Question

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

Short answer

In the case of the isothermal expansion of a gas into a vacuum, we find that $\mathrm{\Delta }E$, 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 ($\mathrm{\Delta }S$) is zero, leading to a change in Gibbs free energy ($\mathrm{\Delta }G$) 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 $\mathrm{\Delta }G=0$.

Step-by-step solution

Define the given values

We are given that: - $\mathrm{\Delta }E=0$ - $q=0$ - $w=0$ Isothermal expansion implies that the temperature remains constant throughout the process.

Find the change in entropy

In an isothermal process, the change in internal energy ($\mathrm{\Delta }E$) can be expressed as: $$\mathrm{\Delta }E=q+w$$ Since, $\mathrm{\Delta }E=0$, 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 ($\mathrm{\Delta }S$) of this process is given by: $$\mathrm{\Delta }S=\frac{q_{rev}}{T}$$ Since $q=0$, it means there is no heat exchange in the process. Therefore, $\mathrm{\Delta }S=0$.

Find the change in Gibbs free energy

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

Determine the spontaneity

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

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, $\mathrm{\Delta }G$, 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 $\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S$.
In this exercise, since both enthalpy change $\mathrm{\Delta }H$ and entropy change $\mathrm{\Delta }S$ are zero, $\mathrm{\Delta }G$ becomes zero as well, indicating the system is at equilibrium, rather than being spontaneous.Understanding $\mathrm{\Delta }G$ 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 $\mathrm{\Delta }S$ can reveal a lot about the nature of the process.

• Entropy change is calculated using the formula $\mathrm{\Delta }S=\frac{q_{rev}}{T}$, 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 $\mathrm{\Delta }S=0$.

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, $\mathrm{\Delta }G$.
• If $\mathrm{\Delta }G$ is negative, the process is spontaneous; if $\mathrm{\Delta }G$ is positive, the process is non-spontaneous.
• If $\mathrm{\Delta }G=0$, the process is in equilibrium and neither favorably spontaneous nor non-spontaneous.

In the given isothermal expansion exercise, since $\mathrm{\Delta }G=0$, 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.