Problem 85 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Which of the thermodynamic quantities \(p, H, q, w,\) and \(G\) are state functions? (b) Consider a system going from state 1 to state 2 in a reversible and an irreversible way. Compare \(q_{\text {rev }}\) and \(q_{\text {irtev }}\) (c) Consider a system going from state 1 to state 2 in a reversible and an irreversible way. Compare \(w_{\text {rev }}\) and \(w_{\text {trev }}\). (d) For a reversible isothermal process, write an expression for \(\Delta H\) and an expression for \(\Delta G\) in terms of \(q, w\) and \(T, p\) and \(\Delta V\).

Step-by-Step Solution

Answer

The state functions among the given thermodynamic quantities are \(p\), \(H\), and \(G\). In comparing heat and work in reversible and irreversible processes, \(q_{rev}\) and \(q_{irrev}\) can have different values, while \(w_{rev}\) is maximum, and \(w_{irrev}\) is less. For a reversible isothermal process, the change in enthalpy is given by \(\Delta H = q_p\), and the change in Gibbs free energy is given by \(\Delta G = -w_{rev} = nRT \ln \frac{V_2}{V_1}\).

Step 1: Part (a): Identifying State Functions

State functions are properties that depend only on the current state of the system and not on the path taken to reach that state. From the given thermodynamic quantities, we can identify state functions as follows: 1. \(p\) - Pressure is a state function, as it depends only on the current state of the system. 2. \(H\) - Enthalpy is a state function, as it depends only on the internal energy, pressure, and volume of the system. 3. \(q\) - Heat is not a state function, as it depends on the path taken by the system during a change of state. 4. \(w\) - Work is not a state function, as it also depends on the path taken by the system during a change of state. 5. \(G\) - Gibbs free energy is a state function, as it depends only on enthalpy, temperature, and entropy.

Step 2: Part (b): Comparing Heats in Reversible and Irreversible Processes

In a reversible process, the system is in thermal and mechanical equilibrium with its surroundings at each moment. In an irreversible process, the system is not in complete equilibrium during the change of state. Since the heat transfer \(q\) depends on the path taken by the system during a change of state, \(q_{rev}\) and \(q_{irrev}\) can have different values.

Step 3: Part (c): Comparing Works in Reversible and Irreversible Processes

Similar to heat, work also depends on the path taken. In a reversible process, the work done (\(w_{rev}\)) is maximum, while in an irreversible process, the work done (\(w_{irrev}\)) is less. This is because there is more energy lost as heat in an irreversible process, which reduces the amount of work done.

Step 4: Part (d): Expressions for ΔH and ΔG in a Reversible Isothermal Process

For a reversible isothermal process, the temperature \(T\) remains constant. The change in enthalpy (\(\Delta H\)) can be expressed as: \[\Delta H = q_p\] where \(q_p\) is the heat transfer at constant pressure. The change in Gibbs free energy (\(\Delta G\)) can be expressed as: \[\Delta G = \Delta H - T \Delta S\] Since the process is isothermal, we can relate the pressure, volume, and temperature changes: \[w = -nRT \ln \frac{V_2}{V_1}\] where \(n\) is the number of moles, \(R\) is the gas constant, and \(V_1\) and \(V_2\) are the initial and final volumes, respectively. As \(\Delta G\) is equal to the maximum non-expansion work done by the system, we have: \[\Delta G = -w_{rev}\] Thus, in a reversible isothermal process, we can write the change in enthalpy as: \[\Delta H = q_p\] and the change in Gibbs free energy as: \[\Delta G = -w_{rev} = nRT \ln \frac{V_2}{V_1}\]

Key Concepts

State Functions

In thermodynamics, state functions are quantities that depend solely on the current state of a system, not on the path taken to reach that state. This means they are properties determined by the state variables such as pressure, volume, temperature, and composition.

Examples of state functions include:

• Pressure (\(p\)): It's a measure of force exerted per unit area and is a defining property of the state.
• Enthalpy (\(H\)): It combines the system's internal energy with the product of pressure and volume. It remains unaffected by the process path.
• Gibbs Free Energy (\(G\)): A crucial function for predicting the feasibility of a process under constant pressure and temperature. It aggregates enthalpy and entropy (the measure of disorder).

Non-state functions like heat (\(q\)) and work (\(w\)) are path-dependent and change with different process routes.

Reversible Processes

A reversible process is a theoretical concept where a system changes its state infinitely slowly, allowing it to stay in equilibrium with its surroundings throughout the transformation. Because it proceeds in such tiny increments, the process can be "reversed" at any point along the path without any net change in the system or the surroundings.

In a reversible process, the heat exchange (\(q_{rev}\)) and the work done (\(w_{rev}\)) are opposite compared to those in irreversible processes. Reversible processes:

• Maximize work done by or on the system.
• Ensure minimal energy loss, as it happens completely between the system and its environment.
• Require infinitely small steps, making them practical only in theoretical scenarios.

In contrast, irreversible processes are often faster and less efficient, resulting in loss of energy as heat and lower work done.

Enthalpy

Enthalpy (\(H\)) is a crucial concept in thermodynamics often associated with heat changes during chemical reactions. It is defined as the sum of the internal energy (\(U\)) and the product of pressure (\(p\)) and volume (\(V\)) of a system:\[H = U + pV\]Enthalpy is specifically useful for processes occurring at constant pressure, where the change in enthalpy (\(\Delta H\)) equates to the heat absorbed or released by the system. In an isothermal process, where the temperature remains constant, the enthalpy change is closely tied to the heat exchange under constant pressure conditions:

• For an isothermal process: \(\Delta H = q_p\) where \(q_p\) denotes heat at constant pressure.

This relationship simplifies the analysis of thermal changes during reactions or phase changes at constant pressure.

Gibbs Free Energy

Gibbs free energy (\(G\)) is a striking thermodynamic function used to predict the direction of chemical processes at constant temperature and pressure. It combines enthalpy, temperature, and entropy (\(S\)):\[G = H - TS\]The change in Gibbs free energy (\(\Delta G\)) during a process indicates whether the process can occur spontaneously:

• If \(\Delta G < 0\), the process is spontaneous.
• If \(\Delta G = 0\), the system is in equilibrium.
• If \(\Delta G > 0\), the process is non-spontaneous.

In reversible isothermal processes, \(\Delta G\) is equal to the maximum work that can be performed by the system, not including expansion work:\[\Delta G = -w_{rev}\]Thus, Gibbs free energy serves as a vital predictor for phase changes, reaction equilibria, and the feasibility of chemical processes.