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

(a) The state functions are Temperature (T), Internal Energy (E), and Entropy (S). (b) Path-dependent functions are Heat (q) and Work (w). (c) There are infinite reversible paths between two states of a system. (d) For a reversible isothermal process: \(\Delta E = q + w\) with \(\Delta E = 0\), and \(\Delta S = \frac{q_{rev}}{T}\).

Step 1: (a) State Functions

A state function is a thermodynamic quantity that depends only on the current state of the system and not on the path taken to reach that state. In the given exercise, the state functions are Temperature (T), Internal Energy (E) and Entropy (S) since their values depend only on the particular state of the system and not on the process that lead to that state.

Step 2: (b) Path-dependent Functions

Path-dependent functions are those thermodynamic quantities that have values that are dependent on the specific process or path taken from one state to another. In the given exercise, Heat (q) and Work (w) are path-dependent functions because their values depend on the specific process that takes the system from one state to another.

Step 3: (c) Reversible Paths

In theory, there are infinite reversible paths between two states of a system. This is because reversible processes are idealized, and any path could be infinitely divided into smaller sub-processes that could be considered reversible. However, it is important to note that in practice, we often work with specific types of reversible processes, such as isothermal, isobaric, or adiabatic processes.

Step 4: (d) Expressions for Reversible Isothermal Processes

For a reversible isothermal process, the change in internal energy \(\Delta E\) is given by the first law of thermodynamics, which states that: \[ΔE = q + w\] Since the process is isothermal, the temperature remains constant, and for an ideal gas, the change in internal energy is zero, so we have: \[\Delta E = 0 = q + w\] The expression for the change in entropy \(\Delta S\) can be found by the definition of entropy change as the heat transfer divided by the temperature for a reversible process: \[\Delta S = \frac{q_{rev}}{T}\] Where \(q_{rev}\) represents the heat transfer for a reversible process and T is the constant temperature of the isothermal process.

Key Concepts

State Functions

When exploring the vast expanse of thermodynamics in chemistry, it's essential to understand state functions. These are like a snapshot of a system's health, similar to how your current location on a map is independent of the path you took to get there. Temperature (T), Internal Energy (E), and Entropy (S) fall under this category. They are like your altitude, energy levels, and randomness at a given point, unfazed by the journey you've made. So, whether you climbed straight up a mountain or took a winding path, your height above sea level—like a state function—remains the same at the top.

Path-dependent Functions

Unlike state functions, path-dependent functions are the journey stories of thermodynamics. These functions, including Heat (q) and Work (w), are like your travel diary, detailing how you crossed each mile, surmounted every obstacle, and expended energy during your hike. They change with every twist and turn of the path and are never the same for any two different treks. The same amount of work can result in different temperatures and internal energy levels depending on the process taken - fast and furious or slow and steady.

Reversible Processes

Picture a reversible process as the perfect dance, every step meticulously planned and executed so that you can return to your starting point without missing a beat. Theoretically, between any two points in a choreography, or in our case states of a system, there are countless ways to perform this dance, i.e., infinite reversible paths. This concept is an ideal scenario often used to simplify the complexity of thermodynamics. In the real world, we approximate these conditions with processes like isothermal (constant temperature) or adiabatic (no heat exchange) transitions, as these can be reversed with the least amount of non-recoverable steps or changes.

Isothermal Process

An isothermal process in thermodynamics is akin to walking in a constant climate. During this process, the temperature (T) remains unchanged, no matter how much heat (q) you add or work (w) you do. Imagine holding a balloon filled with gas; you compress and expand it but keep it at the same temperature throughout. For an ideal gas under these conditions, the process requires meticulous balancing of heat and work to ensure that the internal energy and therefore the temperature, doesn't change. This unique property of isothermal processes allows us to predict and control the behavior of a system in certain conditions.

First Law of Thermodynamics

Thermodynamics' very own version of the 'what goes around comes around' philosophy is encapsulated in the first law of thermodynamics. This fundamental law serves as the universe's balance sheet, stating that energy can neither be created nor destroyed - only transformed. When dealing with an isothermal process, this means that any heat (q) added to the system can subsequently be used to do an equivalent amount of work (w), with the internal energy (E) acting as the intermediary. So if you add energy to a system, that system will either store it or use it to do work on its surroundings.

Entropy Change

Think of entropy (S) as the measure of disorder in a teenager's bedroom—it can only increase or, at best, remain the same (we wish!). In thermodynamics, entropy change (ΔS) quantifies the dispersal of energy within a system. When a reversible isothermal process takes place, the entropy change is precisely the heat transferred (q) divided by the temperature (T). It reflects how much the energy spread out during the process, whether that's a neat transfer or a messy diffusion. If you're adding heat to a system without changing its temperature, you're essentially increasing the disorder or chaos within that system.

Internal Energy

Last but definitely not least is the concept of internal energy, often symbolized as (E). Imagine it as the total energy within your body—available for you to do things. In a closed system, it's the sum of all the potential and kinetic energy of its molecules. Internal energy is tightly intertwined with the first law of thermodynamics, where any change in it is a result of heat exchange (q) and work done (w). During an isothermal process in an ideal gas, even when there's a flurry of activity—heat being absorbed and work being done—the internal energy stays unchanged, highlighting the perfect balancing act that is the core principle of this process.