Why don't we breathe nitrogen while it makes most of the air?
Why do we always tend to breathe oxygen, not hydrogen and nitrogen?
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I'd argue that we do "breathe" all those gases. Air that we inhale (at sea level) is about 78% N$_2$, 20.9% O$_2$, 1% argon, and smaller percentages of CO$_2$, neon, methane, etc. So all those gases are going into the lungs with every breath in.
We take up oxygen preferentially because we have hemoglobin to bind O$_2$. When hemoglobin binds the oxygen, it upsets the balance and pulls more oxygen across the alveolar membrane. This is aided by pulmonary circulation which carries the blood away. Here's a demo of the diffusion process.
Animals use oxygen as a chemical energy source because oxygen gas can react with many other compounds to form oxides, which releases energy and happen spontaneously.
Both carbon and nitrogen can be made to react with oxygen, but otherwise they are pretty inert. So of all the gasses in the air present at over a fraction of a percent, oxygen is the only one we can use for energy.
Hydrogen (and sulfer) are both possible substitutes for oxygen in the role of redox energy source, but are normally pretty small components of our environment. On another planet they might well be the basis of biometabolism.
Of course the fact that plants can use carbon dioxide to fix carbon is a different case of biology using a gas out of the air. Its the defining quality of plants!
The energetics of using CO2 is endothermic - it requires energy input. They have to use sunlight to get the energy to utilize this energy and its very costly energetically. Animals can afford to move and grow because they use oxygen while they eat plants.
Nitrogen is much less reactive than oxygen. Indeed, if I haven't totally forgotten my long-ago chemistry courses, most chemical reactions involving N2 are energy-consuming. Thus you get nitrogen compounds produced by lightning, in auto engines, and other places where there's a lot of energy to spare.
Oxygen reactions, OTOH, are energy-producing. You might think instead of fire: most organic stuff will burn (if dried), but it only combines with the oxygen in the air, not the nitrogen.
PS: Indeed, many nitrogen compounds take so much energy to create that they are explosives. Ammonium nitrate, nitroglycerin, trinitrotolulene (TNT), even the potassium nitrate (saltpeter) used to make gunpowder.
The bond in oxygen molecules is high energy, and ready to undergo an energy-yielding reaction with other molecules like sugar.
The bond in nitrogen not chemical useful to us...other organisms use energy to "fix" nitrogen to make energy rich nitrogen compounds that we can use.
The other answers seem to be missing the role of oxygen in oxidative phosphorylation, as an organism with aerobic metabolism we use oxygen for its electronegativity. Basically, as we break down glucose energy is released in the form of free electrons, these are "transported" for use in the oxidative phosphorylation to create new ATP which is our main form of energy storage. Oxygen eagerly accepts these free electrons at the last step of the oxidative phosphorylation and binds with H+ to form H2O. See here.
So without oxygen there would be an accumulation of electrons, stopping the oxidative phosphorylation and forcing us to break down glucose in the (less efficient) anaerobic manner.
There are two parts to this answer, and several answers have addressed one or both aspects but I figured I'd put it all in one place.
1) We use oxygen for a purpose that nitrogen is chemically useless for
2) While there is a different purpose we might want to use nitrogen for, it is something that is difficult to evolve (only bacteria have done it) and we can manage without.
1) We use oxygen because our metabolism uses it for energy. Our metabolism derives chemical energy from the breakdown of complex carbon molecules; this doesn't happen on its own and you need very reactive molecules to interact with those complex carbon molecules and break them down. All organisms do this step-by-step, using successive "electron acceptors" to basically strip electrons off of simpler-and-simpler molecules and thus break them down. Molecular oxygen is the most reactive molecule and greedy electron acceptor out there, and allows organisms that use it to get the most energy possible out of a given carbohydrate. That's why aerobic respiration is so useful, and that's what we use oxygen for. Molecular nitrogen has completely different chemical properties; it isn't that electronegative (i.e. greedy for electrons) at all. There are other molecules that can be used as electron acceptors, and are used in various forms of anaerobic respiration: nitrate, sulfate, carbon dioxide... but molecular nitrogen isn't one of them.
