the adsorption is more intense, indicating a stronger interaction between the gas and the solid.
• Conversely, a smaller value of \(1/n\) (closer to zero) indicates a lower adsorption intensity and weaker interaction.

The value of \(n\) is typically greater than 1, reflecting a non-linear sensitivity to changes in pressure. This helps explain why the adsorptive capacity does not always increase proportionately with rising pressure.
Understanding this concept is crucial, as it provides insights into the efficiency of the adsorption process, indicating how various materials will interact with gases.

Pressure Variation in Adsorption

Pressure plays a pivotal role in the process of adsorption, especially concerning how it affects the amount of gas a solid can adsorb. The Freundlich adsorption isotherm explains the variation in adsorption based on different pressure levels.
At low pressures, adsorption tends to follow a pattern expressed in option (a), \(\frac{x}{m} \propto P^{1/n}\), where the adsorption is most intense with increasing pressure.
As the pressure increases further into moderate levels, the relationship sometimes aligns with option (b), \(\frac{x}{m} \propto P^{1}\). This linear behavior suggests that an increase in pressure linearly increases the amount of gas adsorbed.
At high pressure levels, though, the surface of the adsorbent begins to saturate. This is where option (c), \(\frac{x}{m} \propto P^0\), becomes applicable, indicating that pressure changes have little to no effect, as the adsorption process reaches a plateau.

• The multiplicity of behavior across the pressure ranges shows the flexibility and applicability of the Freundlich isotherm.
• Understanding these transitions can help in designing and optimizing systems for gas adsorption, where controlling pressure to maintain desired adsorption levels can be critical.

Gas Adsorption by Solids

Adsorption of gases by solid materials is a fascinating and complex phenomenon, crucial for various industrial and chemical processes. The principle here involves a solid adsorbent attracting and holding gas molecules on its surface.
The amount of gas adsorbed depends significantly on several factors, such as the surface area of the solid, the temperature, and the pressure exerted on the gas. According to the Freundlich adsorption isotherm, at constant temperature, the relationship between the amount of gas adsorbed and the pressure is empirical.

• This relationship is non-linear, highlighting that different adsorbents will react uniquely to gases.
• The Freundlich isotherm model accommodates variations across different pressure conditions, supporting its use in practical applications ranging from pollution control to chemical synthesis.

Understanding how gases interact with solid adsorbents allows scientists and engineers to select and design materials for specific purposes, enhancing efficiency and efficacy in processes like air purification, catalysis, and storage of gases.