Question

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 \(0.378 \mathrm{~m}\) solution is prepared by dissolving \(38.4 \mathrm{~g}\) sodium diatrizoate (NaDTZ) in \(1.60 \times 10^{2} \mathrm{~mL}\) water at \(31.2^{\circ} \mathrm{C}\) (the density of water at \(31.2^{\circ} \mathrm{C}\) is \(\left.0.995 \mathrm{~g} / \mathrm{mL}\right)\). What is the molar mass of sodium diatrizoate? What is the vapor pressure of this solution if the vapor pressure of pure water at \(31.2^{\circ} \mathrm{C}\) is \(34.1\) torr?

Short answer

The molar mass of sodium diatrizoate (NaDTZ) is 638 g/mol, and the vapor pressure of the 0.378 m NaDTZ solution at 31.2°C is 33.9 torr.

Step-by-step solution

Calculate number of moles of sodium diatrizoate

Using the given molality and mass of sodium diatrizoate, we can find the number of moles of NaDTZ: \(Malality = \frac{Number ~of~ moles ~of~ NaDTZ}{Mass ~of~ solvent ~in~ kg}\) Rearranging the equation: \(Number ~of~ moles ~of~ NaDTZ = Molality \times Mass ~of~ solvent ~in~ kg\) Given the volume of water (160 mL) and the density (0.995 g/mL) at 31.2 °C, we can find the mass of solvent (water) in grams and then convert it to kilograms: \(Mass ~of~ water = Volume ~of~ water \times Density ~of~ water\) \(Mass ~of~ water = 160 mL \times 0.995 \frac{g}{mL} = 159.2 g\) Converting grams to kilograms: \(Mass ~of~ water = \frac{159.2 g}{1000} = 0.1592 kg\) Now, let's find the number of moles of NaDTZ in the solution using the given molality (0.378 m): \(Number ~of~ moles ~of~ NaDTZ = 0.378 m \times 0.1592 kg = 0.0602 ~moles\)

Calculate molar mass of sodium diatrizoate

Now, we can find the molar mass of NaDTZ using the mass and number of moles: \(Molar ~mass ~of ~NaDTZ = \frac{Mass ~of ~NaDTZ}{Number ~of ~moles ~of ~NaDTZ}\) Given the mass of NaDTZ (38.4 g), we can calculate the molar mass: \(Molar ~mass ~of ~NaDTZ = \frac{38.4 g}{0.0602 ~moles} = 638 \frac{g}{mol}\) So, the molar mass of sodium diatrizoate is 638 g/mol.

Calculate mole fraction of water

To find the vapor pressure of the solution, we first need to determine the mole fraction of water in the solution. The mole fraction of a component is the ratio of the number of moles of that component to the total number of moles in the solution. We know the number of moles of NaDTZ (0.0602 moles) and the mass of water (159.2 g). We can find the number of moles of water by dividing the mass by the molar mass of water (18.015 g/mol): \(Number ~of ~moles ~of ~water = \frac{Mass ~of ~water}{Molar ~mass ~of ~water} = \frac{159.2 g}{18.015 \frac{g}{mol}} = 8.842 ~moles\) Now, we can find the mole fraction of water (X_water) in the solution: \(X_{water} = \frac{Number ~of ~moles ~of ~water}{Total ~number ~of ~moles ~in ~solution} = \frac{8.842 ~moles}{8.842 ~moles + 0.0602 ~moles} = 0.993\)

Calculate the vapor pressure of the solution

Now that we have the mole fraction of water, we can apply Raoult's law to find the vapor pressure of the solution: \(Vapor ~pressure ~of ~solution = X_{water} \times Vapor ~pressure ~of ~pure ~water\) Given the vapor pressure of pure water at 31.2 °C (34.1 torr), we can calculate the vapor pressure of the solution: \(Vapor ~pressure ~of ~solution = 0.993 \times 34.1 ~torr = 33.9 ~torr\) So, the vapor pressure of the 0.378 m sodium diatrizoate solution at 31.2 °C is 33.9 torr.

Key concepts

Sodium Diatrizoate

Sodium Diatrizoate, often abbreviated as NaDTZ, is a chemical compound used primarily as a contrast agent in medical imaging. Due to its solubility in water, it plays an essential role in enhancing the visibility of bodily structures during X-ray examinations. Its ability to absorb X-rays better than most body tissues makes it particularly useful for gastrointestinal tract studies. While it is nonvolatile in nature, meaning it does not easily evaporate, its solubility ensures it can be administered safely in appropriate doses. When preparing solutions containing Sodium Diatrizoate, understanding its chemical and physical properties helps ensure accurate imaging results.

Vapor Pressure

Vapor pressure is a fundamental concept in solution chemistry that refers to the pressure exerted by a vapor in equilibrium with its liquid or solid form. It helps determine how a solution will behave under certain conditions. In the case of sodium diatrizoate solutions, the vapor pressure can be slightly lower than that of pure solvent water due to the presence of solute particles.
Applying Raoult's Law, which states that the vapor pressure of a solvent in a solution is proportional to its mole fraction, allows us to predict changes in vapor pressure. Therefore, adding a nonvolatile solute like sodium diatrizoate to water decreases the overall vapor pressure, because fewer solvent molecules can evaporate.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture, calculated by dividing the number of moles of one component by the total number of moles in the solution. In our solution, determining the mole fraction of water is pivotal to calculating the vapor pressure of the solution.
For instance, if we know the number of moles of both sodium diatrizoate and water, it becomes straightforward to compute the water's mole fraction. This value helps recognize how the presence of solute changes the properties of the solution, such as vapor pressure. The mole fraction offers a dimensionless number that facilitates calculations involving mixtures and solutions.

Molality

Molality is another concentration term used when discussing solution chemistry, defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on volume, molality is based on mass, making it more stable under varying temperature and pressure.
It offers an advantage in precise scientific measurements, especially when dealing with temperature-dependent properties like vapor pressure. In the given problem, knowing the molality allows for the accurate calculation of the number of moles in the sodium diatrizoate solution. Understanding molality is crucial for applications that require careful control of solute concentration, such as medical and pharmaceutical solutions.