A \(2.50 .-\mathrm{mL}\) aerosol can at \(25^{\circ} \mathrm{C}\) and \(1.10\) atm was thrown into an incinerator. When the temperature in the can reached \(712^{\circ} \mathrm{C}\), it exploded. What was the pressure in the can just before it exploded, assuming it reached the maximum pressure possible at that temperature?

Understanding the behavior of gases under various conditions is essential for many scientific and engineering fields. The Gas Laws are a series of equations that describe the relationships between the pressure, volume, and temperature of a gas.

One of the most fundamental of these is the Ideal Gas Law, represented by the equation PV=nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. The law assumes that the gas molecules are point particles in constant, random motion and do not interact with each other.

In the exercise, we assume that the amount and volume of the gas do not change, which simplifies the Ideal Gas Law to a direct relationship between pressure and temperature, for a given amount of gas in a constant volume. This is described by the formula P1/T1 = P2/T2, which allows us to solve for unknown variables when others are held constant or change in a predictable way.

At its core, Thermodynamics is the study of energy, heat, work, and how they interplay within physical systems. It's a fundamental branch of physics that applies to all materials - gases, liquids, and solids. The scenario presented in the textbook problem is an excellent example of applied thermodynamics, where heat transfer results in a significant increase in temperature and pressure inside the aerosol can.

The First Law of Thermodynamics, also known as the Law of Energy Conservation, states that energy cannot be created or destroyed, only transformed from one form to another.

Heat Transfer and Work

In the incinerator, the aerosol can absorbs heat, increasing the energy of the gas molecules inside. As they move faster, they exert more force on the container's walls, leading to an increase in pressure. This is a direct visualization of thermodynamics in action, where heat absorbed is translated into work (expansion against the can walls), eventually leading to the can's failure.

The Pressure-Temperature relationship, often called Gay-Lussac's Law, is a specific case of the Ideal Gas Law and is critical in understanding how the pressure of a gas tends to increase as the temperature increases, when the volume is kept constant.

Mathematically, this relationship is expressed as P/T = k, where P is the pressure, T is the temperature, and k is a constant value when dealing with a fixed amount of gas at constant volume. As per the exercise, when the temperature inside the aerosol can increases drastically, the pressure does as well, due to the proportionality described by the equation P1/T1 = P2/T2. This equation succinctly shows that if the temperature rises, and since the volume of the can is fixed, the pressure must go up to maintain the balance in the equation.

Understanding this relationship is vital not only in academic exercises but also in real-life applications such as pressure-cooking, refrigeration, and understanding the behavior of gases in different weather conditions.