What are the standard approaches to calculating how dissociation constants for acids and bases, $\mathrm{p}K_\mathrm{a}$ and $\mathrm{p}K_\mathrm{b}$, change as the ionic product of water, $\mathrm{p}K_\mathrm{w}$, changes?
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2$\begingroup$ They primarily change with temperature, but they still follow pKa + pKb = pKw $\endgroup$ – Poutnik Jun 28 '20 at 20:06
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$\begingroup$ Is the change generally "linear", if pKw decreases, do pKa and pKb both decrease proportionally (pKa + pKb = pKw)? $\endgroup$ – Asker123 Jun 28 '20 at 20:20
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2$\begingroup$ @Asker123: Show first what you think in details. $\endgroup$ – Mathew Mahindaratne Jun 28 '20 at 20:21
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$\begingroup$ I don't really know how you'd think anyone could show more what they think than "if the equation is pKa + pKb = pKw, will both acid and conjugate base disassociation constant generally change "linearly"" $\endgroup$ – Asker123 Jun 28 '20 at 20:47
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The equation
$$\mathrm{p}K_\mathrm{a} + \mathrm{p}K_\mathrm{b} = \mathrm{p}K_\mathrm{w}\tag{1}$$
has two degrees of freedom, so two values are independent on each other and the third one depends on the other two.
Reaction equilibrium constant $K_\mathrm{a}$ for
$$\ce{HA + H2O <=> H3O+ + A-}\tag{R1}$$
is chemically independent on $K_\mathrm{w}$ for reaction
$$\ce{2 H2O <=> H3O+ + OH-}\tag{R2}$$
OTOH, the equilibrium constant $K_\mathrm{b}$ for the conjugate base is derived from the above two.
$$K_\mathrm{a} = \frac{[\ce{H+}][\ce{A-}]}{[\ce{HA}]} = \frac{K_\mathrm{w}[\ce{A-}]}{[\ce{HA}][\ce{OH-}]} = \frac{K_\mathrm{w}}{K_\mathrm{b}}\tag{2}$$
$$\mathrm{p}K_\mathrm{a} + \mathrm{p}K_\mathrm{b} = \mathrm{p}K_\mathrm{w}$$
Another thing is that both $\mathrm{p}K_\mathrm{a}$ and $\mathrm{p}K_\mathrm{w}$ depend on temperature. It is not dependence
$$\Delta T \to \Delta \mathrm{p}K_\mathrm{w} \to \Delta \mathrm{p}K_\mathrm{a},\tag{3}$$
but
$$ \begin{cases}\tag{4} \Delta T \to \Delta \mathrm{p}K_\mathrm{w} \\ \Delta T \to \Delta \mathrm{p}K_\mathrm{a} \end{cases} $$
For the temperature dependence of equilibrium constants, there is van 't Hoff equation, which calculates the constant value changes from the reaction enthalpy:
$$\ln \frac{K_2}{K_1} = \frac{-\Delta H^\circ}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right),$$
from which:
$$K_2 = K_1 \cdot \exp{\left( \frac{-\Delta H^\circ}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)\right)}\tag{5}$$
