Problem 87 - Chemistry: The Central Science

Question

(a) Which of the thermodynamic quantities \(T, E, q, w\), and \(S\) are state functions? (b) Which depend on the path taken from one state to another? (c) How many reversible paths are there between two states of a system? (d) For a reversible isothermal process, write an expression for \(\Delta E\) in terms of \(q\) and w and an expression for \(\Delta S\) in terms of \(q\) and \(T\).

Step-by-Step Solution

Answer

The thermodynamic state functions are T (temperature), E (internal energy), and S (entropy). The properties that depend on the path taken are q (heat) and w (work). There are an infinite number of reversible paths between two states of a system. For a reversible isothermal process, \(\Delta E\) is given by \(\Delta E = q + w\) with \(\Delta E = 0\), and \(\Delta S\) is given by \(\Delta S = \frac{q_{rev}}{T}\).

Step 1: (a) Identifying state functions

Thermodynamic state functions are those properties that depend only on the current state of a system and not on the path taken to reach that state. From the given thermodynamic quantities, T (temperature), E (internal energy), and S (entropy) are state functions.

Step 2: (b) Properties depending on the path taken

The properties that depend on the path taken from one state to another are q (heat) and w (work).

Step 3: (c) Number of reversible paths between two states

There are an infinite number of reversible paths between two states of a system. This is because reversible paths can be distinguished by their unique sequences of intermediate states, and there are an infinite number of possible sequences for any pair of initial and final states.

Step 4: (d) Expressions for \(\Delta E\) and \(\Delta S\) for a reversible isothermal process

For a reversible isothermal process, the change in internal energy, \(\Delta E\), is given by the first law of thermodynamics: \[\Delta E = q + w\] Since the process is isothermal, there is no change in temperature, which implies that \(\Delta E = 0\). Therefore, we can write the expression for \(\Delta E\) as: \[0 = q + w\] For the change in entropy, \(\Delta S\), in a reversible isothermal process, we use the relation: \[\Delta S = \frac{q_{rev}}{T}\] Where \(q_{rev}\) is the heat transfer for the reversible process at constant temperature T.

Key Concepts

Reversible Isothermal Process

Understanding a reversible isothermal process is paramount in the study of thermodynamics, particularly when examining how systems exchange energy. Isothermal means that the process occurs at a constant temperature, which implies that the system's total temperature T remains unchanged throughout the process.

A reversible process can be thought of as a theoretical idealization where changes occur so slowly that the system remains in equilibrium with its surroundings at all times. Imagine it like gently pressing down on a spring; if you do it slowly enough, the spring's compression can be precisely reversed. In practical terms, no real process is perfectly reversible, but some processes can approximate this behaviour over certain conditions.

In the context of a reversible isothermal process, the work done w by or on the system is intricately linked to the heat q exchanged since the internal energy E remains constant (for an ideal gas, it depends only on temperature and since the temperature is constant, \(\Delta E = 0\)). According to the first law of thermodynamics, which is the law of energy conservation, the energy added to the system as heat must leave the system as work or vice versa, leading to the simplified equation \(w = -q\).

This relationship ensures energy conservation and reflects the perfectly efficient energy conversion without losses. When applied to an ideal gas, the formula for work in a reversible isothermal process may be derived from integration, yielding \(w = -nRT\ln(\frac{V_f}{V_i})\), where n is the number of moles, R is the gas constant, and \(V_f\) and \(V_i\) are the final and initial volumes, respectively.

Thermodynamics First Law

The first law of thermodynamics is fundamental and states that energy cannot be created or destroyed in an isolated system. It can be expressed as \(\Delta E = q + w\), where \(\Delta E\) is the change in internal energy of the system, q is the heat added to the system, and w is the work done by the system.

Internal energy is a state function, meaning it depends only on the current state of the system and not on how the system reached that state. If the system has a higher internal energy after a process, that energy had to come from somewhere - either absorbed as heat q from the surroundings or as work w done on the system.

The significance of the first law lies in its ability to provide a balance sheet for energy transfer. In any thermodynamic process, it gives us a precise accounting of where the system's energy goes or comes from. It validates the conservation of energy and establishes a relationship between heat and work, which are path-dependent quantities. They rely on the specific processes undertaken from one state to another, unlike state functions like temperature, internal energy, or entropy.

Entropy Change

Entropy, symbolized by S, is a measure of the disorder or randomness in a system and is another state function within thermodynamics. The change in entropy \(\Delta S\) quantifies how much the disorder has increased or decreased during a process.

In a reversible isothermal process, the entropy change can be calculated using the equation \(\Delta S = \frac{q_{rev}}{T}\), where \(q_{rev}\) is the reversible heat exchange and T is the absolute temperature at which the process occurs. This calculation assumes that the entire process is reversible and that heat is added or removed uniformly at a constant temperature.

During a reversible process, entropy may increase or decrease, but the overall change will be zero if the process is returned to its initial state. This is because reversible processes are, by definition, capable of returning to their original state without any net change. However, in the real world, where processes are often irreversible, the entropy of the universe tends to increase, as stated by the second law of thermodynamics. This concept highlights the irreversible nature of natural processes and the tendency for systems to move from order to disorder.