# 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 $$W$$?

• 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 $(-2 \rho_{XY} \sigma_X \sigma_Y+2 \rho_{XZ} \sigma_X \sigma_Z-2 \rho_{YZ} \sigma_Y \sigma_Z+\sigma_X^2+\sigma_Y^2+\sigma_Z^2)/(\log{10})^2$.
– JimB
May 26, 2022 at 6:20

$$W$$ 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.
• 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 May 26, 2022 at 6:27