# Why n and p are not additive in law of mass action? or Why is the sum of electron concentration and hole concentration is not constant?

I was asked this question during a viva examination. I tried to give answer through mathematical derivation but I was told to give the answer from intuition.

So why is $$n_0 p_0 = n_i^2$$, and not $$n_0 + p_0 = \text{(some constant)}$$

Why is $$n_0 + p_0$$ not constant ?

• Because it is, in essence, a chemical reaction governed by detailed balance. Just like chemical equilibrium if a reaction is a product, not sum, based reaction constant. – Jon Custer May 29 at 11:47
• Maybe it's not the physics answer, but suppose you had $n > 2n_i$. Would you expect to have $p<0$? – The Photon May 29 at 17:36

## 1 Answer

Consider the reaction

$${A <=> B + C}$$

Taking $$A$$ as the standard state, using the law of mass action or detailed balance from chemistry, the reaction constant is then

$${K = [B][C]}$$

Now, $$A$$ is a filled state in the valence band. $$B$$ is an electron in the conduction band and $$C$$ is a hole in the valence band. As long as there is a path for e-h generation and recombination, the reaction can and will come to equilibrium, and that equilibrium state will obey $$np =$$ some constant. Adding excess $$B$$ or $$C$$ through doping shifts the ratio of $$B$$ to $$C$$, but not the overall equilibrium constant.

• What’s does the square bracket notation mean? – boyfarrell May 29 at 20:05
• In chemistry those mean concentrations, so map directly to $n$ or $p$. – Jon Custer May 29 at 20:54