One mole of an ideal gas at \(300 \mathrm{~K}\) is expanded isothermally from an initial volume of \(1 \mathrm{~L}\) to \(10 \mathrm{~L}\). The change in internal energy, \(\Delta U\), for the gas in this process is (a) \(163.7 \mathrm{cal}\) (b) zero (c) \(1381.1 \mathrm{cal}\) (d) 9 L-atm

The first law of thermodynamics is a version of the law of conservation of energy, tailored for thermodynamic systems. It states that the energy added to a system in the form of heat, minus the energy lost as work done by the system, equals the change in internal energy of the system. Mathematically, it is expressed as \(\Delta U = Q - W\).

Understanding this law is crucial when analyzing processes involving ideal gases, as it helps to determine what will happen to the internal energy when heat and work are exchanged. Isothermal processes are a great example, as no matter how much work is done or heat is transferred, the internal energy remains constant because the temperature does not change. This is tightly correlated with the concept that for ideal gases, the internal energy is a function of temperature alone.

In the context of an ideal gas, the internal energy is directly related to its temperature. This can be inferred from the kinetic molecular theory of gases, which posits that an ideal gas is composed of particles in constant, random motion. The energy of these particles, hence the internal energy of the gas, is dependent on their kinetic energy which is a function of temperature.

During an isothermal process, like the one given in the exercise, the temperature and thus the internal energy does not change. The formula for the internal energy of an ideal gas, \(U = \frac{3}{2}nRT\), where \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature, explains why the internal energy remains unchanged if the temperature remains constant. For the exercise's isothermal expansion, where temperature stays at 300 K, the internal energy change \(\Delta U\) is zero.

When a gas expands, it does work on its surroundings. In thermodynamics, work done by a gas is an energy transfer from the gas to the surroundings. This is because to expand, the gas particles must exert a force over the distance they move against an external pressure.

In an isothermal process for an ideal gas, Henry Charles's Law helps calculate the work done. The work done by an isothermal expansion is given by \(W = nRT \ln(\frac{V_2}{V_1})\), where \(V_1\) and \(V_2\) are the initial and final volumes, respectively. Even though work is being done by the gas in the exercise (expanding from 1 L to 10 L), this energy transfer doesn't alter the internal energy. It does, however, reflect as heat added to the system (\(Q\)) to maintain constant temperature, demonstrating the interplay of work and heat in an isothermal process.

It's essential to understand that while work done by the gas might not change its internal energy in an isothermal process, it is a critical component of the system's energy exchange, and therefore of our understanding of how gases behave under different conditions.