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It is well known that the A in both DNA and RNA stands for acid, but where is the acid in chemical formula for the compound, and it is classified so based on what acid-base theory? Like Arrhenius, Brønsted-Lowry or Lewis.

Just to better explain my question, when I say DNA I am essencially thinking about nucleotides linked together, am I missing something in the structure that contains my answer or this is the way to go?

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The phosphate ester groups which connect the nucleotides contain one acidic proton at their OH group, and two at the end of each strand. As you can see here, here and in the picture below (with the groups marked turquoise), most of these groups are deprotonated depending on pH. They act as Brønsted-Lowry acids.

enter image description here

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Just to better explain my question, when I say DNA I am essencially thinking about nucleotides linked together, am I missing something in the structure that contains my answer or this is the way to go?

Yes, you do. *NA consist of nucleotides, linked by phosphodiester fragments. This leaves one hydroxyl of the phosphate fragment intact, so it can act as an acid.

Look closely:

enter image description here

There is a $\ce{P-O-}$ fragment, which means that there is a cation somewhere nearby. So, *NA are Brønsted acids, usually occuring as salts.

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It refers to phosphoric acid. There are three aspects that might make this confusing.

  1. Phophoric acid is a triprotic acid, i.e. it can lose 3 protons.
  2. The phosphoric acid comes in the form of a diester (two ester bonds linking the phosphoric acid with a a sugar-hydroxyl), so now it only has one proton that can dissociate.
  3. At pH = 7, it already lost the proton, so what we see in DNA and RNA at neutral pH is the conjugate base of the phosphate diester.

The resulting negative charge on the phosphate oxygen makes DNA and RNA a polyanion and explains some of its properties (high solubility in water, metal binding, direction it travels in gel electrophoresis, to name just three).

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