Air at 1.00 atm is inside a cylinder \(20.0 \mathrm{~cm}\) in radius and \(20.0 \mathrm{~cm}\) in length that sits on a table. The top of the cylinder is sealed with a movable piston. A \(20.0-\mathrm{kg}\) block is dropped onto the piston. From what height above the piston must the block be dropped to compress the piston by \(1.00 \mathrm{~mm} ? 2.00 \mathrm{~mm} ? 1.00 \mathrm{~cm} ?\)

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It's most commonly associated with the principles that govern the energy conversions within a system, particularly focused on temperature, energy, and work interactions. When we think of thermodynamics in the context of gases, we often refer to the four laws of thermodynamics that lay the foundation for thermodynamic processes.

Boyle's Law, which features in our exercise, is closely related to the first law of thermodynamics. The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. Boyle's Law illustrates this principle through the interplay of pressure and volume in a gas at a constant temperature. As one changes, the other adjusts in an inversely proportional manner, underlining the conversion of potential energy into work without a change in internal energy (since temperature is held constant).

The pressure-volume relationship, a central aspect of the ideal gas law, is beautifully demonstrated by Boyle’s Law. Boyle’s Law states that for a given mass of an ideal gas at a constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, this is expressed as \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume of the gas, and \( P_2 \) and \( V_2 \) are the final pressure and volume after a change has occurred.

In the exercise given, students witness how increasing the pressure exerted by a dropped weight onto the piston reduces the volume the gas occupies, in a quantifiable way using Boyle’s Law. Understanding this relationship is crucial for further studies in physics and engineering, where it applies to everything from breathing to the operation of internal combustion engines.

The concept of work done by gas is a fundamental aspect of the study of thermodynamics. Work is a measure of energy transfer and occurs when a force is applied to an object over a distance. In the context of a gas in a piston, as with our example exercise, work is done by the gas when it expands or is compressed.

In the process of compression, when a force is applied to the piston, the gas inside does work against this external force to change its volume. This is calculated using the formula \( W = P\triangle V \) where W is the work, P is the pressure, and \( \triangle V \) is the change in volume. Importantly, this illustrates how energy is transferred and underscores the broader principle that, in thermodynamics, work, and heat are the two means by which energy can be transferred into or out of a system.

In the worked example, students calculate the work done during each compression by using an integrated form of the work equation that accounts for changing pressure. Understanding the work done by gas helps students appreciate the interactivity of energy within systems, a concept having a variety of practical applications from HVAC systems to automotive engines.