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When a graph is plotted between \[\log {\text{x / m}}\] and \[\log {\text{p}}\], it is a straight line with an angle \[{45^ \circ }\] and intercept \[0.3010\] on y-axis. If the initial pressure is \[0.3{\text{ atm}}\], what will be the amount of gas adsorbed per gram of the adsorbent?
A.\[0.4\]
B.\[0.6\]
C.\[0.8\]
D.\[0.1\]

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Last updated date: 20th Jun 2024
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Answer
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Hint: The answer to this question we need to use the formula given by Freundlich adsorption isotherm. Taking log with base 10 both sides we will have the required relation. On comparing the equation with the equation for straight line we will get the value of the amount of gas adsorbed. Tangent of angle also gives the value of slope.
Formula used:
\[\dfrac{{\text{x}}}{{\text{m}}} = {\text{k}}{{\text{P}}^{{\text{1/n}}}}\] here \[\dfrac{{\text{x}}}{{\text{m}}}\] is amount of gas adsorbed per gram of solvent, k is proportionality constant, P is pressure and n is also a constant.

Complete step by step answer:
We will take the log on both sides on the formula given by Freundlich isotherm as
\[\dfrac{{\text{x}}}{{\text{m}}} = {\text{k}}{{\text{P}}^{{\text{1/n}}}}\]
Taking logarithm both sides,
\[{\text{log }}\dfrac{{\text{x}}}{{\text{m}}} = {\text{log (k}}{{\text{P}}^{{\text{1/n}}}})\]
\[{\text{log }}\dfrac{{\text{x}}}{{\text{m}}} = {\text{log k}} + {\text{ log }}{{\text{P}}^{{\text{1/n}}}}\]
We will further simplify the above equation:
\[{\text{log }}\dfrac{{\text{x}}}{{\text{m}}} = {\text{log k}} + {\text{ }}\dfrac{1}{{\text{n}}}{\text{log P}}\]
The above equation represents the linear straight line equation, \[{\text{y}} = {\text{mx}} + {\text{c}}\]. When a graph is plotted against y and x then m is the slope and c is the intercept.
In our question y is \[\dfrac{{\text{x}}}{{\text{m}}}\] and x is P . So the slope becomes \[\dfrac{1}{{\text{n}}}\]. Instead of the value of slope an angle has been given. If we calculate the tangent of the angle, we will get the slope.
\[{\text{slope}} = \dfrac{1}{{\text{n}}} = \tan {45^ \circ }\]
\[ \Rightarrow {\text{slope}} = 1\]
\[\log {\text{k}}\] is the intercept whose value is given to us as \[0.3010\]. The value of k will come out to be,
\[k = {10^{0.3010}} = 2\]
Initial pressure is \[0.3{\text{ atm}}\]. Substituting all the given values in the formula we will get:
\[\dfrac{{\text{x}}}{{\text{m}}} = 2{(0.3)^{{\text{1/1}}}} = 0.6\]

Hence, the correct option is B.

Note:
An isotherm is that which tells the variation of amount of substance adsorbed on the surface of adsorbent with change in pressure, keeping the value of temperature constant.