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What exactly is the difference between the following equations? $$p = kx \qquad\text{and}\qquad m = kp$$ Is the $k$ in both same, because I have seen same $k$ in textbooks for both equations?

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    $\begingroup$ Both are different Henry's law constants. See: en.wikipedia.org/wiki/Henry%27s_law $\endgroup$ May 20, 2019 at 20:48
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    $\begingroup$ Henry's law constants also reflect an implicit scale: molality, mole fraction, concentration, etc. (with a given reference state defined as such). $\endgroup$
    – Rolf
    May 20, 2019 at 22:40

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While both equations imply a relation between the partial pressure of the solute and its concentration in solution, clearly $k$ means different things in each.

The first, $$x=H^\mathrm{xp} p$$ is Henry's law with $H^\mathrm{xp}=1/k$ the Henry's law constant in units of mole fraction concentration per unit pressure. The second $$m=H^\mathrm{mp} p$$ is an alternative form with $H^\mathrm{mp} = k$ the Henry's law constant in units of molal concentration per unit pressure.

Both are valid at low solute concentrations given that a conversion between the concentration scales is possible using

$$H^\mathrm{xp} = H^\mathrm{mp} M_\ce{H2O}$$

as explained in the Wikipedia.

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