Question

How is a change in the internal energy defined in terms of the initial and final internal energies?

Short answer

The change in internal energy, \( \Delta U \), is defined as \( \Delta U = U2 - U1 \), where \( U1 \) is the initial internal energy and \( U2 \) is the final internal energy.

Step-by-step solution

Identify Initial and Final Internal Energies

The change in internal energy of a system can be defined by identifying the initial internal energy, typically denoted as U1, and the final internal energy, denoted as U2.

Calculate the Change in Internal Energy

The change in internal energy (\( \Delta U \)) is the difference between the final internal energy (U2) and the initial internal energy (U1). Therefore, the change in internal energy can be calculated using the formula: \[ \Delta U = U2 - U1 \].

Key concepts

Initial and Final Internal Energies

When we think about the energy content of a system in thermodynamics, we refer to it as 'internal energy'. Each state of the system has a specific amount of internal energy associated with it. The initial internal energy is the energy that the system possesses at the start of a process, and we typically denote it as \( U1 \). Similarly, the final internal energy, denoted as \( U2 \), is the energy the system holds at the end of the process.

For instance, imagine heating a pot of water. The initial internal energy is the total energy contained in the water and pot system at room temperature. As heat is applied, the energy of the system increases, which is reflected in the increased temperature of the water. When we measure the energy content of the now hot water and pot, that's the final internal energy. By comparing these two states, we can determine how much energy has been added to or released from the system during the heating process.

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. The core idea is that energy can be transformed from one form to another and transferred between systems, but it cannot be created or destroyed—this is known as the first law of thermodynamics or the law of energy conservation.

In the context of internal energy, thermodynamics enables us to understand how energy moves and changes forms. If our pot of water is heated on a stove, thermodynamics will describe how the heat from the stove is converted into increased internal energy of the water. A key aspect of thermodynamics is that these energy changes are predictable and quantifiable, which allows scientists and engineers to design systems and processes that efficiently use energy.

Energy Conservation

The principle of energy conservation is the cornerstone of thermodynamics and indeed all of physics. It states that the total energy of an isolated system remains constant—it is said to be conserved over time. Energy can neither be created nor destroyed but can change forms, such as when chemical energy is converted to kinetic energy or vice versa.

Let's go back to our example of heating water. The conservation of energy principle tells us that the energy input into the water (via heating) must go somewhere. It doesn't just disappear; it transforms into the internal energy of the water, resulting in a higher temperature. When we calculate the change in internal energy (\( \Delta U \)), we are observing this principle in action by measuring how much energy has moved into or out of the system during a specific process, aligning perfectly with the energy conservation law.