Chapter 11, Problem 117AE

Patients undergoing an upper gastrointestinal tract laboratory test are typically given an X-ray contrast agent that aids with the radiologic imaging of the anatomy. One such contrast agent is sodium diatrizoate, a nonvolatile water-soluble compound. A 0.378-m solution is prepared by dissolving 38.4 g sodium diatrizoate (NaDTZ) in.l.60 × 10² mL water at 3 1.2°C (the density of water at 31.2°C is 0.995 g/cm³). What is the molar mass of sodium diatrizoate? What is the vapor pressure of this solution if the vapor pressure of pure water at 31.2°C is 34.1 torr?

Interpretation Introduction

Interpretation: The molar mass of sodium diatrizoate and vapor pressure of given sodium diatrizoate solution in given condition should be calculated.

Concept introduction:

• Raoult's law:

  The mole fraction of a solute is related to the vapor pressure of the solution thus,

                                 $\begin{array}{l}{\text{P}}_{\text{solution}}{\text{=P°}}_{\text{solvent}}{\text{X}}_{\text{solvent}}\mathrm{......}\text{(1)}\\ \text{P}\text{is}\text{vapor pressure}\text{of the solution}\\ {\text{P°}}_{\text{solvent}}\text{pressure}\text{of the solvent}\\ {\text{X}}_{\text{solvent}}\text{ mole fraction of}\text{solvent}\end{array}$

Answer to Problem 117AE

Answer

• The molar mass of Sodium diatrizoate is $\text{639}\text{g/mol}$.
• The vapor pressure of Sodium diatrizoate solution is $\text{33}\text{.7}\text{torr}$.

Explanation of Solution

Record the given data,

                         Mass of Sodium diatrizoate = 38.4 g

                         Density of water at $\text{3 1}\text{.2°C}$ = $\text{0}{\text{.995 g/cm}}^{\text{3}}$

                         Pressure $\text{= 34}\text{.1 torr}$

                         Mole of Sodium diatrizoate solution = $\text{0}\text{.378}\text{m}$

                         Volume of water $\text{= 1}{\text{.60 X 10}}^{\text{2}}\text{ mL}$

To calculate mass of water.

                         $\begin{array}{l}\text{=}\text{volume}\text{(mL)×}\frac{\text{0}\text{.995}}{\text{mL}}\\ \text{=}\text{160}\text{mL×}\frac{\text{0}\text{.995}}{\text{mL}}\text{=159}\text{g}\\ \text{=}\text{0}\text{.159}\text{kg}\end{array}$

• The given values are plugging in to above equation to give the mass of water at $\text{31}\text{.2°C}$.
• the mass of water at $\text{31}\text{.2°C}$ is $\text{0}\text{.159}\text{kg}$.

To calculate the molar mass of Sodium diatrizoate solution.

             $\begin{array}{l}\text{Mole}\text{of}\text{Sodium diatrizoate}\text{=}\text{0}\text{.159}\text{Kg×}\frac{\text{0}\text{.378}\text{mol}}{\text{Kg}}\\ \text{=}\text{0}\text{.0601}\text{mol}\\ \text{Mass}\text{of}\text{Sodium diatrizoate}\text{=}\frac{\text{38}\text{.4}}{\text{0}\text{.0601}\text{mol}}\\ \text{=}\text{639}\text{g/mol}\end{array}$

• The calculated mass of water is multiplied by mole of Sodium diatrizoate solution to give mass of Sodium diatrizoate.
• The molar mass of Sodium diatrizoate is $\text{639}\text{g/mol}$.

To calculate the mole of water.

                        $\begin{array}{l}\text{Mole}\text{of}\text{water}\text{=Massof}\text{water}\frac{\text{1}\text{mol}}{\text{18}\text{.02}\text{g}}\\ \text{=}\text{159g×}\frac{\text{1}\text{mol}}{\text{18}\text{.02}\text{g}}\\ \text{=}\text{8}\text{.82}\text{mol}\end{array}$

• The calculated mass of water is plugging in above equation to give mole of water.
• The mole of water is $\text{8}\text{.82}\text{mol}$.

To calculate vapor pressure of solution.

                                    $\begin{array}{l}{\text{χ}}_{\text{water}}\text{=}\frac{\text{8}\text{.82}\text{mol}}{\text{0}\text{.120}\text{mol+}\text{8}\text{.82}\text{mol}}\\ \text{=}\text{0}\text{.987}\\ {\text{P}}_{\text{solution}}\text{=}\text{0}\text{.987}\text{×}\text{34}\text{.1}\text{torr}\\ \text{=}\text{33}\text{.7}\text{torr}\end{array}$

• The Sodium diatrizoate dissolve in water and dissociate to give ${\text{Sodium}}^{+}$ and ${\text{diatrizoate}}^{\text{-}}$ ions so the total ions are two.
• The mole fraction of Sodium diatrizoate is calculate plugging in values equation (1) to give mole fraction of Sodium diatrizoate is $\text{0}\text{.987}$.
• The calculated mole fraction is multiplied by given pressure to give vapor pressure of Sodium diatrizoate solution.
• The vapor pressure of Sodium diatrizoate solution is $\text{33}\text{.7}\text{torr}$.

Conclusion

• The molar mass of Sodium diatrizoate and vapor pressure of given sodium diatrizoate solution was calculated by using mass of water and molar mass of Sodium diatrizoate.
• The vapor pressure of Sodium diatrizoate solution is $\text{33}\text{.7}\text{torr}$