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The First Law of Thermodynamics is a fundamental concept in physics that relates to the energy changes within a system. It is often phrased as "energy cannot be created or destroyed; it can only be transformed from one form to another." This principle applies to ideal gas processes and many other thermodynamic scenarios. The law can be mathematically expressed as:

• \( \Delta U = Q - W \)

where:

• \( \Delta U \) is the change in internal energy of the system,
• \( Q \) is the heat added to the system,
• \( W \) is the work done by the system (positive if work is done by the system and negative if work is done on the system).

Understanding this equation helps us analyze energy transactions in various processes.
In the context of an isothermal process, as described in the exercise, the internal energy remains constant, simplifying calculations.

Internal energy refers to the total energy contained within a system, such as an ideal gas, due to both its kinetic and potential energies. For ideal gases, internal energy is directly related to temperature, which is crucial for understanding reactions and processes involving gases.
In the given exercise, the temperature is held constant during compression, indicating no change in internal energy. Thus, the \( \Delta U = 0 \text{ J} \).
This is because, in an ideal gas, if there is no change in temperature between states, the kinetic energy of gas molecules remains constant, resulting in unchanged internal energy. This simplifies analysis when applying the First Law of Thermodynamics.

Heat transfer is the process by which thermal energy reaches or exits a system, influencing its internal energy but often through different pathways than mechanical work. The first law helps in determining the heat exchanged with the environment by accounting for changes in internal energy and work done by or on the system.
Using the exercise, since the internal energy change \( \Delta U = 0 \text{ J} \), the heat transferred can be directly equated to the work done on the system, resulting in

• \( Q = 75 \text{ J} \)

This positive value of \( Q \) indicates that heat energy was absorbed by the gas. Understanding heat balance is essential in practical applications like engines and HVAC systems.

An isothermal process occurs when a system undergoes changes at a constant temperature, which is a common scenario in ideal gases. This type of process is significant because it features simple characteristics:

• No change in internal energy (\( \Delta U = 0 \)), since temperature directly determines the internal energy of ideal gases.
• Work done and heat transfer are equal in magnitude, because all the work done results in heat added to or eliminated from the system.

Isothermal processes are often reversible and can be represented graphically by a hyperbolic curve on a PV (Pressure-Volume) graph.
In the exercise, the gas being compressed to one-third its original volume and the calculation of work at constant temperature illustrate the application of this concept, leading to straightforward equations for energy calculations.