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Background

The Henderson-Hasselbalch equation is given by

$$\text{pH} = \text{pK}_{\text{a}} + \log_{10} \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)$$

where $\text{pH} = -\log [\text{H}^+]$. Solving for the $\text{pK}_{\text{a}}$ we obtain the equality

$$\text{pK}_{\text{a}} = \text{pH} - \log_{10} \left( \frac{[\text{Base}]}{[\text{Acid}]} \right).$$

Just to make our notation simpler, let us substitute:

  • $W :=\text{pK}_{\text{a}}$
  • $X := [\text{H}^+]$
  • $Y := [\text{Acid}]$
  • $Z := [\text{Base}]$

giving us the expression

$$W = - \log_{10}X - \log_{10} \left( \frac{Z}{Y} \right)$$

which is equivalent to

$$W = \log_{10} \left( \frac{Y}{XZ} \right).$$

Those familiar with using the equation may raise an eyebrow at treating $\text{pK}_{\text{a}}$ as a random variable since it is often taken to be a constant. Here I am supposing that variability enters into our calculation of the $\text{pK}_{\text{a}}$ through the variables $X,Y,Z$ that are themselves not known with perfect precision.

Assume that $X,Y,Z$ have a joint log-normal distribution. Statistical independence cannot be assumed.

Question

What is the distribution of $W$?

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  • $\begingroup$ The answer below is correct but I assume you are also looking for the detailed description of the mean and variance. If $\log X$, $\log Y$, and $\log Z$ follow a multivariate normal with the usual parameters, then $W$ is normal with mean $(\mu_Y-\mu_X-\mu_Z)/\log (10)$ and variance $(-2 \rho_{XY} \sigma_X \sigma_Y+2 \rho_{XZ} \sigma_X \sigma_Z-2 \rho_{YZ} \sigma_Y \sigma_Z+\sigma_X^2+\sigma_Y^2+\sigma_Z^2)/(\log{10})^2$. $\endgroup$
    – JimB
    May 26, 2022 at 6:20

1 Answer 1

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$W$ is Normal:

$$W=-\log_{10} X -\log_{10} Z+\log_{10} Y$$ and by assumption the three quantities on the right-hand side are multivariate Normal, so their sum is Normal.

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  • $\begingroup$ I am aware of the stability property of the normal distribution which requires independence. How is the result obtained in the dependent case? $\endgroup$
    – Galen
    May 26, 2022 at 6:06
  • $\begingroup$ These are multivariate Normal distributions -- that's what it means for $(X,Y,Z)$ to be multivariate logNormal. So any linear combination of them is also Normal $\endgroup$ May 26, 2022 at 6:27

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