How to get the approximation $$ \frac{15}{k}\approx\frac{4R}{(1-R)^2} $$ given that $$ \frac{15}{k}=\frac{X(1-X)}{Y(Y-X)} $$ and $$ R=\frac{X(1-Y)}{Y(1-X)} $$ where $k$ and $R$ are constants, and $Y(t)=1-\exp(-kt)$ with $t>0$.
I was stuck with this approximation when I was studying the theory of chromatography - a series of research papers wrote by E. Glueckauf. One can check the origin here.
$R$ is defined as the distribution factor of an adsorption isotherm $Y=f(X)$. For Langmuir isotherms $R$ is a constant as shown before. According to the constant pattern condition which suggests $\bar Y=X$ and by taking the linear diffusion equation which gives $d \bar Y/dt=15(Y-\bar Y)$, the author finally obtained this approximate relation. $\bar Y$ is the actual uptake at concentration $X$. $\bar Y(0)=0$. $Y$ is the equilibrium uptake at concentration $X$. Concentration $X$ varies with time $t$.
But he didn't demonstrate the derivation process, which obsessed me for quite a while. What the author was trying to do is to link the output shape characterized by $k$ to the input shape characterized by $R$. I hope someone could help disperse the mist around me.
My thought is as below:
Suppose $k=4\pi^2$, one can give a plot of $Y$ against $X$:
This isotherm $Y=f(X)$ should be characterized by the Langmuir equation:
$$Y=\frac{X}{R+(1-R)X}$$
So by fitting the isotherm, $R$ can be obtained. Thus an input $k$ gives an output $R$. $R$ is highly related to $k$. By plenty of calculations, one will obtain a plot of $k$ against $R$. That plot may be characterized by
$$ k \approx \frac{15(1-R)^2}{4R}$$
as shown below:
