Calculate the change in entropy when the pressure of \(70.9 \mathrm{~g}\) of methane gas is increased isothermally from \(7.00 \mathrm{kPa}\) to \(350.0 \mathrm{kPa}\). Assume ideal behavior.

When studying thermodynamics, an isothermal process is vital. It represents a change that occurs at a constant temperature, meaning there's no variation in temperature throughout the entire process. Isothermal changes are particularly relevant when examining the behavior of gases. For instance, imagine a piston containing a gas; if we slowly compress or expand the gas such that the temperature remains constant by allowing heat exchange with the environment, we're observing an isothermal compression or expansion. According to the First Law of Thermodynamics, the internal energy for an ideal gas in an isothermal process does not change, because any work done on or by the system is exactly balanced by heat transfer with the surroundings.

In the context of our exercise, when the pressure of the methane gas increases isothermally, although the pressure changes, the temperature stays the same. This characteristic has significant repercussions on properties such as entropy, a measure of randomness or disorder, and is used to determine the amount of energy in a physical system that is not available to do work.

The ideal gas law is a cornerstone for understanding how gases behave under various conditions. It unifies the relationship of pressure (P), volume (V), temperature (T), and the number of moles of gas (n) into one equation, represented as PV = nRT, where R is the universal gas constant. This equation assumes that the gas particles are in constant random motion and that their size is negligible compared to the container volume. Furthermore, it is presumed that there are no intermolecular forces acting upon these particles.

In practical use, such as in the given exercise, this law allows us to approach a complex problem by breaking it down into known quantities and relationships. Understanding the ideal gas law is crucial because it forms the basis for other thermodynamic equations, including those used for calculating entropy changes in an isothermal process. This intrinsic connection is why being adept at manipulating the ideal gas law is indispensable for students keen on mastering thermodynamics.

The concept of natural logarithm, commonly represented by ln, emerges frequently in mathematical equations related to growth and decay processes, and is especially prevalent in areas like physics and engineering. The natural logarithm is the power to which the constant e (approximately 2.71828, called Euler's number) must be raised to produce a certain number. It provides a means of transforming multiplicative relationships into additive ones, which is incredibly useful for simplifying complex equations.

In thermodynamics, the natural logarithm often appears in equations involving entropy changes, such as in our exercise where it is used to relate the initial and final pressures during an isothermal process. It is essential to handle the natural logarithm with care during calculations, particularly regarding signs, as getting this wrong can lead to significant errors in estimating entropy changes. The natural logarithm's ability to express ratios as differences makes it invaluable for calculating entropy, an inherently comparative measure.