For a solution of $\ce{HA-}$, I have seen the following approximations for the pH:

1. $\textrm{pH}=\frac{1}{2}(\textrm{pKa1}+\textrm{pKa2})$

2. $\textrm{pH}=\sqrt{\frac{\textrm{K1}\times\textrm{K2}\times\ce{[HA-]}+\textrm{K1}\times\textrm{Kw}}{\textrm{K1}+\ce{[HA^-]}}}$

How do they differ? Is the second approximation always more accurate than the first? I know that the second is derived from the mass and charge balance equations and the first is the same as the second, with several assumptions.

For a solution of $\ce{HA-}$, I have seen the following approximations for the $\mathrm{pH}:$

$$\mathrm{pH} = \frac{1}{2}(\mathrm{p}K_\mathrm{a1} + \mathrm{p}K_\mathrm{a2})\tag{1}$$

$$\mathrm{pH} = \sqrt{\frac{K_1K_2[\ce{HA-}] + K_1K_\mathrm{w}}{K_1 + [\ce{HA-}]}}\tag{2}$$

How do they differ? Is the second approximation always more accurate than the first? I know that the second is derived from the mass and charge balance equations and the first is the same as the second, with several assumptions.

For a solution of $\text{HA-}$, I have seen the following approximations for the pH:

1. $\text{pH}=\frac{1}{2}(\text{pKa1}+\text{pKa2})$

2. $\text{pH}=\sqrt{\frac{(\text{K1}*\text{K2}*\text{[HA-]}+\text{K1}*\text{Kw})}{(\text{K1}+\text{[HA-]})}}$

How do they differ? Is the second approximation always more accurate than the first? I know that the second is derived from the mass and charge balance equations and the first is the same as the second, with several assumptions.

For a solution of $\ce{HA-}$, I have seen the following approximations for the pH:

1. $\textrm{pH}=\frac{1}{2}(\textrm{pKa1}+\textrm{pKa2})$

2. $\textrm{pH}=\sqrt{\frac{\textrm{K1}\times\textrm{K2}\times\ce{[HA-]}+\textrm{K1}\times\textrm{Kw}}{\textrm{K1}+\ce{[HA^-]}}}$

How do they differ? Is the second approximation always more accurate than the first? I know that the second is derived from the mass and charge balance equations and the first is the same as the second, with several assumptions.