[H⁺] x [HCO₃^-] = K x [CO₂] x [H₂O]

Hasselbalch modified Henderson's elegant idea by regarding the water concentration as constant and taking logarithms of the remaining components (pK is the negative logarithm of "K"). This resulted in the Henderson-Hasselbalch Equation:

pH = pK + log ( [HCO₃^-] / [CO₂] )

The consequence of using negative logarithms is that "everything is upside down" and incomprehensible to most physicians; it contains the same information as Henderson's simple equilibrium equation. It could have been so much easier; the conversion could have been applied to the whole equation at once. The first step is to write Henderson's equation in the right order with the water concentration omitted as a constant.

[H⁺] = K x [CO₂] / [HCO₃^-]

The second is to take the negative logarithm:

pH = -Log ( K x [CO₂] / [HCO₃^-] )

The "K" is still "K" and the equation is still recognizable. Why, then, have generations of medical students been taught the Henderson-Hasselbalch version? Why, in fact, were we taught it at all? Were our teachers so mathematically naive that they failed to recognize that the two equations were mathematically equivalent. If so, did they succumb to the temptation to teach us - and therefore test us - using the more complex version?

Part of the reason lies outside medicine; chemists find knowing the negative logarithm of "K" (pK) is a useful shorthand way of writing a long number. In addition, the same logarithmic version is in widespread use, although it is known by other names in other places. At the Royal Veterinary and Agricultural University of Copenhagen it is known as the "Bjerrum equation" in honor of Professor Bjerrum who worked there; and, in the chemical world it is generally known as the buffer equation.. There is no need in physiology for us to use this equation. It is part of history's legacy. The Modified Henderson Equation is recommended

Henderson without Hassel(balch)

Modified Henderson Equation

Henderson's equation quantitates the relationship between [H⁺], [HCO₃^-], and PCO₂. In other words it performs precisely the same function as the more intimidating Henderson-Hasselbalch Equation. It does it, however, without using negative logarithms. It is, accordingly, much easier to understand:
 

[H⁺] [HCO₃^-] = k x PCO₂

The Interactive Henderson Equation: Use the Equation Only to enter values and observe the automatic interpretation. Then try to Test Yourself and your ability to interpret the data you enter. Select the version you need - it will open in a separate window:

Henderson without the Hassel

The year 1908 was a good year for acid-base balance - thanks to Henderson. In that year he appreciated the buffering power of CO₂ and went on to apply the law of mass action to produce his simple formula which, rearranged, looks like this:
 

[H⁺] [HCO₃^-] = K [CO₂] [H₂O]

Simplification is possible because [H₂O] remains constant and physicians are much more familiar with PCO₂. This gives us the version used in the Interactive Equation:

[H⁺] [HCO₃^-] = k x PCO₂

Easy to understand - most school children could grasp its meaning - which probably means that even medical students could, too. The simplicity, however, also guaranteed that it was of little "interest". Next year, 1909, was the start of a downhill slide: Sorensen introduced the negative logarithm (pH). This paved the way for Hasselbalch. In 1916 he combined Henderson's excellent equation with Sorensen's pH to produce the dreaded Henderson-Hasselbalch equation.

Acid-base balance became instantly incomprehensible and, therefore, fascinating to teach. The new equation contained no extra information for physicians; it solved no medical problem; and it added nothing to our sum of acid-base physiology except obfuscation and unnecessary exam questions.

Three generations of doctors have endured the Henderson-Hasselbalch equation. It's time to make a stand. Enough! Play with the Modified Henderson Equation. Appreciate that it is the whole truth. If you must convert between pH and [H⁺], then do so, but do it in two simple steps. This way you may understand what you are doing:

1. Calculate the [H⁺] using the modified Henderson equation:

   [H⁺] = K [CO₂] x [H₂O] / [HCO₃^-]

2. Take the negative logarithm of the [H⁺]
    

   pH = - log [H⁺]