For various kinds of anaerobic respiration, see :
2) There IS a purpose for which one could use molecular nitrogen, which is to use it to build nitrogen-based molecules that our body depends on - like DNA, RNA and proteins, which basically do everything in a living organism. No organism uses molecular nitrogen as a source for these; it's much easier to use organic nitrogen compounds like nitrates and ammonia. It can seem silly that such compounds are so limiting, when nitrogen makes up most of the atmosphere! This is less of an issue for carnivores since we get all of our nitrogen needs from eating nitrogen-filled animals, but it's a huge issue for plants. The need for such compounds (and, to a lesser extent, phosphates) is why agriculture needs fertilizer. So why can very very few organisms break down molecular nitrogen ? Because it is a very stable molecule; if you've done chemistry you might know that the two nitrogen atoms in the nitrogen molecule are connected by a triple bond, which is very strong and hard to break. This may be a big reason why the metabolism to break that bond evolved only in bacteria, and all Eukaryotes get by using the bacteria themselves (nitrogen-fixing plants), absorbing nitrogen-filled organisms (carnivores, carnivorous plants - it's the reason they're carnivorous!) or getting by on the organic nitrogen that naturally occurs in the ground thanks to nitrogen-fixing bacteria.
As an aside, the fertilizer humans make uses the Haber process, which converts molecular nitrogen in the atmosphere to ammonia. If you look at the Wikipedia page by the way you'll get an idea of how hard it is to break that triple bond, between the catalysts and the high temperatures and pressures... But through that process you could argue that humanity as a species does "breathe" nitrogen.
So basically, tl;dr:
1) we don't need to breathe nitrogen
2) if we did our bodies still wouldn't because it's really hard to do; no eukaryote does it, except maybe humans themselves but only through technology.
As others have pointed out, we do breathe atmospheric nitrogen but we cannot do anything useful with it.
The problem is that the triple-bonded N2 is very unreactive and almost all animals and plants cannot convert it to anything else. Humans do not have the ability to reduce it to NH3, for example, but is would be great if we could.
All forms of life need nitrogen. It is necessary to make protein and DNA, for example. It is also very plentiful. N2 constitutes about 78% of air by volume, but it is in a form (N2) not usable by most forms of life. And therein lies the problem: in order to use atmospheric nitrogen, it needs to be 'fixed', ie converted to a form available to 'normal' metabolic transformation. This is usually taken to mean that N2 needs to be converted to NH3. Very few forms of life have the ability to do this. The 'fixation' of N2 is also a great industrial problem.
As outlined by Rozenn Keribin, the first artificial process to successfully 'fix' N2 is the Haber-Bosch process, which was developed in the early part of the 20th century, and uses a metal catalyst and high pressures to achieve the following transformation:
N2 + 3 H2 → 2 NH3
This process remains the main industrial source of NH3 today, and is a 'mainstay' of the fertilizer industry. During the First World War, it was also an source of NH3 for the production of ammunition by Germany. For his work in this area, Haber was awarded the 1918 Nobel prize in Chemistry, a (controversial) honour that has surely stood the test of time.
(I will not deal with leguminous plants, which can also 'fix' N2 using a symbiotic relationship with bacteria in root nodules, as I do not know enough about it).
In nature, [the] ability to fix N2 is restricted to a small but diverse group of diazotrophic microorganisms that contain the enzyme nitrogenase (Burgess & Lowe, 1996)
One type of nitrogenase (that contains molybdenum) catalyzes the following reaction:
N2 + 8 H+ + 8 e- + 16 ATP → 2 NH3 + H2 + 16 ADP + 16 Pi
Let's analyze this one: that's an eight-electron reduction that uses 16 ATPs just to make two NH3 from one N2 !
Nitrogenases may be classified into 3 general types depending on metal content: Molybdenum nitrogenase, vanadium nitrogenase and iron-only nitrogenase. (All forms contain iron).
Biological nitrogen fixation was discovered by Martinus Beijerinck and t Hermann Hellriegel. In addition, Beijerinck discovered that tobacco mosaic disease was caused by a virus. Neither received a Nobel prize.
Burgess, B,K. & Lowe, D. J. (1996) Mechanism of Molybdenum Nitrogenase Chem. Rev. 96, 2983−3011
Basically when air fills our alveoli, by the process of diffusion, only oxygen in the air is taken into the blood stream while the other gases along with the waste CO2 is exhaled. So you do breathe in nitrogen, but it is exhaled as it is by the body. The whole process of the respiratory system is explained here with diagrams.