Question

Using the Langmuir isotherm. The data given below are for the adsorption of CO on charcoal at 273 k. Confirm that they fit the Langmuir isotherm and find the constant K and the volume corresponding to complete coverage.
In each case V has been corrected to 1.00 atm.
P/kPa: 13.3 26.7 40.0 53.3 66.7 80.0 93.3
V/cm³: 10.2 18.6 25.5 31.5 36.9 41.6 46.1

Answers

Answered by GPT-4o mini
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:
1. \( V_m = \text{slope} \times K \)
2. \( V_m = \text{intercept} \times K \)

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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