To determine which two solutions have similar solubilities at [tex]$40^{\circ}C$[/tex], it is necessary to compare the solubility values of the given substances at this temperature. Here is the solubility data for the substances listed:
1. [tex]\(Na_2SO_4\)[/tex] - 42.7 g/100 g water
2. [tex]\(NaCl\)[/tex] - 36 g/100 g water
3. [tex]\(Na_2HAsO_4\)[/tex] - 4.22 g/100 g water
4. [tex]\(Ba(NO_3)_2\)[/tex] - 7.5 g/100 g water
5. [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex] - 3.2 g/100 g water
Now, let's match the solubilities:
- [tex]\(Na_2SO_4\)[/tex] and [tex]\(NaCl\)[/tex]:
- [tex]\(Na_2SO_4\)[/tex]: 42.7 g/100 g water
- [tex]\(NaCl\)[/tex]: 36 g/100 g water
- Difference: 6.7 g/100 g water
- [tex]\(Na_2HAsO_4\)[/tex] and [tex]\(NaCl\)[/tex]:
- [tex]\(Na_2HAsO_4\)[/tex]: 4.22 g/100 g water
- [tex]\(NaCl\)[/tex]: 36 g/100 g water
- Difference: 31.78 g/100 g water
- [tex]\(Na_2HAsO_4\)[/tex] and [tex]\(Na_2SO_4\)[/tex]:
- [tex]\(Na_2HAsO_4\)[/tex]: 4.22 g/100 g water
- [tex]\(Na_2SO_4\)[/tex]: 42.7 g/100 g water
- Difference: 38.48 g/100 g water
- [tex]\(Ba(NO_3)_2\)[/tex] and [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex]:
- [tex]\(Ba(NO_3)_2\)[/tex]: 7.5 g/100 g water
- [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex]: 3.2 g/100 g water
- Difference: 4.3 g/100 g water
The pairs with the most similar solubilities are:
- [tex]\(Ba(NO_3)_2\)[/tex] and [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex]: Difference is 4.3 g/100 g water.
Therefore, the two solutions with the most similar solubilities at [tex]\(40^{\circ}C\)[/tex] are [tex]\(Ba(NO_3)_2\)[/tex] and [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex].
