Question

Benzene, a known carcinogen, was once widely used as a solvent. \(\mathrm{A}\) sample of benzene vapor in a flask of constant volume exerts a pressure of \(325 \mathrm{~mm} \mathrm{Hg}\) at \(80^{\circ} \mathrm{C}\). The flask is slowly cooled. (a) Assuming no condensation, use the ideal gas law to calculate the pressure of the vapor at \(50^{\circ} \mathrm{C} ;\) at \(60^{\circ} \mathrm{C}\). (b) Compare your answers in (a) to the equilibrium vapor pressures of benzene: \(269 \mathrm{~mm} \mathrm{Hg}\) at \(50^{\circ} \mathrm{C}, 389 \mathrm{~mm} \mathrm{Hg}\) at \(60^{\circ} \mathrm{C}\). (c) On the basis of your answers to (a) and (b), predict the pressure exerted by the benzene at \(50^{\circ} \mathrm{C} ;\) at \(60^{\circ} \mathrm{C}\).

Short answer

Answer: The predicted actual pressures of benzene vapor are 269 mm Hg at 50°C and 304 mm Hg at 60°C.

Step-by-step solution

Initial condition and ideal gas law equation

First, let's note down the initial pressure and temperature of the benzene vapor. The initial pressure P1 is given as 325 mm Hg and the initial temperature T1 is 80°C. We need to convert the given temperatures from Celsius to Kelvin by adding 273.15 to the temperature, so T1 = 80 + 273.15 = 353.15 K. We'll be using the Ideal Gas Law equation to find the pressure at different given temperatures.

Calculate the pressure at given temperatures

Now, let's find the pressure (Pa) at 50°C (T2a = 323.15 K) and 60°C (T2b = 333.15 K) using the Ideal Gas Law equation, which states P1/T1 = P2/T2, where P1 and P2 are the initial and final pressures, and T1 and T2 are the initial and final temperatures, respectively. (a) For 50°C: P1/T1 = Pa/T2a Pa = P1*T2a/T1 Pa = 325 * 323.15 / 353.15 Pa ≈ 297 mm Hg (b) For 60°C: P1/T1 = Pb/T2b Pb = P1*T2b/T1 Pb = 325 * 333.15 / 353.15 Pb ≈ 304 mm Hg

Comparing the calculated pressures with equilibrium vapor pressures

Now we'll compare our calculated pressures from step 2 with the given equilibrium vapor pressures for the respective temperatures. (a) 50°C: Calculated pressure: 297 mm Hg Equilibrium vapor pressure: 269 mm Hg (b) 60°C: Calculated pressure: 304 mm Hg Equilibrium vapor pressure: 389 mm Hg

Predicting the actual pressure

(a) For 50°C, since the calculated pressure (297 mm Hg) is greater than the equilibrium vapor pressure (269 mm Hg), the vapor pressure will be equal to the equilibrium vapor pressure of 269 mm Hg, because condensation will occur until the pressure is reduced to the equilibrium vapor pressure value. (b) For 60°C, since the calculated pressure (304 mm Hg) is less than the equilibrium vapor pressure (389 mm Hg), no condensation will occur and the vapor pressure will be equal to the calculated value, which is 304 mm Hg.

Key concepts

Vapor Pressure

Vapor pressure is a critical concept in understanding how substances transition between their liquid and gaseous phases. Essentially, vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The pressure arises from the collisions of vapor molecules against the walls of the container.

For instance, when benzene, or any substance, is introduced into a flask and some of the liquid evaporates, the vapor particles will collide with the walls of the flask, exerting pressure. Over time, as more liquid turns into vapor, the pressure increases until it reaches a point where there is a dynamic balance between the molecules entering and leaving the vapor phase – this is what we call equilibrium vapor pressure.

Equilibrium Vapor Pressure

Equilibrium vapor pressure occurs when the rate of evaporation of a liquid matches the rate of condensation of its vapor. At this point, the vapor pressure remains constant since the system has reached a state of equilibrium. It is essential to note that this pressure is specific to each substance and varies with temperature.

In the case of benzene, the equilibrium vapor pressures at two different temperatures were provided, which we compared with the pressures calculated using the ideal gas law. Equilibrium vapor pressure is a vital concept because it allows us to predict at what point a substance will boil or condense under a certain temperature and pressure, as seen in our textbook exercise.

Benzene as a Carcinogen

Benzene is known to be a carcinogenic substance, meaning that it has the potential to cause cancer. It is a colorless liquid that occurs naturally and can also be produced by human activities. Benzene used to be widely utilized in the industry as a solvent and in the production of various chemicals.

As an educator, it is important to remind students that while learning about the chemical properties of benzene, safety should always be a priority. Use of benzene in school laboratories is often strictly regulated, and alternatives are commonly suggested for educational purposes. Understanding the health risks associated with chemicals like benzene is crucial for students developing a mindset of safety and responsibility in the scientific field.

Pressure-Temperature Relationship

The pressure-temperature relationship, often represented by Gay-Lussac's law in the case of gases held at a constant volume, states that the pressure of a gas is directly proportional to its temperature, provided the volume does not change. When temperature increases, the kinetic energy of the molecules increases, leading to more frequent and forceful collisions with the container's walls, thus increasing the pressure.

In the context of our exercise, we utilized this relationship while assuming no change in volume to calculate the benzene vapor pressure at different temperatures. For example, when the temperature dropped from 80°C to 50°C, we predicted a decrease in pressure through the Ideal Gas Law. This relationship is fundamental to understanding how gases behave under various thermal conditions and is widely applied in various scientific and engineering calculations.