To determine which two solutions have similar solubilities at [tex]\(40^{\circ} C\)[/tex], let's compare the solubility values of the different salts and calculate the differences between their solubilities. We are given the solubilities at [tex]\(40^{\circ} C\)[/tex]:
- [tex]\(Na_2SO_4\)[/tex]: 45 g/100g [tex]\(H_2O\)[/tex]
- NaCl: 39 g/100g [tex]\(H_2O\)[/tex]
- [tex]\(Na_2HAsO_4\)[/tex]: 41 g/100g [tex]\(H_2O\)[/tex]
- [tex]\(Ba(NO_3)_2\)[/tex]: 5 g/100g [tex]\(H_2O\)[/tex]
- [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex]: 2 g/100g [tex]\(H_2O\)[/tex]
We need to determine the absolute differences in solubility between each pair:
1. Difference between [tex]\(Na_2SO_4\)[/tex] and NaCl:
[tex]\[
|45 - 39| = 6
\][/tex]
2. Difference between [tex]\(Na_2HAsO_4\)[/tex] and NaCl:
[tex]\[
|41 - 39| = 2
\][/tex]
3. Difference between [tex]\(Na_2HAsO_4\)[/tex] and [tex]\(Na_2SO_4\)[/tex]:
[tex]\[
|41 - 45| = 4
\][/tex]
4. Difference between [tex]\(Ba(NO_3)_2\)[/tex] and [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex]:
[tex]\[
|5 - 2| = 3
\][/tex]
Now, let's interpret these differences. The pair of salts that has the smallest difference in solubility values will be the most similar in terms of solubility at [tex]\(40^{\circ} C\)[/tex].
Among the calculated differences:
- [tex]\(Na_2SO_4\)[/tex] and NaCl: difference of 6
- [tex]\(Na_2HAsO_4\)[/tex] and NaCl: difference of 2
- [tex]\(Na_2HAsO_4\)[/tex] and [tex]\(Na_2SO_4\)[/tex]: difference of 4
- [tex]\(Ba(NO_3)_2\)[/tex] and [tex]\(Ce_2(SO_4)_3 \cdot 9H_2O\)[/tex]: difference of 3
The smallest difference is [tex]\(2\)[/tex], which indicates that the solutions with the most similar solubilities at [tex]\(40^{\circ} C\)[/tex] are [tex]\(Na_2HAsO_4\)[/tex] and NaCl.
