Question

\(p\) -Dichlorobenzene, \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\), can be one of the ingredients in mothballs. Its vapor pressure at \(20^{\circ} \mathrm{C}\) is \(0.40 \mathrm{~mm} \mathrm{Hg}\). (a) How many milligrams of \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\) will sublime into an evacuated 750-mL flask at \(20^{\circ} \mathrm{C}\) ? (b) If \(5.0 \mathrm{mg}\) of \(p\) -dichlorobenzene were put into an evacuated 750 -mL flask, how many milligrams would remain in the solid phase? (c) What is the final pressure in an evacuated 500 -mL flask at \(20^{\circ} \mathrm{C}\) that contains \(2.00 \mathrm{mg}\) of \(p\) -dichlorobenzene? Will there be any solid in the flask?

Short answer

How much will remain solid after a certain mass is put in an evacuated flask? What will be the final pressure if 2.00 mg of p-Dichlorobenzene is placed in another evacuated 500-mL flask at 20°C? Answer: (a) 3.16 mg of p-Dichlorobenzene will sublime into the 750-mL flask at 20°C. (b) If initially, there are 5.0 mg of p-Dichlorobenzene, 1.84 mg will remain in the solid phase after the sublimation takes place in the 750-mL flask. (c) The final pressure in the evacuated 500-mL flask containing 2.00 mg of p-Dichlorobenzene is approximately 0.27 mm Hg, and there will be no solid left in the flask at 20°C.

Step-by-step solution

(Step 1: Find the molar mass of p-Dichlorobenzene)

(Based on the molecular formula \(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{Cl}_{2}\), we can calculate its molar mass using the molar masses of carbon (12.01 g/mol), hydrogen (1.008 g/mol), and chlorine (35.45 g/mol). Molar mass = (6 × 12.01) + (4 × 1.008) + (2 × 35.45) = 72.06 + 4.032 + 70.90 = 146.992 g/mol)

(Step 2: Convert vapor pressure to atmospheres)

(The vapor pressure of p-Dichlorobenzene at 20°C is given in mm Hg. We need to convert this pressure to atm to use it in the Ideal Gas Law equation. 0.4 mm Hg × (1 atm / 760 mm Hg) ≈ 0.000526 atm) (a)

(Step 3a: Find the number of moles of p-Dichlorobenzene that will sublime into the flask)

(Using the Ideal Gas Law, we'll solve for n (number of moles) since we have the pressure, volume, and temperature. Volume is given in mL, so convert it to L by dividing by 1000: 750 mL / 1000 = 0.75 L. Also, we'll use 20°C as temperature in Kelvin: 20°C + 273.15 = 293.15 K. So the equation becomes: 0.000526 atm × 0.75 L = n × (0.0821 L atm / (mol K)) × 293.15 K Solving for n, n ≈ 0.0000215 mol)

(Step 4a: Convert moles to milligrams)

(Now, we'll convert the number of moles we found to milligrams using the molar mass we calculated before. 0.0000215 mol × 146.992 g/mol × (1000 mg / 1 g) ≈ 3.16 mg) (b)

(Step 5b: Find the amount remaining solid in the flask)

(Given that initially there are 5.0 mg of p-Dichlorobenzene in the flask, and we found that 3.16 mg is the amount that would sublime at 20°C, we can find the amount that remains solid by subtracting the two amounts. 5.0 mg - 3.16 mg ≈ 1.84 mg) (c)

(Step 6c: Find the number of moles for 2.00 mg p-Dichlorobenzene)

(We are given 2.00 mg of p-Dichlorobenzene, and we'll convert this amount to moles using the molar mass: 2.00 mg × (1 g / 1000 mg) × (1 mol / 146.992 g) ≈ 0.0000136 mol)

(Step 7c: Find the final pressure in the flask)

(Using the Ideal Gas Law again, we need to solve for the pressure with the given number of moles, volume (converted to L): 500 mL / 1000 = 0.5 L, and temperature in Kelvin. P × 0.5 L = 0.0000136 mol × (0.0821 L atm / (mol K)) × 293.15 K Solving for P, P ≈ 0.000356 atm)

(Step 8c: Convert pressure to mm Hg and determine the presence of solid)

(We'll convert the calculated pressure to mm Hg to compare it with the vapor pressure at 20°C and determine if there is any solid present in the flask. 0.000356 atm × (760 mm Hg / 1 atm) ≈ 0.27 mm Hg Since the calculated pressure (0.27 mm Hg) in the flask is less than the vapor pressure of p-Dichlorobenzene at 20°C (0.40 mm Hg), it means that all the p-Dichlorobenzene has turned into vapor and no solid is left in the flask.) The answers to the given exercise are: (a) 3.16 mg of p-Dichlorobenzene will sublime into the 750-mL flask at 20°C. (b) 1.84 mg of p-Dichlorobenzene will remain in the solid phase after 5.0 mg were put in the evacuated 750-mL flask. (c) The final pressure in the evacuated 500-mL flask containing 2.00 mg of p-Dichlorobenzene is approximately 0.27 mm Hg, and there will be no solid left in the flask at 20°C.

Key concepts

Understanding Sublimation

Sublimation is a fascinating process where a substance transitions from a solid directly to a gas without passing through the liquid phase. This might seem like an unusual behavior, but it's actually seen in everyday life. For example, dry ice, which is solid carbon dioxide, sublimates at room temperature, turning directly into carbon dioxide gas.

In the exercise, p-Dichlorobenzene is used to demonstrate sublimation when exposed to certain conditions inside an evacuated flask. When a solid substance like p-Dichlorobenzene sublimes, its molecules gain enough energy to overcome the forces holding them in a solid structure, thus facilitating the transition into the gas phase. The vapor pressure that is mentioned in the exercise is the pressure exerted by the vapor released during sublimation.

Practical Implications of Sublimation

Not only does sublimation have practical applications in creating items like freeze-dried foods or mothballs, but it also plays a critical role in materials science. Understanding sublimation allows scientists to predict and control the behavior of substances under different temperature and pressure conditions.

Applying the Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical 'ideal' gas. It is a good approximation for the behavior of many gases under various conditions. The law is usually stated as \(PV = nRT\), where \(P\) stands for pressure, \(V\) is the volume, \(n\) represents the number of moles of gas, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

In our exercise, the Ideal Gas Law helps to determine how much of p-Dichlorobenzene turns into gas from a solid state within the flask at a known temperature and volume. By rearranging the Ideal Gas Law to solve for \(n\) (the number of moles), we can predict the behavior of the gas and calculate theoretical yields in chemical experiments.

Real-World Relevance

Industries ranging from pharmaceuticals to aerospace use the Ideal Gas Law to understand and manipulate the behavior of gases in product development and manufacturing. Students and professionals alike rely on this law to make accurate predictions about gas behavior in a controlled environment.

The Significance of Molar Mass

Molar mass is a physical property defined as the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. It is expressed in units of grams per mole (\(g/mol\)).

In the context of our exercise, the molar mass of p-Dichlorobenzene is calculated to convert the mass of the substance into moles, a step necessary to apply the Ideal Gas Law. The accuracy of this calculation is critical because it allows us to relate a real-world mass of a sample to the theoretical world of moles, the base unit of measurement in chemistry.

Importance in Chemical Reactions

Molar mass is essential in stoichiometry, where it aids in predicting the outcomes of chemical reactions based on the quantities of reactants and products. In practical applications, such as formulating medications or manufacturing industry-standard materials, knowing the molar mass ensures that precise amounts of compounds are used and desired reactions are achieved.