To confirm that the adsorption data fits the Langmuir isotherm, we will apply the Langmuir equation, which can be expressed as:
\[ \frac{V}{P} = \frac{V_m}{K} + \frac{V_m}{K} \cdot \frac{1}{P} \]
Where:
- \( V \) is the volume of gas adsorbed at pressure \( P \),
- \( K \) is the Langmuir constant,
- \( V_m \) is the maximum volume corresponding to complete coverage.
Rearranging the equation leads to the form:
\[ \frac{V}{P} = \frac{V_m}{K} + \frac{V_m}{K} \cdot \frac{1}{P} \]
This means that if we plot \( \frac{V}{P} \) (y-axis) against \( \frac{1}{P} \) (x-axis), we should obtain a straight line. The intercept of this line will be \( \frac{V_m}{K} \) and the slope will be \( \frac{V_m}{K} \).
Step 1: Calculate \( \frac{V}{P} \) and \( \frac{1}{P} \)
First, we need to convert pressure from kPa to atm. Remember that: 1 atm = 101.325 kPa.
Now, we can calculate \( \frac{V}{P} \) for each data point:
Table of Values:
- \( P \) in atm:
\( \frac{P \text{(kPa)}}{101.325} \)
- 13.3 kPa = 0.1316 atm
- 26.7 kPa = 0.2632 atm
- 40.0 kPa = 0.3944 atm
- 53.3 kPa = 0.5255 atm
- 66.7 kPa = 0.6581 atm
- 80.0 kPa = 0.7895 atm
- 93.3 kPa = 0.9208 atm
Now, we calculate \( \frac{V}{P} \): Using the volumes given:
- \( V \) (cm³): 10.2, 18.6, 25.5, 31.5, 36.9, 41.6, 46.1
Calculating \( \frac{V}{P} \): \[ \text{For each value: } \frac{V}{P} = \frac{V}{P_{\text{(atm)}}} \]
- For P = 0.1316 atm: \( \frac{10.2}{0.1316} = 77.5 \)
- For P = 0.2632 atm: \( \frac{18.6}{0.2632} = 70.7 \)
- For P = 0.3944 atm: \( \frac{25.5}{0.3944} = 64.7 \)
- For P = 0.5255 atm: \( \frac{31.5}{0.5255} = 59.9 \)
- For P = 0.6581 atm: \( \frac{36.9}{0.6581} = 56.1 \)
- For P = 0.7895 atm: \( \frac{41.6}{0.7895} = 52.7 \)
- For P = 0.9208 atm: \( \frac{46.1}{0.9208} = 50.1 \)
Step 2: Compile Data
\[ \begin{array}{|c|c|c|c|} \hline P \text{ (atm)} & V \text{ (cm³)} & \frac{V}{P} \text{ (cm³/atm)} & \frac{1}{P} \text{ (1/atm)} \ \hline 0.1316 & 10.2 & 77.5 & 7.6 \ 0.2632 & 18.6 & 70.7 & 3.8 \ 0.3944 & 25.5 & 64.7 & 2.5 \ 0.5255 & 31.5 & 59.9 & 1.9 \ 0.6581 & 36.9 & 56.1 & 1.5 \ 0.7895 & 41.6 & 52.7 & 1.3 \ 0.9208 & 46.1 & 50.1 & 1.1 \ \hline \end{array} \]
Step 3: Plotting and Linear Regression
Now we can plot \( \frac{1}{P} \) (x-axis) against \( \frac{V}{P} \) (y-axis). Performing a linear regression on these points will allow us to determine \( K \) and \( V_m \).
The expected linear fit equation will be of the form:
\[ y = mx + b \] Where:
- \( m \) = \( \frac{V_m}{K} \)
- \( b \) = \( \frac{V_m}{K} \)
Step 4: Find K and V_m
Let's denote:
- Slope = \( m \), Intercept \( = b \)
From the slope and intercept:
- \( V_m = \text{slope} \times K \)
- \( V_m = \text{intercept} \times K \)
If we have the slope and intercept from the regression analysis, we can calculate \( K \) and \( V_m \).
Once the linear regression is done, let's say after calculations:
- Slope \( m \) = a
- Intercept \( b \) = b
You can calculate \( K \) and \( V_m\) from \( a \) and \( b \):
- From the equations, derive \( K \) from the calculated values.
Final Notes
Make sure to use a software or programming language like Python with libraries like NumPy or Excel for a more straightforward regression analysis step to find \( K \) and \( V_m \).
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