Problem 26 - Chemistry The Central Science

Question

(a) Does the entropy of the surroundings increase for spontaneous processes? (b) In a particular spontaneous process the entropy of the system decreases. What can you conclude about the sign and magnitude of \(\Delta S_{\text { surr. }} ?(\mathbf{c})\) During a certain reversible process, the surroundings undergo an entropy change, \(\Delta S_{\text { surr }}=-78 \mathrm{J} / \mathrm{K}\) . What is the entropy change of the system for this process?

Step-by-Step Solution

Answer

(a) The entropy change of the surroundings for spontaneous processes cannot be generalized; it may increase, decrease, or remain constant as long as the total entropy change is positive. (b) The sign of the entropy change for the surroundings is positive, and the magnitude of the entropy change must be more significant than the decrease in the system's entropy to maintain spontaneity. (c) During a certain reversible process with a given entropy change of the surroundings, \(-78 \frac{J}{K}\), the entropy change of the system is \(78 \frac{J}{K}\).

Step 1: Entropy and Spontaneous Processes

By the Second Law of Thermodynamics, for a spontaneous process, the total entropy of the universe (system + surroundings) must increase, or we can write: \[ \Delta S_{universe} = \Delta S_{system} + \Delta S_{surr} > 0 \]. (a) Therefore, if we only consider the surroundings, we cannot generalize the sign of the entropy change for spontaneous processes. It may increase, decrease, or remain constant as long as the total entropy change is positive.

Step 2: Entropy Change of Surroundings

In this case, we know that the spontaneous process causes a decrease in the system's entropy. We need to determine the sign and magnitude of the entropy change for the surroundings, which will lead to the theoretical probability that the overall process is spontaneous. As mentioned earlier, for a spontaneous process, the total entropy must increase. In this case, the system entropy is decreasing, so to maintain overall spontaneity, the surroundings' entropy must increase by a magnitude larger than the decrease in the system's entropy. This result allows the total entropy change to remain positive. (b) Therefore, we can conclude that the sign of the entropy change for the surroundings is positive, and the magnitude of the entropy change must be more significant than the decrease in the system's entropy.

Step 3: Entropy Change in a Reversible Process

For a reversible process, the total entropy change must be zero, which can be written as: \[ \Delta S_{universe} = \Delta S_{system} + \Delta S_{surr} = 0 \]. (c) To determine the entropy change of the system, we can rearrange the formula as follows: \[ \Delta S_{system} = -\Delta S_{surr} \]. Given that the surroundings undergo an entropy change of \(-78 \frac{J}{K}\), we can substitute the value into the equation: \[ \Delta S_{system} = -(-78 \frac{J}{K}) = 78 \frac{J}{K} \]. Hence, the entropy change of the system during this reversible process is \(78 \frac{J}{K}\).

Key Concepts

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle that governs the direction of thermal energy transfer and the efficiency of heat engines. Simply put, the law states that entropy in an isolated system can only remain the same or increase over time—it cannot decrease. Entropy is a measure of the disorder or randomness within a system. The Second Law implies that systems naturally evolve towards a state of greater entropy. For instance, heat spontaneously flows from a hot object to a cold one, never the reverse, leading to an increase in the overall entropy of the combined system.

In the context of spontaneous processes, such as a piece of ice melting in a warm room, the Second Law dictates that the entropy of the universe—the system plus its surroundings—must increase. While the entropy of the melting ice (the system) decreases as it becomes more ordered in its liquid state, the entropy of the surroundings increases as heat is dispersed, thus obeying the Second Law.

Total Entropy of the Universe

The concept of the total entropy of the universe encompasses the combined entropy changes of both the system under observation and its surroundings. According to the Second Law of Thermodynamics, the sum of these changes in a spontaneous process must result in an increase in the overall entropy. This principle gives us a universal criterion for spontaneity of processes: a process will be spontaneous if it leads to an increase in the total entropy of the universe.

When we apply this concept to any thermodynamic process, we understand that the universe's entropy serves as the ultimate balance sheet. Any decrease in entropy within a system must be offset by a larger increase in the entropy of the surroundings, ensuring that the net change is positive. This provides the framework for analyzing whether processes are spontaneously moving towards greater disorder and randomness on a universal scale.

Entropy Change of the Surroundings

When considering the entropy change of the surroundings in thermodynamic processes, it's crucial to recognize that it's intrinsically linked to the entropy change of the system. If the system loses entropy, like a gas condensing to a liquid, the surroundings must gain a corresponding amount of entropy for the process to be spontaneous. The gain in the surroundings' entropy often comes from the heat flow from the system to the surroundings.

As seen in the exercise, if the entropy of the system decreases, the Second Law ensures us that the entropy change of the surroundings, \(\Delta S_{\text{surr.}}\), must be positive and sufficient enough to result in a net increase in the total entropy. This balance between the system and its surroundings emphasizes the interconnectedness of all thermodynamic processes and how the surroundings play a compensatory role when system entropy changes occur.

Reversible Process

A reversible process in thermodynamics is idealistic and represents a series of equilibrium states wherein the system can be returned to its original state without any net change in the universe. This sort of process is characterized by a balance in entropy change; the entropy gained or lost by the system is exactly equal to the entropy lost or gained by the surroundings, making the net entropy change of the universe zero.

In reality, perfectly reversible processes don’t occur, but they serve as an important model for understanding the efficiency limits of engines and other systems. For example, in the exercise, a reversible process results in an entropy change of the surroundings of \( -78 \frac{J}{K} \). To maintain reversibility, the entropy change of the system must be equal in magnitude but opposite in sign, thus confirming \( \Delta S_{system} = 78 \frac{J}{K} \), illustrating the precise balance that defines a reversible process.