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 \(\Delta U\) for the gas for this process?

The first law of thermodynamics is a fundamental principle that acts as the conservation of energy law for thermodynamics. Simply put, it states that energy cannot be created or destroyed within an isolated system. Instead, energy can only change from one form to another. In terms of a gas undergoing an expansion or any process, the first law is articulated through the equation \( \Delta U = Q - W \), where \(\Delta U\) represents the change in internal energy, \(Q\) stands for the heat added to the system, and \(W\) is the work done by the system on its surroundings.

During an isothermal expansion, this equation tells us how the internal energy of the gas is affected by the heat added to it and the work it performs. It's through this lens we can analyze the behavior of an ideal gas under such conditions.

Internal energy is the total energy contained within a system due to the kinetic and potential energies of the molecules. In the context of an ideal gas, the internal energy is directly proportional to the temperature of the gas. This relationship hinges on the fact that the internal energy of an ideal gas is a sum of its molecules' kinetic energies, which is an outcome of their random and continuous motion. As the temperature of the gas increases, so does the speed of these molecules, thereby increasing the internal energy.

During isothermal processes, such as the one our exercise describes, the temperature of the ideal gas is kept constant, which implies that the internal energy does not change. The equation \( \Delta U = 0 \) denotes this state of no change in internal energy throughout the isothermal expansion.

When discussing thermodynamics and gases, 'work done' is a term used to describe the gas's exertion of force over a distance. In the case of an isothermal expansion against atmospheric pressure, as the gas expands, it pushes against the external pressure exerted by the atmosphere doing work. This work can be calculated using the formula \( W = P_{ext} \Delta V \), where \(P_{ext}\) is the external pressure and \(\Delta V\) is the change in volume.

Because we are analyzing an isothermal process, even though work is being done by the gas as it expands, this does not result in a change in internal energy (\(\Delta U = 0\)) due to the simultaneous heat exchange that supplies energy equal to the work done.

Heat exchange is the process of energy transfer due to a temperature difference between a system and its surroundings. In an isothermal expansion of an ideal gas, heat must be added to the system to maintain the constant temperature while the gas does work on the surroundings. This input of heat compensates for the energy expended as work, therefore upholding the temperature of the gas.

Without this heat exchange in the direction of heat entering the system, the energy lost as work would result in cooling the gas, which contradicts the isothermal condition of constant temperature. The nature of this heat exchange is critical because it ensures that the internal energy does not change, as described by the equation \( \Delta U = Q - W \) under the first law of thermodynamics.