The addition of a dilute hydrobromic acid would clearly distinguish solutions of

a. barium nitrate and sodium sulfate
b. lead nitrate and silver nitrate
c. mercury(I) nitrate and silver nitrate
d. silver nitrate and calcium sulfate
e. calcium nitrate and barium nitrate

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To determine which solutions would be clearly distinguished by the addition of dilute hydrobromic acid, we need to understand the reactions that may occur between the given compounds and hydrobromic acid.

Hydrobromic acid (HBr) is a strong acid that dissociates in water to produce H+ ions and Br- ions. The addition of hydrobromic acid to a solution can result in various reactions depending on the nature of the compounds present.

Let's analyze each given pair of compounds:

a. Barium nitrate (Ba(NO3)2) and sodium sulfate (Na2SO4)
When hydrobromic acid is added to a solution of barium nitrate, it will react with barium ions (Ba2+) to form insoluble barium sulfate (BaSO4) and hydrochloric acid (HCl):

Ba(NO3)2 + HBr → BaBr2 + 2HNO3
BaBr2 + H2SO4 → BaSO4 + 2HBr

On the other hand, hydrobromic acid will not react with sodium ions (Na+) or sulfate ions (SO42-) to produce any precipitate or significant change. Therefore, the addition of dilute hydrobromic acid would clearly distinguish solutions of barium nitrate and sodium sulfate as a white precipitate of barium sulfate would form only in the presence of barium ions.

b. Lead nitrate (Pb(NO3)2) and silver nitrate (AgNO3)
Hydrobromic acid will react with lead ions (Pb2+) to form insoluble lead bromide (PbBr2) and nitric acid (HNO3):

Pb(NO3)2 + 2HBr → PbBr2 + 2HNO3

Similarly, hydrobromic acid will react with silver ions (Ag+) to form insoluble silver bromide (AgBr) and hydrochloric acid (HCl):

AgNO3 + HBr → AgBr + HNO3

In this case, the addition of dilute hydrobromic acid would not clearly distinguish solutions of lead nitrate and silver nitrate since both compounds would produce a yellow precipitate of their respective bromides.

c. Mercury(I) nitrate (Hg2(NO3)2) and silver nitrate (AgNO3)
Hydrobromic acid will react with mercury(I) ions (Hg2^2+) to form insoluble mercury(I) bromide (Hg2Br2) and nitric acid (HNO3):

Hg2(NO3)2 + 2HBr → Hg2Br2 + 2HNO3

Similarly, as mentioned in the previous case, hydrobromic acid will react with silver ions (Ag+) to form insoluble silver bromide (AgBr) and hydrochloric acid (HCl):

AgNO3 + HBr → HBr2 + HNO3

Therefore, the addition of dilute hydrobromic acid would clearly distinguish solutions of mercury(I) nitrate and silver nitrate as a white precipitate of mercury(I) bromide would form only in the presence of mercury(I) ions.

d. Silver nitrate (AgNO3) and calcium sulfate (CaSO4)
Hydrobromic acid will not react with calcium ions (Ca2+) or sulfate ions (SO42-) to produce any precipitate or significant change. Thus, the addition of dilute hydrobromic acid would not clearly distinguish solutions of silver nitrate and calcium sulfate.

e. Calcium nitrate (Ca(NO3)2) and barium nitrate (Ba(NO3)2)
Similarly, hydrobromic acid will not react with calcium ions (Ca2+) or barium ions (Ba2+) to form any precipitate or significant change. Therefore, the addition of dilute hydrobromic acid would not clearly distinguish solutions of calcium nitrate and barium nitrate.

In conclusion, the solutions that would be clearly distinguished by the addition of dilute hydrobromic acid are:
- Solutions of barium nitrate and sodium sulfate
- Solutions of mercury(I) nitrate and silver nitrate