To determine the constants \( K \) and the volume corresponding to complete coverage \( V_{\text{max}} \) for the Langmuir isotherm using the given data, we can follow these steps:
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Langmuir Isotherm Equation: The Langmuir isotherm is given by:
\[ \frac{p}{V} = \frac{K}{V_{\text{max}}} + \frac{p}{V_{\text{max}}} \]
Rearranging gives us a linear form:
\[ \frac{p}{V} = \frac{K}{V_{\text{max}}} + \frac{1}{V_{\text{max}}} p \]
Here, if we plot \( \frac{p}{V} \) against \( p \), we can obtain a linear relationship where:
- The slope of the line represents \( \frac{1}{V_{\text{max}}} \)
- The y-intercept represents \( \frac{K}{V_{\text{max}}} \)
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Data Conversion: We need to compute \( \frac{p}{V} \) for each set of values:
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For \( p = 13.3 \) kPa and \( V = 10.2 \) cm³: \[ \frac{p}{V} = \frac{13.3}{10.2} \approx 1.3039 , \text{kPa/cm}^3 \]
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For \( p = 26.7 \) kPa and \( V = 18.6 \) cm³: \[ \frac{p}{V} = \frac{26.7}{18.6} \approx 1.4339 , \text{kPa/cm}^3 \]
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For \( p = 40.0 \) kPa and \( V = 25.5 \) cm³: \[ \frac{p}{V} = \frac{40.0}{25.5} \approx 1.5686 , \text{kPa/cm}^3 \]
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For \( p = 53.3 \) kPa and \( V = 31.5 \) cm³: \[ \frac{p}{V} = \frac{53.3}{31.5} \approx 1.6952 , \text{kPa/cm}^3 \]
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For \( p = 66.7 \) kPa and \( V = 36.9 \) cm³: \[ \frac{p}{V} = \frac{66.7}{36.9} \approx 1.8066 , \text{kPa/cm}^3 \]
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For \( p = 80.0 \) kPa and \( V = 41.6 \) cm³: \[ \frac{p}{V} = \frac{80.0}{41.6} \approx 1.9226 , \text{kPa/cm}^3 \]
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For \( p = 93.3 \) kPa and \( V = 46.1 \) cm³: \[ \frac{p}{V} = \frac{93.3}{46.1} \approx 2.0244 , \text{kPa/cm}^3 \]
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Creating the Data Set: The data points are:
| p (kPa) | \( \frac{p}{V} \) (kPa/cm³) | |---------|--------------------------| | 13.3 | 1.3039 | | 26.7 | 1.4339 | | 40.0 | 1.5686 | | 53.3 | 1.6952 | | 66.7 | 1.8066 | | 80.0 | 1.9226 | | 93.3 | 2.0244 |
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Linear Regression: To find the constants \( K \) and \( V_{\text{max}} \), plot \( \frac{p}{V} \) on the y-axis and \( p \) on the x-axis, and perform a linear regression to fit the line. The slope of this line gives \( \frac{1}{V_{\text{max}}} \) and the intercept gives \( \frac{K}{V_{\text{max}}} \).
Using a software or statistical package to perform linear regression, you will obtain values for the slope and intercept.
- Calculating Constants: Assume the linear regression obtained:
- Slope = \( m = \frac{1}{V_{\text{max}}} \)
- Intercept = \( b = \frac{K}{V_{\text{max}}} \)
From the slope, you can calculate \( V_{\text{max}} \):
\[ V_{\text{max}} = \frac{1}{m} \]
From the intercept, you can calculate \( K \):
\[ K = b \cdot V_{\text{max}} \]
- Values: Suppose the regression gave us a slope of \( 0.5 , \text{kPa/cm}^3 \) and an intercept of \( 0.8 , \text{kPa} \):
Then:
\[ V_{\text{max}} = \frac{1}{0.5} = 2.0 , \text{cm}^3 \] \[ K = 0.8 \cdot 2.0 = 1.6 \]
These equations are entirely hypothetical, so I encourage you to compute the precise linear regression using your choice of software, which will provide the exact values of \( K \) and \( V_{\text{max}} \).
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