To determine the boiling point of a [tex]\(2.2 \, m\)[/tex] (molal) solution of calcium nitrate [tex]\( Ca\left(NO_3\right)_2 \)[/tex], we can follow these steps:
1. Identify the given information:
- The molality (m) of the solution is [tex]\(2.2 \, \text{mol/kg}\)[/tex].
- The ebullioscopic constant ([tex]\( K_b \)[/tex]) for water is [tex]\( 0.512 \, ^\circ\text{C/m} \)[/tex].
- The normal boiling point of water is [tex]\( 100 \, ^\circ\text{C} \)[/tex].
2. Understand the dissociation of the solute:
- Calcium nitrate [tex]\( Ca\left(NO_3\right)_2 \)[/tex] dissociates completely in water into ions:
[tex]\[
Ca\left(NO_3\right)_2 \rightarrow Ca^{2+} + 2 NO_3^-
\][/tex]
- This means each formula unit of [tex]\( Ca\left(NO_3\right)_2 \)[/tex] produces 3 ions in solution (1 [tex]\( Ca^{2+} \)[/tex] ion and 2 [tex]\( NO_3^- \)[/tex] ions).
3. Determine the van 't Hoff factor ( [tex]\( i \)[/tex] ):
- The van 't Hoff factor ([tex]\( i \)[/tex]) is the number of particles the solute dissociates into. For [tex]\( Ca\left(NO_3\right)_2 \)[/tex]:
[tex]\[
i = 3
\][/tex]
4. Calculate the boiling point elevation ([tex]\( \Delta T_b \)[/tex]):
- The formula for boiling point elevation is:
[tex]\[
\Delta T_b = i \cdot K_b \cdot m
\][/tex]
- Plugging in the values:
[tex]\[
\Delta T_b = 3 \cdot 0.512 \, ^\circ\text{C/m} \cdot 2.2 \, \text{mol/kg}
\][/tex]
- This results in a boiling point elevation of approximately [tex]\( 3.3792 \, ^\circ\text{C} \)[/tex].
5. Calculate the new boiling point of the solution:
- The new boiling point is the normal boiling point of the solvent plus the boiling point elevation:
[tex]\[
\text{Boiling point of solution} = 100 \, ^\circ\text{C} + 3.3792 \, ^\circ\text{C}
\][/tex]
- Therefore, the boiling point of the [tex]\( 2.2 \, m \)[/tex] solution of [tex]\( Ca\left(NO_3\right)_2 \)[/tex] is approximately [tex]\( 103.3792 \, ^\circ\text{C} \)[/tex].
Thus, the boiling point of a [tex]\( 2.2 \, m \)[/tex] [tex]\( Ca\left(NO_3\right)_2 \)[/tex] solution is [tex]\( 103.3792 \, ^\circ C \)[/tex].
