Question

To determine the entropy change for an irreversible process between states 1 and \(2,\) should the integral \(\int_{1}^{2} \delta Q / T\) be performed along the actual process path or an imaginary reversible path? Explain.

Short answer

Answer: When calculating the entropy change for an irreversible process, we should use an imaginary reversible path between the initial and final states. This is because entropy is a state function that does not depend on the path taken, and using a reversible path allows us to properly evaluate the integral without dealing with the complexities involved in the heat exchange of an irreversible process.

Step-by-step solution

Definition of entropy change and reversible paths

Entropy change, denoted by ΔS, can be expressed in the formula \(\Delta S = \int_{1}^{2} \frac{\delta Q_{rev}}{T}\), where \(\delta Q_{rev}\) represents the heat exchanged during a reversible process and T is the absolute temperature. Notice that the formula specifies a reversible process, which implies that we need to determine the correct path involved (reversible or irreversible) to evaluate the entropy change.

Entropy as a state function

Entropy is a state function because it only depends on the initial and final states of the system and not the path taken to reach the final state. Since the entropy change has the same value regardless of whether the process is reversible or irreversible, it's crucial to consider this when deciding which path to use for the integral calculation.

Consequence of using an irreversible process path

When evaluating the integral for an irreversible process, we would have to account for the various factors causing irreversibility (such as friction, viscous dissipation, or chemical reactions) in the heat exchange \(\delta Q\). In general, this would make the integral difficult or even impossible to evaluate.

Using a reversible process path for calculating the entropy change

Since the actual process, in this case, is irreversible, it's not straightforward to determine the heat exchange and integrate it over the process path. However, because the entropy change is the same for both reversible and irreversible processes between the same states, we can design an imaginary reversible process between states 1 and 2. This will allow us to calculate ΔS using a reversible path, which should be more mathematically tractable.

Conclusion: Choosing the correct path

In conclusion, we should perform the integral \(\int_{1}^{2} \frac{\delta Q}{T}\) along an imaginary reversible path, not the actual irreversible process path. By doing so, we ensure that we can properly evaluate the entropy change without dealing with the complexities and uncertainties involved in the heat exchange resulting from an irreversible process.

Key concepts

Reversible Process

A reversible process in thermodynamics is one that can be reversed without leaving any trace on the surroundings. In other words, both the system and the environment return to their original states at the end of the reversible cycle. This idealized concept is essential because it provides a benchmark for calculating certain thermodynamic properties.

Mathematical Representation

For a process to be deemed reversible, the change must occur infinitesimally slowly, allowing the system to remain in a state of equilibrium with its surroundings. Mathematically, during a reversible process, the formula for entropy change, \( \Delta S \), could be represented as \( \Delta S = \int_{1}^{2} \frac{\delta Q_{rev}}{T} \), where \( \delta Q_{rev} \) is the small amount of heat exchanged reversibly and \( T \) is the absolute temperature.

This is an idealized concept, as all real processes are irreversible to some extent, but understanding reversible processes allows us to calculate maximum efficiency for systems like engines.

State Function

Entropy, symbolized as \( S \), is a fundamental thermodynamic quantity that, unlike heat or work, is a state function. State functions are properties that depend only on the current state of a system, not on how that state was reached.

Characteristic of State Functions

These functions are path-independent; that means the value of a state function at any given state is fixed, no matter the process the system underwent to arrive at that state. For example, when considering the elevation of a mountain peak, it doesn't matter whether you hiked up or took a helicopter, the elevation (state function) remains the same. Similarly, the change in entropy \( \Delta S \) from state 1 to state 2 is constant, irrespective of whether the process was reversible or irreversible.

This property allows us to use the reversible path to evaluate entropy changes in real-world, irreversible processes because the end value will be identical for the same initial and final states.

Irreversible Process

An irreversible process is one where the system cannot return to its initial state without a change in the external environment. In everyday life, most processes are irreversible due to factors like friction, inelastic collision, or natural phenomena such as spontaneous heat flow from hot to cold.

Complexity in Calculating Entropy

For these processes, calculating entropy change is not straightforward. It involves non-equilibrium states and often requires knowledge about the intricacies of the process, which might be unknown or complex. Moreover, because the state function should remain the same despite the path taken, using an irreversible process to calculate entropy change would not yield a general expression due to the irreversibilities.

To overcome this difficulty, we imagine a reversible process that takes the system between the same initial and final states. In doing so, we can use the simplified reversible process to accurately calculate the entropy change for the actual irreversible process, thereby simplifying the complex integration that would otherwise be required.