Following of which can be an empirical relationship between the quantity of adsorbed by unit mass of solid adsorbent and pressure at a particular temperature. X=mass of the gas adsorbed on a mass 'm' of the adsorbent at a pressure 'P'.K and Constants,which depend on the nature of the adsorbent and the gas at a partial temperature.
log x+log m=log k+\(\frac{1}{n}\) log P
log x+log m=log k-\(\frac{1}{n}\) log P
log x+log m=-log k+\(\frac{1}{n}\) log P
log x-log m=log k+\(\frac{1}{n}\) log P
log x-log m=log k-\(\frac{1}{n}\) log P
The Correct Option is D
Approach Solution - 1
Adsorption isotherm analysis:
The Freundlich adsorption isotherm gives the empirical relationship: \[ \frac{x}{m} = kP^{1/n} \]
Taking logarithms: \[ \log\left(\frac{x}{m}\right) = \log k + \frac{1}{n}\log P \] \[ \log x - \log m = \log k + \frac{1}{n}\log P \]
Comparison with options: This matches exactly with option (D).
Thus, the correct option is (D): \(\log x - \log m = \log k + \frac{1}{n} \log P\).
Approach Solution -2
1. Empirical relationship for adsorption (Freundlich isotherm):
The Freundlich adsorption isotherm provides an empirical relationship between the quantity of gas adsorbed per unit mass of adsorbent (\(x/m\)) and the pressure (\(P\)) at a particular temperature:
\[\frac{x}{m} = k P^{1/n}\]
where:
- \(x\) is the mass of gas adsorbed
- \(m\) is the mass of the adsorbent
- \(P\) is the pressure
- \(k\) and \(n\) are constants that depend on the nature of the adsorbent and the gas, as well as the temperature
2. Take the logarithm of both sides:
Taking the logarithm of the Freundlich equation:
\[\log \left(\frac{x}{m}\right) = \log k + \frac{1}{n} \log P\]
3. Simplify using logarithmic properties:
Using the property of logarithms \(\log(a/b) = \log a - \log b\), we can rewrite the equation as:
\[\log x - \log m = \log k + \frac{1}{n} \log P\]
4. Compare with the given options:
The derived equation matches option (D):
\[\log x - \log m = \log k + \frac{1}{n} \log P\]
5. Final answer:
The correct option is (D).
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Concepts Used:
Adsorption
Heinrich Kayser, the German physicist was the first to coin the term adsorption. Adsorption can be explained as a surface phenomenon where particles remain attached on the top of a material. Generally, it comprises the molecules, atoms, liquid, solid in a dissolved stage, even the ions of a gas that are attached to the surface. Much to our surprise, the consequence of surface energy i.e. adsorption is present in biological, physical, chemical, and natural systems and are used in many industrial applications.
