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Write an equation to show how the $HSO_4^ - $ ion can act as an Arrhenius acid and write an equation to show how $HSO_4^ - $ ion can act as a bronsted-lowry base?

Answer
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Hint: A substance which dissociates in water and forms ${H^ + }$ ion is called Arrhenius acid. And a substance which is able to accept protons from another molecule is called bronsted-lowry base.

Complete step by step answer:
As through the definition we can understand the difference between Arrhenius acid and bronsted-lowry base. A substance which dissociates in water and forms ${H^ + }$ ion is called Arrhenius acid. And a chemical species which is able to accept protons from another molecule is called bronsted-lowry base.
In order to show the equation of $HSO_4^ - $ ion as an Arrhenius acid, when $HSO_4^ - $ is dissociate in water is gives ${H^ + }$ ions and $SO_4^{2 - }$ ion. This formation of ${H^ + }$ ion makes it an Arrhenius acid. The equation can be written as:
$HSO_4^ - \rightleftharpoons {H^ + } + SO_4^{2 - }$
Now, In order to show the equation of $HSO_4^ - $ ion as a bronsted-lowry base, when $HSO_4^ - $ is treated with ${H^ + }$ ions then it will accept that proton and form a new molecule that is ${H_2}S{O_4}$ . This accepting of proton makes it a bronsted-lowry acid. The equation can be written as:
$HSO_4^ - + {H^ + } \to {H_2}S{O_4}$
Hence, the above given two equations can show the ion as an Arrhenius acid and a bronsted-lowry base.

Additional information:
The equilibrium sign $ \rightleftharpoons $ used in Arrhenius acid equation because the reaction can occur in both forward and backward directions. The acid can lose a proton to become its conjugate base and the base can accept a proton to become its conjugate acid.

Note:
The fundamental concept of bronsted-lowry base theory states that when an acid and a base are treated with each other, the acid will form its conjugate base, and the base will form its conjugate acid by exchange of a proton. This theory is a generalization of the Arrhenius theory.