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The adsorption of a gas follows the Langmuir isotherm with K = 1.25 kPa–1 at 25°C. The pressure (in Pa) at which surface coverage is 0.2 is..............
Correct answer is '200'. Can you explain this answer?
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The adsorption of a gas follows the Langmuir isotherm with K = 1.25 kP...
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The adsorption of a gas follows the Langmuir isotherm with K = 1.25 kP...
**Langmuir Isotherm**
The Langmuir isotherm is a mathematical model that describes the adsorption of a gas onto a solid surface. It is based on the assumption that adsorption occurs through the formation of a monolayer of gas molecules on the surface. The Langmuir isotherm equation is given by:
\[ \theta = \frac{{Kp}}{{1 + Kp}} \]
Where:
- θ is the surface coverage (the fraction of surface sites covered by adsorbate molecules)
- K is the equilibrium constant for adsorption
- p is the pressure of the gas
**Given Information**
In the given problem, we are told that the adsorption of a gas follows the Langmuir isotherm with K = 1.25 kPa^(-1) at 25°C. We are asked to find the pressure at which the surface coverage is 0.2.
**Solving for Pressure**
We can rearrange the Langmuir isotherm equation to solve for the pressure:
\[ \theta = \frac{{Kp}}{{1 + Kp}} \]
Multiply both sides by (1 + Kp):
\[ \theta (1 + Kp) = Kp \]
Expand the brackets:
\[ \theta + \theta Kp = Kp \]
Rearrange the terms:
\[ \theta = Kp - \theta Kp \]
Combine like terms:
\[ \theta = p(K - \theta K) \]
Divide both sides by (K - θK):
\[ \frac{{\theta}}{{K - \theta K}} = p \]
Substitute the given values:
\[ p = \frac{{0.2}}{{1.25 - (0.2)(1.25)}} \]
Simplify:
\[ p = \frac{{0.2}}{{1.25 - 0.25}} \]
\[ p = \frac{{0.2}}{{1}} \]
\[ p = 0.2 \times 10^3 \, \text{Pa} \]
\[ p = 200 \, \text{Pa} \]
Therefore, the pressure at which the surface coverage is 0.2 is 200 Pa.
The Langmuir isotherm is a mathematical model that describes the adsorption of a gas onto a solid surface. It is based on the assumption that adsorption occurs through the formation of a monolayer of gas molecules on the surface. The Langmuir isotherm equation is given by:
\[ \theta = \frac{{Kp}}{{1 + Kp}} \]
Where:
- θ is the surface coverage (the fraction of surface sites covered by adsorbate molecules)
- K is the equilibrium constant for adsorption
- p is the pressure of the gas
**Given Information**
In the given problem, we are told that the adsorption of a gas follows the Langmuir isotherm with K = 1.25 kPa^(-1) at 25°C. We are asked to find the pressure at which the surface coverage is 0.2.
**Solving for Pressure**
We can rearrange the Langmuir isotherm equation to solve for the pressure:
\[ \theta = \frac{{Kp}}{{1 + Kp}} \]
Multiply both sides by (1 + Kp):
\[ \theta (1 + Kp) = Kp \]
Expand the brackets:
\[ \theta + \theta Kp = Kp \]
Rearrange the terms:
\[ \theta = Kp - \theta Kp \]
Combine like terms:
\[ \theta = p(K - \theta K) \]
Divide both sides by (K - θK):
\[ \frac{{\theta}}{{K - \theta K}} = p \]
Substitute the given values:
\[ p = \frac{{0.2}}{{1.25 - (0.2)(1.25)}} \]
Simplify:
\[ p = \frac{{0.2}}{{1.25 - 0.25}} \]
\[ p = \frac{{0.2}}{{1}} \]
\[ p = 0.2 \times 10^3 \, \text{Pa} \]
\[ p = 200 \, \text{Pa} \]
Therefore, the pressure at which the surface coverage is 0.2 is 200 Pa.
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