\(\Delta U=q+w\), is mathematical expression for (a) first law of thermodynamics (b) second law of thermodynamics (c) third law of thermodynamics (d) zeroth law of thermodynamics.

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 the heart of thermodynamics lie four fundamental laws that describe how these physical quantities are related and how they affect matter.

The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. This law is crucial for the study of energy transfer and efficiency in any physical or chemical process. Essentially, it tells us that the total amount of energy in the universe remains constant.

Breaking down complex topics within thermodynamics, like understanding a heat engine or predicting the behavior of gases under different conditions, always circles back to these core laws. They form the foundation that helps us understand not only basic scientific phenomena but also complex engineering and environmental processes.

The internal energy of a system is the total of all the energy contained within it. This includes both the kinetic energy of the particles moving and vibrating, as well as the potential energy that arises from the interactions between these particles.

When discussing the internal energy change, denoted as \(\Delta U\), in thermodynamics, we are essentially talking about the change in this energy content. It can occur due to heating, cooling, work being done by or on the system, or a combination of these processes.

\textbf{Example:} If you were to heat up a gas in a sealed container, its internal energy would increase as the particles move faster and the temperature rises; conversely, if the gas expands and does work on its surroundings, the internal energy decreases. Understanding these changes is essential in various applications, such as in designing engines or refrigerators where energy efficiency is key.

Heat exchange, a topic often misunderstood or oversimplified, involves the transfer of thermal energy between different systems or between a system and its environment. It occurs due to a temperature difference and can take place in various ways: through conduction, convection, or radiation.

In thermodynamics, the heat exchange is represented by the variable 'q'. It's crucial in calculating the changes in the state of a system when it interacts with its surroundings. For instance, in the given exercise \(\Delta U=q+w\), 'q' would represent the heat added to or removed from the system.

\textbf{Practical Application:} Whether it's in your coffee cup cooling down or in the complex workings of a power plant, heat exchange is everywhere. In practical applications, understanding heat exchange helps in creating more efficient systems for heating and cooling, like better insulation for buildings or optimal operating conditions for industrial processes.