The Freundlich adsorption isotherm is an empirical relationship that describes the adsorption of gases or solutes onto a surface. It is represented by: \[ \frac{x}{m} = k p^{1/n} \] where $x$ is the mass of the adsorbate, $m$ is the mass of the adsorbent, $p$ is the pressure, and $k$ and $n$ are constants. The graphical representations are:
A: Correct, shows $\frac{x}{m}$ vs $p$ with a logarithmic relationship.
B: Correct, shows $\log \frac{x}{m}$ vs $\log p$ as a straight line.
D: Correct, shows a logarithmic relationship $\frac{x}{m}$ vs $p^{1/n}$.
C: Incorrect, as it shows a constant $\frac{x}{m}$, not related to Freundlich's isotherm. \end{itemize}
Thus, the correct representations are A, B, and D.
If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to: