A \(40-\) W heat source is applied to a gas sample for 25 s, during which time the gas expands and does 750 J of work on its surroundings. By how much does the internal energy of the gas change?

Understanding the first law of thermodynamics is essential when we talk about energy changes within a system, particularly when dealing with internal energy. The first law, also known as the law of energy conservation, establishes that energy cannot be created or destroyed in an isolated system. It can only change form or be transferred from one part of the system to another.

The mathematical expression for the first law of thermodynamics is \(\Delta U = Q - W\), where \(\Delta U\) is the change in internal energy of the system, \(Q\) is the heat added to the system, and \(W\) is the work done by the system on its surroundings. This equation tells us that when heat is added to a system, it can do two things: it can change the internal energy of the system, or it can result in work being done on the environment.

In our particular example, when the 40 W heat source was applied to the gas for 25 seconds, the amount of heat added was responsible for both changing the internal energy and doing work. The balance between these two effects determines the total change in the system's internal energy.

Heat transfer is a fundamental concept in thermodynamics that describes the movement of heat energy from one place to another. It can occur through various mechanisms, including conduction, convection, and radiation. When we consider a 40 W heat source applied to a gas sample, we're dealing with heat transfer from the heat source to the gas.

In our exercise, calculating the quantity of heat transferred, \(Q\), involves multiplying the power of the heat source, \(P\), which is energy per unit of time, by the time, \(t\), for which it is applied. This gives us \(Q = P \times t\), and in this context, \(Q\) stands for the total heat energy that has been added to the gas. Heat transfer is the starting point for understanding how the system's internal energy and its ability to do work are affected.

The concept of 'work done by gas' is particularly intriguing within the domain of thermodynamics. It typically refers to the work performed by a gas as it expands or contracts within its container. This expansion or contraction can push against external forces, such as a piston in an engine, thereby doing work on the environment.

In the case of the given exercise, the gas expands and does 750 J of work on its surroundings. This quantifiable work is represented by \(W\) in our first law of thermodynamics equation. It is important to note that when a gas does work on its surroundings, it releases energy, and this release of energy is considered a positive amount of work done by the system. Conversely, if work is done on the gas (compressing it, for instance), it would be considered as having a negative value for \(W\) because energy is being transferred to the gas.