What is the entropy change when 1 mole oxygen gas expands isothermally and reversibly from an initial volume of \(10 \mathrm{~L}\) to \(100 \mathrm{~L}\) at \(300 \mathrm{~K} ?\) (a) \(19.14 \mathrm{JK}^{-1}\) (b) \(109.12] \mathrm{K}^{-1}\) (c) \(29.12 \mathrm{JK}^{-1}\) (d) \(10 \mathrm{~K}^{-1}\)

In thermodynamics, an isothermal process is one that occurs at a constant temperature. This means that the system we are examining exchanges heat with its surroundings to maintain a uniform temperature throughout the process. When dealing with an ideal gas, like oxygen in this problem, an isothermal process is described by Boyle's Law, which states that the product of pressure and volume is a constant when temperature is held steady.

An isothermal expansion, such as the oxygen gas expanding from 10L to 100L, involves the gas doing work against the external pressure, and thus absorbing an identical amount of heat from the surroundings to preserve its temperature. The significance of an isothermal process in entropy calculations lies in the fact that, unlike adiabatic or polytropic processes, calculating the entropy change only involves the volumes and the gas constant, making the computation more straightforward.

Entropy, in simple terms, is a measure of the disorder or randomness of a system. It plays a pivotal role in determining the feasibility and direction of a thermodynamic process. For an ideal gas undergoing an isothermal process, the entropy change is directly related to the change in volume.

The key equation for entropy change in an ideal gas during an isothermal and reversible expansion or compression is \[\[\begin{align*}\Delta S = nR \ln\left(\frac{V_f}{V_i}\right)\end{align*}\]\], where:\begin{itemize}\item\(\Delta S\) is the entropy change\item\(n\) is the number of moles\item\(R\) is the universal gas constant\item\(V_i\) and \(V_f\) are the initial and final volumes, respectively.\end{FOOTER}.\end{itemize} To better understand entropy, imagine spreading a drop of ink into a glass of water. Initially, the ink is concentrated, but over time it disperses throughout the water, creating a more disordered state. In the context of gases, when a gas expands, its molecules become more dispersed, increasing the entropy of the system.

In thermodynamics, the gas constant, denoted as \(R\), is a fundamental parameter that appears in various equations relating the physical properties of gases. It features prominently in the ideal gas law, which relates the pressure, volume, temperature, and number of moles of an ideal gas. The value of the universal gas constant is approximately \(8.314\) J/(mol K).

Understanding the gas constant is crucial when performing calculations involving gases. It connects the macroscopic and microscopic worlds - relating quantities like temperature and volume at a scale we can observe directly, to the kinetic activity of molecules, which we cannot see. In entropy calculations, the gas constant allows us to quantify the entropy change associated with the expansion or compression of the gas, as it serves as a scaling factor between temperature and molecular randomness.