Problem 31 - Chemistry: The Central Science

Question

For the isothermal expansion of a gas into a vacuum, \(\Delta E=0, q=0\), and \(w=0\). (a) Is this a spontaneous

Step-by-Step Solution

Answer

In the case of the isothermal expansion of a gas into a vacuum, we find that \(\Delta E\), q, and w are all equal to zero, and so no heat is exchanged nor work is done in this process. We also find that the change in entropy (\(\Delta S\)) is zero, leading to a change in Gibbs free energy (\(\Delta G\)) of zero as well. Since the change in Gibbs free energy is not negative, this process is neither spontaneous nor non-spontaneous. Instead, it is considered an equilibrium process with \(\Delta G = 0\).

Step 1: Define the given values

We are given that: - \(\Delta E = 0\) - \(q = 0\) - \(w = 0\) Isothermal expansion implies that the temperature remains constant throughout the process.

Step 2: Find the change in entropy

In an isothermal process, the change in internal energy (\(\Delta E\)) can be expressed as: \[\Delta E = q + w\] Since, \(\Delta E = 0\), we get \(q = -w\). However, given that both \(q\) and \(w\) equal to zero, no heat is exchanged nor work is done in this process. The change in entropy (\(\Delta S\)) of this process is given by: \[\Delta S = \frac{q_{rev}}{T}\] Since \(q = 0\), it means there is no heat exchange in the process. Therefore, \(\Delta S = 0\).

Step 3: Find the change in Gibbs free energy

The change in Gibbs free energy (\(\Delta G\)) determines the spontaneity of a process. It is given by: \[\Delta G = \Delta H - T\Delta S\] For an isothermal process, the change in enthalpy (\(\Delta H\)) can be expressed as: \[\Delta H = q\] Since \(q = 0\), we get \(\Delta H = 0\). Now we can calculate the change in Gibbs free energy (\(\Delta G\)) as follows: \[\Delta G = 0 - T(0) = 0\]

Step 4: Determine the spontaneity

For a process to be spontaneous, the change in Gibbs free energy (\(\Delta G\)) must be negative. However, in this case, we obtained \(\Delta G = 0\). Therefore, (a) this isothermal expansion of a gas into a vacuum is neither spontaneous nor non-spontaneous. It is considered an equilibrium process since \(\Delta G = 0\).

Key Concepts

Gibbs Free Energy

In thermodynamics, Gibbs free energy is a concept that helps us understand if a process can occur spontaneously. Gibbs free energy, often represented by the symbol \(G\), combines enthalpy, entropy, and temperature into one value.

• It is defined as \(G = H - TS\), where \(H\) is enthalpy, \(T\) is temperature, and \(S\) is entropy.
• The change in Gibbs free energy, \(\Delta G\), is of particular interest because it tells us about the process's spontaneity.

For an isothermal gas expansion, where temperature and internal energy remain constant, we find that \(\Delta G = \Delta H - T\Delta S\).
In this exercise, since both enthalpy change \(\Delta H\) and entropy change \(\Delta S\) are zero, \(\Delta G\) becomes zero as well, indicating the system is at equilibrium, rather than being spontaneous.Understanding \(\Delta G\) is crucial, as it not only helps determine spontaneity but also gives insight into the energy available for doing work beyond the system's own internal energy requirements.

Entropy Change

Entropy is a measure of disorder or randomness within a system. In an isothermal process such as the one described in this exercise, the entropy change \(\Delta S\) can reveal a lot about the nature of the process.

• Entropy change is calculated using the formula \(\Delta S = \frac{q_{rev}}{T}\), where \(q_{rev}\) is the reversible heat exchange and \(T\) is the temperature.
• For the isothermal expansion of a gas into a vacuum, both the heat exchange \(q\) and temperature \(T\) are zero, hence \(\Delta S = 0\).

This means there is no change in disorder, making the process entropy-neutral.
Even though no change occurs directly, entropy is a foundational concept for understanding energy distribution. It helps explain why some reactions or processes occur naturally while others do not. In this context, a zero entropy change implies that no work or heat is driving the system away from equilibrium.

Spontaneity

Spontaneity in chemical and physical processes refers to whether a reaction can happen on its own. A spontaneous process proceeds by itself under given conditions without needing external energy or influence.

• The main determinant of spontaneity is the sign of the change in Gibbs free energy, \(\Delta G\).
• If \(\Delta G\) is negative, the process is spontaneous; if \(\Delta G\) is positive, the process is non-spontaneous.
• If \(\Delta G = 0\), the process is in equilibrium and neither favorably spontaneous nor non-spontaneous.

In the given isothermal expansion exercise, since \(\Delta G = 0\), the process is at equilibrium.
Thus, it is neither spontaneous nor does it require external force to maintain its state.Understanding spontaneity helps predict if and how a process can occur naturally and is crucial in both chemistry and physics for designing reactions and systems.