To determine the freezing point of a 0.51 molal solution of benzoic acid in chloroform, we need to use the concept of freezing point depression. The formula we use is:
[tex]\[
\Delta T_f = K_f \cdot m
\][/tex]
where:
- [tex]\(\Delta T_f\)[/tex] is the change in the freezing point,
- [tex]\(K_f\)[/tex] is the freezing point depression constant of the solvent,
- [tex]\(m\)[/tex] is the molality of the solution.
Given values are:
- Molality ([tex]\(m\)[/tex]) = 0.51 mol/kg
- Freezing point depression constant ([tex]\(K_f\)[/tex]) = 4.70 [tex]\(^{\circ}C \cdot kg / mol\)[/tex]
- Freezing point of pure chloroform = [tex]\(-63.5^{\circ}C\)[/tex]
First, we need to calculate the change in the freezing point [tex]\(\Delta T_f\)[/tex]:
[tex]\[
\Delta T_f = K_f \cdot m
\][/tex]
Substitute the given values into the formula:
[tex]\[
\Delta T_f = 4.70 \cdot 0.51
\][/tex]
The result of this calculation gives us the change in the freezing point:
[tex]\[
\Delta T_f = 2.4^{\circ}C
\][/tex]
Next, we determine the freezing point of the solution by subtracting the change in freezing point from the freezing point of the pure chloroform:
[tex]\[
\text{Freezing point of the solution} = \text{Freezing point of pure chloroform} - \Delta T_f
\][/tex]
Substitute the known values:
[tex]\[
\text{Freezing point of the solution} = -63.5^{\circ}C - 2.4^{\circ}C
\][/tex]
Performing the subtraction gives us:
[tex]\[
\text{Freezing point of the solution} = -65.9^{\circ}C
\][/tex]
Therefore, the freezing point of the 0.51 molal solution of benzoic acid in chloroform is [tex]\(-65.9^{\circ}C\)[/tex]. Rounded to the nearest tenth:
[tex]\[
\boxed{-65.9^{\circ}C}
\][/tex]
