Given the six solutions, the solution that will separate aqueous barium ion and aqueous lead(II) ion is ammonium chloride.
In qualitative analysis, it is common to separate substances in solution by introducing a solution that preferentially precipitates one of the ions in solutions while leaving the other one in solution.
In this case, we know that all chlorides are soluble except those of lead, mercury and silver according to the solubility rules. Hence, if ammonium chloride is added to a solution of aqueous barium ion and aqueous lead(II) ion, lead II chloride is preferentially precipitated hence the ions are separated.
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