Question

A stockroom supervisor measured the contents of a partially filled 25.0 -gallon acetone drum on a day when the temperature was \(18.0^{\circ} \mathrm{C}\) and atmospheric pressure was \(750 \mathrm{mmHg}\), and found that 15.4 gallons 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 gallons. What is the total pressure inside the drum after the accident? The vapor pressure of acetone at \(18.0^{\circ} \mathrm{C}\) is \(400 \mathrm{mmHg}\).

Short answer

The total pressure inside the drum after the accident is the sum of the final pressure of the air and the vapor pressure of acetone (400 mmHg). Calculate the final pressure of air using Boyle's law.

Step-by-step solution

Calculation of the Initial Volume and Pressure

First, let's calculate the initial volume of air in the drum. This is equal to the total volume of the drum (25.0 gallons) minus the volume of acetone in it (15.4 gallons). \Therefore, the initial volume of air (\(V_i\)) is \(25.0-15.4 = 9.6\) gallons. Next, the initial pressure inside the drum can be considered equivalent to the atmospheric pressure. Hence, the initial pressure (\(P_i\)) is 750 mmHg.

Calculate Final Volume

The drum was altered and its internal volume reduced to 20.4 gallons. The acetone volume has not changed (still 15.4 gallons), so the final volume of air (\(V_f\)) can be calculated as \(20.4-15.4 = 5.0\) gallons.

Apply Boyle's Law

We can use Boyle's law to calculate the final pressure of the air in the drum. Using the formula \(P_i*V_i = P_f*V_f\), solve for \(P_f\) (final pressure): so, \(P_f = (P_i*V_i) / V_f\) which translates to \(P_f = (750 mmHg * 9.6 gallons) / 5.0 gallons\). This will give the pressure in mmHg.

Calculate Total Pressure Inside Drum

The total pressure inside the drum is the sum of the pressure exerted by the air and the vapor pressure of acetone. The vapor pressure of acetone at \(18.0^{\circ} \mathrm{C}\) is 400 mmHg, and is independent of the volume change. So, just add this value (400 mmHg) to the final pressure of the air calculated in step 3 to get the total pressure in the drum.

Key concepts

Understanding Gas Laws

Gas laws are crucial for studying the behavior of gases under different physical conditions. They describe how various properties of gases—such as volume, pressure, and temperature—interrelate and change when one is altered.

Among these laws, Boyle's Law, which focuses on the pressure-volume relationship at a constant temperature, is a fundamental concept in chemistry. It states that for a given amount of gas at a constant temperature, the pressure of the gas is inversely proportional to its volume. This means that if you increase the volume of a gas, its pressure will decrease, and vice versa, as long as the temperature remains unchanged.

This principle is widely applied in various real-life situations, including the functioning of lungs during breathing, the syringe mechanism in medicine, and even in the behavior of weather balloons as they ascend or descend in the atmosphere.

The Role of Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form. It's an essential property of liquids, particularly in the context of solutions and closed systems. At a given temperature, every liquid has a characteristic vapor pressure that's independent of the surrounding pressure but dependent on the nature of the substance and temperature.

The significance of vapor pressure becomes apparent in the study of mixtures and when considering the evaporation or boiling of a liquid. High vapor pressure usually indicates a liquid that will easily evaporate, like acetone, while low vapor pressure implies the liquid is less prone to evaporation. Understanding vapor pressure is crucial for predicting and explaining several phenomena in both natural and industrial processes, such as weather patterns and the preservation of perishable products.

Pressure-Volume Relationship in Boyle's Law

The pressure-volume relationship, as described by Boyle's Law, is an inverse relationship that provides a mathematical way to predict how the pressure of a gas changes with volume. The law can be expressed mathematically as: \( P_i V_i = P_f V_f \), where \( P_i \) and \( V_i \) are the initial pressure and volume of the gas, and \( P_f \) and \( V_f \) are the final pressure and volume after a change has occurred, provided temperature is held constant.

In practical scenarios such as the exercise with the acetone drum, when the volume of the drum is reduced by an impact, Boyle's Law allows us to calculate the new pressure inside. The law can also apply to more complex situations involving gases in chemical reactions, human-made containers, or natural settings. A clear understanding of this relationship is vital for anyone studying the behavior of gases and can aid in resolving problems that involve changes in gas volume and pressure.