The internal energy of an ideal gas depends only on its temperature. Do a first-law analysis of this process. A sample of an ideal gas is allowed to expand at constant temperature against atmospheric pressure. (a) Does the gas do work on its surroundings? (b) Is there heat exchange between the system and the surroundings? If so, in which direction? (c) What is \(\Delta E\) for the gas for this process?

The internal energy of an ideal gas is a concept intrinsic to understanding how gases interact with their environment. Simply put, this energy is related to the microscopic kinetic energy of the gas molecules moving randomly within the container. For an ideal gas, this energy is solely dependent on the temperature of the system.

This implies a pivotal principle: if there’s no change in temperature, there’s also no change in internal energy, denoted by \( \Delta E = 0 \). This is crucial because it helps predict how an ideal gas will behave when subjected to different thermodynamic processes. For instance, during an isothermal process (which occurs at a constant temperature), the internal energy remains constant irrespective of the volume changes or work done by the gas.

Isothermal expansion is a thermodynamic process where a gas changes volume while maintaining a constant temperature throughout. To maintain this constant temperature, the system must engage in heat exchange with its surroundings.

Imagine holding a balloon filled with gas over a flame. As it heats, it expands, but if you were to move the balloon around to keep the temperature steady, you would be demonstrating isothermal expansion. In reality, to achieve this situation, the system must absorb just the right amount of heat to offset the work done by the gas as it expands.

This process perfectly illustrates two critical concepts: the conservation of energy and the intricate balance between work and heat in thermodynamic systems.

When a gas expands, it pushes against the pressure exerted by its surroundings, which can be other gases or a physical barrier like the walls of a container. This exertion of force over a distance is what we call work done by the gas.

In the context of our ideal gas undergoing an isothermal expansion, work done \( (W) \) can be visualized as the product of pressure and the change in volume, represented by the equation \( W = P \Delta V \). Here, \( P \) stands for the external pressure and \( \Delta V \) is the change in volume. Since the gas does work on its surroundings, it naturally needs energy – this is where heat comes in.

Heat transfer in thermodynamics refers to the movement of thermal energy between a system and its surroundings. It is a cornerstone for understanding how energy flows and is transformed within a thermodynamic system.

During isothermal expansion, the work done by a gas is directly related to heat transfer \( (Q) \) from the surroundings into the system. This energy influx must be exactly balanced out by the energy expended in doing work for the internal energy to remain constant. To put it another way, the heat absorbed by the gas enables it to do work on its surroundings.

In practical situations, methods of heat transfer include conduction, convection, and radiation. For our scenario, the gas absorbs heat from the surroundings via these mechanisms to maintain its temperature and allow for expansion.