Question

(a) How can we calculate $\mathrm{\Delta }S$ foran isothermal process? (b) Does $\mathrm{\Delta }S$ for a process depend on the path taken from the initial to the final state of the system? Explain.

Short answer

(a) For an isothermal process, we can calculate the change in entropy ($\mathrm{\Delta }S$) using the formula $\mathrm{\Delta }S=nR\ln \frac{V_f}{V_i}$, where n is the number of moles, R is the gas constant, $V_f$ is the final volume, and $V_i$ is the initial volume. This is only applicable to an ideal gas. (b) Since entropy (S) is a state function, $\mathrm{\Delta }S$ is independent of the path taken from the initial to the final state of the system. It solely depends on the system's state variables and not on the process by which the system changes.

Step-by-step solution

Part (a): Calculating $\mathrm{\Delta }S$ for an isothermal process

In an isothermal process, the temperature T remains constant throughout the process. To calculate the change in entropy ($\mathrm{\Delta }S$) for an isothermal process, we can use the following formula, $$\mathrm{\Delta }S=nR\ln \frac{V_f}{V_i}$$ where n is the number of moles of the gas involved in the process, R is the gas constant (8.314 J/(mol K)), $V_f$ is the final volume of the gas, and $V_i$ is its initial volume. Note that this formula applies only to an ideal gas undergoing an isothermal process.

Part (b): Dependence of $\mathrm{\Delta }S$ on path taken

Entropy (S) is a state function, which means its value depends only on the current state of the system (e.g., temperature, pressure, and volume) and not on how the system arrived at that state. As a consequence, the change in entropy, $\mathrm{\Delta }S$, should be independent of the path taken from the initial to the final state of the system. The change in entropy can be calculated for any path between two states, and the result will be the same for all paths, provided that the initial and final states are specified. This is because entropy is a state function, and the value of a state function depends only on the state variables of the system and not on the process by which the system changes. As a result, $\mathrm{\Delta }S$ for a process is independent of the path taken from the initial to the final state of the system.

Key concepts

Isothermal Process

An isothermal process is characterized by constant temperature throughout the entire operation. This type of process is vital in thermodynamics because it simplifies the study of gas behavior and entropy—a measure of disorder or randomness in a system. During an isothermal process involving an ideal gas, the heat transfer into or out of the system does work but does not change the temperature.

In practical scenarios where an ideal gas expands or compresses isothermally, we can apply the ideal gas law to understand how its pressure and volume change in relation to each other. Since the temperature remains the same, any increase in volume would cause a corresponding decrease in pressure, and vice versa, maintaining the product of pressure and volume constant. This behavior can be seen, for example, in the operation of a heat engine during its expansion phase.

One crucial aspect of an isothermal process is its reversibility. Because the system is always infinitesimally close to equilibrium, an isothermal expansion or compression can be reversed by an infinitesimal change in external conditions. This reversibility is key to calculating the change in entropy for such processes, where entropy change can be represented as $$\mathrm{\Delta }S=nR\ln \frac{V_f}{V_i}$$, highlighting how entropy increases when the system undergoes an expansion ($V_f>V_i$) and decreases when it is compressed ($V_f<V_i$).

Entropy as a State Function

Entropy, denoted as $S$, is a fundamental concept in thermodynamics. It is a state function, meaning its value is determined solely by the present state of the system, not by the path the system took to reach that state. This characteristic of entropy allows for the calculation of entropy change, $$\mathrm{\Delta }S$$, between two states independently of the process undergone by the system.

For instance, whether an ideal gas is compressed quickly or slowly, through a single step or multiple stages, the overall change in entropy from its initial to final state will be the same. This intrinsic property makes entropy a powerful tool for evaluating the thermodynamic efficiency and spontaneity of processes. One interesting ramification of entropy being a state function is the formulation of the Second Law of Thermodynamics, which states that for a closed system, the total entropy can never decrease over time for any spontaneous process, reflecting the natural tendency of systems to progress towards disorder.

Understanding that the change in entropy is path-independent is not just a theoretical subtlety; it's a practical aspect that simplifies calculations in both engineering and natural processes, fortifying the relevance of entropy as a key quantity in thermodynamics.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and the amount of gas in moles in a simple, yet remarkably accurate formula:$$PV=nRT$$where $P$ represents the pressure, $V$ is the volume, $n$ is the number of moles of the gas, $R$ is the ideal gas constant (8.314 J/(mol K)), and $T$ is the temperature in kelvins. Although the law is predicated on the assumption of an 'ideal' gas, which means the gas particles have no volume and there are no intermolecular forces between them, it provides an excellent approximation for the behavior of real gases under many conditions.

Moreover, the ideal gas law is crucial in understanding and predicting how gases will respond to changes in conditions, such as during isothermal or adiabatic processes. Since the law considers all the state variables that define the condition of a gas, it can be used in conjunction with the concept of entropy as a state function to analyze thermodynamic cycles and processes. The ideal gas law underscores the interconnectivity between pressure, volume, and temperature, guiding scientific and engineering applications as diverse as calculating the buoyancy of balloons to the thermodynamics of the Earth's atmosphere.