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I am reviewing the book Biochemistry Concepts and Connections by Appling, Cahill, and Mathews and I cannot understand why they divide by the hydrogen concentration by $10^{-7}$. Why not just leave it at the antilog(-8.1) over $\pu{1 M}$ like the other concentrations? I literally have see nothing else on the internet like this.

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The textbook is precisely correct. The equilibrium constant $K$ which the logarithm is taken of is dimensionless, and includes activities or fugacities, and not concentrations and pressures. In practice this is achieved by using standard states which refer to the pure materials: standard concentration $c^⦵$ and standard pressure $p^⦵$. One must be very fastidious with units when finding the equilibrium constant. For example, the reaction

$$\ce{aA + bB <=> cC + dD}$$

equilibrium constant $K_c$ is exactly

$$K_c = \frac{([\ce{C}]/c^⦵)^c\cdot ([\ce{D}]/c^⦵)^d}{([\ce{A}]/c^⦵)^a\cdot ([\ce{B}]/c^⦵)^b}$$

For pure water in its standard state $c^⦵ = [\ce{H+}] = \pu{1e-7 M}$. It also correlates with so-called biological standard state of $\mathrm{pH} = 7$. You probably haven't seen it before because many authors use sloppy notations omitting mentioning standard states since they can often be cancelled out. In this case those cannot be cancelled out, and must be written explicitly.


In fact, your own textbook contains extensive explanation [1, p. 91]:

For chemical reactions the standard state for solutes is defined as $\pu{1 M}$; however, in living cells the concentration of $[\ce{H+}]$ is roughly $10^{-7}~\mathrm M$, much lower than the standard value of $\pu{1 M}$. It is therefore appropriate to define the reference concentration of $\ce{H+}$ in biochemical reactions relative to the $\ce{H+}$ concentration found in the living state (i.e., $10^{-7}~\mathrm M$), rather than the value $\pu{1 M}$ defined by the chemical standard state. Recall that when a solute in a dilute solution has a concentration of $\pu{1 M}$, the activity of that solute is unity. For the biochemical standard state we define the activity of $\ce{H+}$ to be unity when $[\ce{H+}] = 10^{-7}~\mathrm M$.

[...]

  1. The mass action expression $Q$ is unitless. We strip the units from each concentration term in $Q$ by dividing each by its proper standard concentration (e.g., $\pu{1 M}$ for all solutes
    except $\ce{H+}$; $10^{-7}~\mathrm M$ for $\ce{H+}$; $\pu{1 bar}$ for gases, etc.).

Refrences

  1. Appling, D. R.; Anthony-Cahill, S. J.; Mathews, C. K. Biochemistry: Concepts and Connections (Global Edition); Pearson: Boston, 2015. ISBN 978-1-292-11210-7.
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    $\begingroup$ There are two standard states for [H+]. In the absence of a prime after the $^\circ$, it is 1 M, and with the prime (biochemical standard state), it is $10^{-7}$ M. $\endgroup$ – Karsten Theis Mar 4 at 13:52
  • $\begingroup$ Thanks, I have the second edition and this explanation is not on this page 91. Is it from Chapter 3? If so I will find it. $\endgroup$ – user75312 Mar 4 at 21:29
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    $\begingroup$ Nevermind, it is on page 59 of the second edition for those with the second edition. $\endgroup$ – user75312 Mar 4 at 21:55
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For an explanation you may want to inspect Recommendations for Biochemical Equilibrium Data 1, which states:

Buffer and pH. If only a limited number of measurements are to be made, they should be carried out at pH = 7.0 and, if possible, also at a pH value at which the apparent equilibrium constant $K_c^\prime$, has little or no dependence on pH. ($K_c^\prime$ is defined in a later section.) If direct measurements at pH = 7.0 are not practicable, the calculated values for this pH should be reported. The procedure used in making these calculations must be carefully described. Care should be taken that the solution is adequately buffered so that the pH is well defined throughout the experiment. It is desirable to determine the effect of varying the nature and concentration of the buffer in order to identify buffer effects. Buffers that are known to interact with reactants (including macromolecules) or salts, such as phosphate or pyrophosphate in the presence of divalent metal ions, should be avoided.

The highlighted portion means that reported values for biochemical reactions are (or should be) referenced to pH 7.0. The data in Table 3.6 refer to this biochemical standard state.

Another question on the subject of equilibrium constants also addresses the importance of properly considering reference states.

Reference

1 The Journal of Biological Chemistry (1976), Vol. 261, No. 22, pp. 6859-6885.

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