I've just been getting into organic and been reading about H-H related to $\mathrm{NMR}$. I've been seeing articles state how recasting of H-H is necessary and want to know how equations like below are derived.
$$σ_\mathrm{peak} = σ_\ce{HA} + \frac{Δσ}{1 + 10^{(\mathrm{p}K_\mathrm{a} - \mathrm{pH})}}\tag{1}$$
$$y = y_\mathrm{min} + \frac{y_\mathrm{max} - y_\mathrm{min}}{1 + 10^{(\mathrm{pH} - \mathrm{p}K_\mathrm{a})}}\tag{2}$$
Eq. 1: https://www.nature.com/articles/srep43748
"In the equation, $\sigma_\mathrm{peak}$ is the measured $\mathrm{NMR}$ chemical shift of the peak of interest, $\sigma_\ce{HA}$ is the chemical shift of the protonated form and $\Delta\sigma$ is the difference between $\sigma_\ce{HA}$ and the chemical shift of the deprotonated form of $\ce{HA}$ ($\sigma_\ce{A}$)."
Eq. 2: https://pubs.acs.org/doi/full/10.1021/ed3000028?mobileUi=0
"$y$ is the experimentally measured parameter (chemical shift, in this case), and $(y_{max} - y)$ and $(y - y_{min})$ are proportional to the concentrations of the deprotonated and protonated forms of the acid, respectively."