Question

A stockroom supervisor measured the contents of a 25.0-gal drum partially filled with acetone on a day when the temperature was \(18.0^{\circ} \mathrm{C}\) and atmospheric pressure was \(750 \mathrm{mmHg}\), and found that 15.4 gal of the solvent remained. After tightly sealing the drum, an assistant dropped the drum while carrying it upstairs to the organic laboratory. The drum was dented, and its internal volume was decreased to 20.4 gal. What is the total pressure inside the drum after the accident? The vapor pressure of acetone at \(18.0^{\circ} \mathrm{C}\) is \(400 \mathrm{mmHg}\). (Hint: At the time the drum was sealed, the pressure inside the drum, which is equal to the sum of the pressures of air and acetone, was equal to the atmospheric pressure.)

Short answer

The total pressure inside the drum after the accident is 1072 mmHg.

Step-by-step solution

Determine the Initial Conditions

Start by identifying the initial conditions before the drum was sealed. The total pressure inside the drum was equal to the atmospheric pressure, which is given as \(750\,\mathrm{mmHg}\). This pressure consists of the sum of the air pressure and the vapor pressure of acetone.

Calculate Initial Air Pressure

The vapor pressure of acetone at \(18.0^{\circ} \mathrm{C}\) is \(400\,\mathrm{mmHg}\). Therefore, the initial pressure of the air inside the drum, before it was sealed, is:\[ P_{\text{air, initial}} = P_{\text{total, initial}} - P_{\text{acetone}} = 750\,\mathrm{mmHg} - 400\,\mathrm{mmHg} = 350\,\mathrm{mmHg} \]

Apply Perfect Gas Law for Air

After the volume of the drum is reduced, the volume of the air is reduced as well. Assuming temperature remains constant, use Boyle's Law: \[ P_{\text{air, initial}} \cdot V_{\text{air, initial}} = P_{\text{air, final}} \cdot V_{\text{air, final}} \] The initial air volume is the difference between the drum volume and the acetone volume: \[ V_{\text{air, initial}} = 25.0\,\mathrm{gal} - 15.4\,\mathrm{gal} = 9.6\,\mathrm{gal} \] The final air volume, after the dent, is:\[ V_{\text{air, final}} = 20.4\,\mathrm{gal} - 15.4\,\mathrm{gal} = 5.0\,\mathrm{gal} \]

Calculate Final Air Pressure

Rearrange Boyle's Law to solve for the final air pressure: \[ P_{\text{air, final}} = \frac{P_{\text{air, initial}} \cdot V_{\text{air, initial}}}{V_{\text{air, final}}} = \frac{350\,\mathrm{mmHg} \cdot 9.6\,\mathrm{gal}}{5.0\,\mathrm{gal}} \]Calculate it to find:\[ P_{\text{air, final}} = 672\,\mathrm{mmHg} \]

Determine Final Total Pressure

The total pressure inside the drum after the volume change is the sum of the final air pressure and the constant vapor pressure of the acetone:\[ P_{\text{total, final}} = P_{\text{air, final}} + P_{\text{acetone}} = 672\,\mathrm{mmHg} + 400\,\mathrm{mmHg} \]Thus, the total pressure is:\[ P_{\text{total, final}} = 1072\,\mathrm{mmHg} \]

Key concepts

Vapor Pressure

Vapor pressure is a crucial concept to understand when dealing with volatile liquids like acetone. But what exactly is vapor pressure? Imagine a liquid in a closed container. Over time, some molecules of the liquid escape into the space above the liquid and become vapor. As more and more molecules escape, they begin to accumulate in the space, exerting pressure. This pressure is known as vapor pressure. It reflects the tendency of the liquid's molecules to escape into the vapor phase.

Certain factors affect vapor pressure:

• Nature of the liquid: Different liquids have varying tendencies to evaporate, affecting their vapor pressures. Substances like acetone, which evaporate easily, have high vapor pressures.
• Temperature: As temperature increases, more molecules have enough energy to escape into the vapor phase, resulting in higher vapor pressure.

In the scenario with the drum, the vapor pressure of acetone was constant at 400 mmHg, even after the drum was dented. This constancy plays a crucial role in calculating the total pressure in the drum.

Boyle's Law

Boyle's Law is a fundamental principle in chemistry that describes how the pressure of a gas tends to increase as the volume of the container decreases, provided the temperature remains constant. It's understood by the equation:\[ P_1 \times V_1 = P_2 \times V_2 \]
Where:

• \( P_1 \) and \( P_2 \) are the initial and final pressures of the gas.
• \( V_1 \) and \( V_2 \) are the initial and final volumes the gas occupies.

This means that pressure and volume are inversely proportional to each other as long as temperature does not change. If the volume decreases, the pressure increases.

In the exercise, after the drum was dented, the internal volume decreased, causing an increase in air pressure according to Boyle's Law. Calculations showed this increased the air pressure inside from 350 mmHg to 672 mmHg.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air above us. It affects many everyday phenomena, including the behavior of gases inside a container. It is important in our problem because it was the reference pressure when the drum was originally recorded as being sealed.

At sea level, atmospheric pressure averages about 760 mmHg, but it can vary with weather conditions and altitude. In this exercise, the mentioned atmospheric pressure was slightly lower at 750 mmHg, aligning with specific initial conditions of the experiment. This atmospheric pressure represents the sum of pressures inside the drum before it was sealed.

Understanding atmospheric pressure helped calculate the change in pressure inside the drum after its volume was altered. It also played a role in ensuring that the total pressure initially inside the drum equaled the pressure of the surrounding air before the accident happened.