Re: DNA is an acidic molecule, then why does it degrade at acidic pH?

Yes, DNA is a weak acid, an organic and very weak acid, but it does exist in the cell as a salt, like other acids. Hence it reacts with some weak bases and exists as salt in the ionic form. Hence it is stable in slightly alkaline pH (pH 8.0). DNA does not always degrade with all the acids, and at acidic pH, like for DNA precipitation, we add 1/10 volume of sodium acetate (pH 5-6 range), which is actually acidic, and acetic acid reduces the pH below 7.0 in sodium acetate buffer.

DNA chemically reacts with strong acids (mainly inorganic acids) and gets degraded or sometimes generates derivatives. It is not necessary that two acids will not react. Two strong acids, nitric acid (HNO3) and hydrochloric acid (HCL), are mixed to form "Aqua Regia", a chemical used for melting gold. But the properties of HCL and HNO3 are lost due to chemical reaction between them. Details of Aqua Regia are given below (taken from enclopedia) to show how two acids react.

Aqua Regia: [Lat.,=royal water], corrosive, fuming yellow liquid prepared by mixing one volume of concentrated nitric acid with three to four volumes of concentrated hydrochloric acid. It was so named by the alchemists because it dissolves gold and platinum, the “royal” metals, which do not dissolve in nitric or hydrochloric acid alone. Its fumes and yellow color are caused by reaction of nitric acid, HNO3, with hydrogen chloride, HCl, to form nitrosyl chloride, NOCl, chlorine, Cl2, and water; both chlorine and nitrosyl chloride are yellow-colored and volatile. The nitrosyl chloride further decomposes to nitric oxide, NO, and chlorine. Nitric acid is a powerful oxidizing agent (see oxidation and reduction), but the chemical equilibrium for its reaction with gold, Au, only permits formation of a tiny amount of Au+3 ion, so the amount of gold dissolved in pure nitric acid is undetectable. The presence of chloride ion, Cl-, allows formation of the stable chloraurate complex ion, AuCl4-. Because of the high concentration of chloride ion in aqua regia, the Au+3 is reacted almost as soon as it is formed, keeping its concentration low; this allows oxidation of more Au to Au+3, and the gold is dissolved. The gold may also react directly with the free chlorine in aqua regia, since chlorine is a powerful oxidizing agent.

Hence, being a organic molecule, DNA (and any other acidic organic molecule) may degraded while reacting with a strong acid. More over as a general concept, you may please remember that an acid is stable in its salt form, and for that it need to react with some base (Sometimes its conjugate base).