Question
Use natural logarithm to show that Freundlich isotherm graph can be changed into a linear graph
Answers
Answered by
GPT-4o mini
The Freundlich adsorption isotherm equation is expressed as:
\[
q_e = K_f C_e^{1/n}
\]
where:
- \( q_e \) is the amount of adsorbate per unit mass of adsorbent at equilibrium,
- \( C_e \) is the equilibrium concentration of the adsorbate in the solution,
- \( K_f \) and \( n \) are constants specific to the system.
To linearize this equation using natural logarithms, we can perform the following steps:
1. **Take the natural logarithm of both sides**:
\[
\ln(q_e) = \ln(K_f C_e^{1/n})
\]
2. **Use the property of logarithms** to separate the terms on the right side:
\[
\ln(q_e) = \ln(K_f) + \frac{1}{n} \ln(C_e)
\]
3. **Rearrange the equation**:
\[
\ln(q_e) = \ln(K_f) + \frac{1}{n} \ln(C_e)
\]
4. **Identify the linear terms**: This equation is now in the form of \( y = mx + b \), where:
- \( y = \ln(q_e) \)
- \( x = \ln(C_e) \)
- \( m = \frac{1}{n} \) (the slope)
- \( b = \ln(K_f) \) (the y-intercept)
Thus, the Freundlich isotherm can be represented as a linear graph by plotting \( \ln(q_e) \) against \( \ln(C_e) \). The slope of this line will be equal to \( \frac{1}{n} \), and the y-intercept will be \( \ln(K_f) \). This linearization of the Freundlich isotherm allows for easier determination of the parameters \( K_f \) and \( n \) through linear regression analysis of experimental data.
\[
q_e = K_f C_e^{1/n}
\]
where:
- \( q_e \) is the amount of adsorbate per unit mass of adsorbent at equilibrium,
- \( C_e \) is the equilibrium concentration of the adsorbate in the solution,
- \( K_f \) and \( n \) are constants specific to the system.
To linearize this equation using natural logarithms, we can perform the following steps:
1. **Take the natural logarithm of both sides**:
\[
\ln(q_e) = \ln(K_f C_e^{1/n})
\]
2. **Use the property of logarithms** to separate the terms on the right side:
\[
\ln(q_e) = \ln(K_f) + \frac{1}{n} \ln(C_e)
\]
3. **Rearrange the equation**:
\[
\ln(q_e) = \ln(K_f) + \frac{1}{n} \ln(C_e)
\]
4. **Identify the linear terms**: This equation is now in the form of \( y = mx + b \), where:
- \( y = \ln(q_e) \)
- \( x = \ln(C_e) \)
- \( m = \frac{1}{n} \) (the slope)
- \( b = \ln(K_f) \) (the y-intercept)
Thus, the Freundlich isotherm can be represented as a linear graph by plotting \( \ln(q_e) \) against \( \ln(C_e) \). The slope of this line will be equal to \( \frac{1}{n} \), and the y-intercept will be \( \ln(K_f) \). This linearization of the Freundlich isotherm allows for easier determination of the parameters \( K_f \) and \( n \) through linear regression analysis of experimental data.