Chapter 13, Problem 1PS

You dissolve 2.56 g of succinic acid, C₂H₄(CO₂H)₂, in 500. mL of water. Assuming that the density of water is 1.00 g/cm³, calculate the molality, mole fraction, and weight percent of acid in the solution.

Interpretation Introduction

Interpretation: The mole fraction, molality, and weight percent of succinic acid has to be identified.

Concept introduction:

Mole fraction: Amount of that component divided by the total amount of all of the components of the mixture $\left({\text{ n}}_{\text{A}}{\text{ + n}}_{\text{B}}{\text{ + n}}_{\text{C }}\text{+ }\mathrm{...}\right)$

Molality: The amount of solute (mol) per kilogram of solvent.

$\text{Concentration (C, mol/kg) = molality of solute = }\frac{\text{Amount of solute (mol)}}{\text{Mass of solvent (Kg)}}$

Weight percent: The mass of one component divided by the total mass of the mixture, multiplied by $100\%$

$\text{weight \% A = }\frac{\text{Mass of A}}{\text{ Mass of A + Mass of B + Mass of C + }\mathrm{...}}\text{×100\%}$

Answer to Problem 1PS

The calculated molality of succinic acid is $\text{0}\text{.0434 m}$

The mole fraction of succinic acid is $\text{0}\text{.000781}$

The weight percent of succinic acid is $\text{0}\text{.509 \%}$

Explanation of Solution

• The Mole fraction:

Given:

$\begin{array}{l}\text{Mass of}\text{Succinic acid = 2}\text{.56 g}\\ \text{Volume of water = 500 ml}\end{array}$

$\text{Mass}\text{of}\text{water}\text{=}\left(\text{500 ml}\right)\left(\text{1}\text{.00g/ml}\right)\text{=}\text{500}\text{g}$

Number of moles can be calculated by the equation given below:

$\text{No}\text{.}\text{of}\text{moles}\text{=}\frac{\text{mass}}{\text{molar}\text{mass}}$

$\begin{array}{l}\text{no}\text{.of moles (succinic acid) = }\frac{\text{2}\text{.56 g}}{\text{ 118}\text{.09 g/mol}}\text{ = 0}\text{.0217 mol}\\ \text{no}\text{.of moles (Water) = }\frac{500g}{\text{ 18 g/mol}}\text{ = 27}\text{.78 mol}\end{array}$

Mole fraction can be determined using the general equation:

${\text{Mole fraction of A (χ}}_{\text{A}}\text{)= }\frac{{\text{n}}_{\text{A}}}{{\text{ n}}_{\text{A}}{\text{ + n}}_{\text{B}}{\text{ + n}}_{\text{C }}\text{+ }\mathrm{...}}$

$\begin{array}{l}{\text{n}}_{succinicacid}\text{= 0}\text{.0217 mol}\\ {\text{n}}_{\text{Water}}\text{= 27}\text{.78 mol}\\ \text{Mole fraction of succinic acid:}\\ {\text{ (χ}}_{\text{succinic acid}}\text{) = }\frac{{\text{n}}_{succinicacid}}{{\text{ n}}_{succinicacid}{\text{ + n}}_{\text{Water}}}\text{ = }\frac{\text{0}\text{.0217 mol}}{\text{0}\text{.0217 mol + 27}\text{.78 mol}}\text{ = 0}\text{.000781}\end{array}$

Convert the mass of succinic acid and water into unit of moles and substitute in the respective mole fraction equation as shown above. Hence, the mole fraction of succinic acid is $\text{0}\text{.000781}$

• The molality:

  $\text{Molality of solute = }\frac{\text{Amount of solute (mol)}}{\text{Mass of solvent (Kg)}}$

  $\begin{array}{l}{\text{n}}_{succinicacid}\text{ = 0}\text{.0217 mol}\\ \text{Amount of solvent = 500 g}\end{array}$

Molality is calculated as given below:

$\begin{array}{c}\frac{\text{Amount of solute (mol)}}{\text{Mass of solvent (Kg)}}\text{= }\frac{\text{0}\text{.0217 mol}}{\text{0}\text{.500 kg}}\\ \text{= 0}\text{.0434 m}\end{array}$

The molality of succinic acid is calculated as shown above. Hence, the calculated molality of succinic acid $\text{0}\text{.0434 m}$

• The Weight percent:

$\begin{array}{l}\text{Given:}\\ \text{Mass of}\text{Succinic acid = 2}\text{.56 g}\end{array}$

$\text{weight \% A = }\frac{\text{Mass of A}}{\text{ Mass of A + Mass of B + Mass of C + }\mathrm{...}}\text{×100\%}$

$\begin{array}{c}\text{weight \% succinic acid = }\frac{\text{Mass of succinic acid}}{\text{ Mass of succinic acid + Mass of water }}\text{×100\%}\\ \text{= }\frac{\text{2}\text{.56 g}}{\left(\text{2}\text{.56 g}\right)\text{+}\left(\text{500 g}\right)}\text{×100\%}\\ \text{=}\text{ = 0}\text{.509 \%}\end{array}$

The weight percent of the succinic acid is calculated as shown above. Hence, the weight percent obtained is $\text{0}\text{.509 \%}$

Conclusion

The mole fraction, molality, and weight percent of succinic acid were identified.