Q145P - Chemistry: Molecular Nature Of Matter And Change

Question

A solution of $\mathbf{\text{5.0 g}}$ of benzoic acid $\mathbf{\left(}{\mathbf{\text{C}}}_{\mathbf{\text{6}}}{\mathbf{\text{H}}}_{\mathbf{\text{5}}}\mathbf{\text{COOH}}\mathbf{\right)}$ in $\mathbf{\text{100.0 g}}$ of carbon tetrachloride has a boiling point of $\mathbf{77}\mathbf{.}{\mathbf{5}}^o\mathbf{C}$.

(a) Calculate the molar mass of benzoic acid in the solution.

(b) Suggest a reason for the difference between the molar mass based on the formula and that found in part (a).

Step-by-Step Solution

Answer

(a) The molar mass of benzoic acid in the solution is $251.25\raisebox{1ex}{$\text{g}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$.

(b) Change in temperature is small, hence the value of calculated molar mass of benzoic acid is larger than the actual molar mass.

A concept:

Temperature is directly proportional to the molality and molality depends on the number of moles of solute.

Given the boiling point is $77.5°$, it went to $76.5°$ and mass of benzoic acid is $5\text{ g}$ and mass of carbon tetrachloride is $\text{100 g}$.

Molar mass of benzoic acid:

Applying the formula of boiling point elevation.

$$\begin{gathered} \Delta T_b=K_b\times m \\ (77.5-76.5)°\text{C}=5.03\raisebox{1ex}{${\displaystyle \text{°C}}$}\!\left/ \!\raisebox{-1ex}{$\text{m}$}\right.\times m \end{gathered}$$ 

$\begin{array}{rcl}m& =& \frac{1°\text{C}}{5.03{\displaystyle \raisebox{1ex}{$\text{°C}$}\!\left/ \!\raisebox{-1ex}{$\text{m}$}\right.}}\\ & =& 0.199\raisebox{1ex}{${\displaystyle \text{mol}}$}\!\left/ \!\raisebox{-1ex}{$\text{kg}$}\right.\end{array}$ 

Where, $m$ is molality.

Molality is defined as the moles of solute dissolved in per kg of solution. 

$\text{molality}=\frac{\text{moles\hspace{0.17em}of solute}}{\text{mass\hspace{0.17em}of\hspace{0.17em}solution }\left(\text{kg}\right)}$ 

Here solute is benzoic acid and the solution is carbon tetrachloride.

Hence,

 $$\begin{gathered} 0.199\raisebox{1ex}{${\displaystyle \text{mol}}$}\!\left/ \!\raisebox{-1ex}{$\text{kg}$}\right.=\frac{\text{moles\hspace{0.17em}of\hspace{0.17em}solute}}{\text{0.1 kg}} \\ \text{moles\hspace{0.17em}of solute}=\text{0.0199 mol} \end{gathered}$$

Molar mass of benzoic can be calculated by dividing the mass of solute by moles of solute. 

$\begin{array}{rcl}\text{molar\hspace{0.17em}mass}& =& \frac{\text{mass \hspace{0.17em}of\hspace{0.17em}solute}}{\text{moles\hspace{0.17em}of\hspace{0.17em}solute}}\\ & =& \frac{\text{5 g}}{\text{0.0199 mol}}\\ & =& 251.25\raisebox{1ex}{${\displaystyle \text{g}}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.\end{array}$ 

Thus, the calculated molar mass of benzoic acid is $251.25\raisebox{1ex}{${\displaystyle \text{g}}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$.

(b) Difference between calculated molar mass and actual molar mass:

Actual molar mass of benzoic acid is $\text{122.12 }\raisebox{1ex}{${\displaystyle \text{g}}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$. As defined, the molar mass is much greater than the actual molar mass. The reason of this difference is temperature or boiling point. Temperature is directly proportional to the molality and molality depends on the number of moles of solute.

So, in calculated molar mass, the number of moles is less than the number of moles in the actual molar and thus the large molality of actual molar mass.

Since the difference between the temperatures was less in the calculated molar mass, the molality value was small.

Hence, the calculated molar mass is greater than the actual molar mass.