(a) Write an equation that expresses the first law of thermodynamics in terms of heat and work. (b) Under what conditions will the quantities q and w be negative numbers?

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. At its core, it examines the movement and conversion of energy. The universe can be seen as a vast energy transformation stage where thermodynamics plays a critical role in understanding how systems interact with energy and work.

Understanding the fundamental principles of thermodynamics is essential for various scientific and engineering fields, including mechanical engineering, chemical engineering, and environmental science. It helps us to design engines, predict chemical reactions, and understand the limits of heat engines and refrigerators, contributing significantly to technological advancements.

Internal energy, denoted as U, is a measure of the energy contained within a system. This energy includes the kinetic energy of particles moving and vibrating inside the matter, as well as the potential energy stored within the chemical bonds between atoms.

The change in internal energy, symbolized as \( \Delta U \), is a fundamental concept to grasp when dealing with thermodynamic processes. For instance, when gas in a container is heated, the particles move faster, indicating an increase in internal energy. It's essential to understand that we cannot measure the total internal energy of a system, but we can measure changes in internal energy resulting from heat transfer and work done.

Heat transfer in thermodynamics refers to the movement of thermal energy from one place to another or from one form to another. The process occurs in three primary ways: conduction, convection, and radiation.

Understanding heat transfer is crucial because it helps explain how heat energy flows within and between systems. For instance, in cooking, heat transfers from a stove to a pot via conduction, while in the Earth's atmosphere, ocean currents transport heat by convection. Heat transfer can also occur without the need for a medium, as in the case of the sun warming the Earth through radiation.

In thermodynamics, work (\(w\)) refers to the energy transferred when a force moves an object over a distance. However, in a thermodynamic context, it is more specifically about the energy transfer that happens when a system expands or contracts against external pressure.

As an example, when a gas is compressed in a piston, work is done on the gas, increasing its internal energy. The concept of work is crucial in understanding how energy is conserved in systems as it demonstrates one way that energy can be added to or taken away from a system.

The conservation of energy principle asserts that energy cannot be created or destroyed, only transformed from one form to another. This tenet is the foundation of the first law of thermodynamics, linking internal energy, heat transfer, and work.

The first law can be seen as a restatement of the conservation of energy, tailored for thermodynamic systems. In practical applications, this principle guides engineers and scientists in predicting the outcome of energy transformations and ensuring that energy efficiency is maximized in systems, such as in electrical power plants or automotive engines.