Question

When heat is added to the surroundings, the entropy of the surroundings increases. How does \(\Delta S_{\text {surr }}\) depend on the temperature of the surroundings? Explain.

Short answer

\(\Delta S_{\text{surr}} = \frac{q}{T}\); lower temperature leads to greater entropy change.

Step-by-step solution

Understanding Entropy Change

The change in entropy for the surroundings, denoted as \(\Delta S_{\text{surr}}\), is related to the amount of heat exchanged with the surroundings and the temperature at which this exchange happens. In thermodynamics, entropy is often described by the formula \(\Delta S = \frac{q}{T}\), where \(q\) is the heat exchanged, and \(T\) is the temperature.

Applying the Entropy Formula

The formula indicates that the change in entropy \(\Delta S\) of the surroundings is inversely proportional to the temperature \(T\). This means as the temperature of the surroundings increases, the incremental increase in entropy due to a given amount of heat \(q\) becomes smaller.

Explaining Temperature's Influence

When heat is added to a system at a lower temperature, the increase in entropy \(\Delta S_{\text{surr}}\) is relatively large compared to if the same amount of heat were added at a higher temperature. This is because at lower temperatures, the system is 'less organized,' so the same energy dispersal leads to a greater change in entropy.

Key concepts

Entropy Change

Entropy change is a fundamental concept in thermodynamics. It describes how the disorder or randomness within a system shifts when energy is transferred. In simple terms, entropy change (\(\Delta S\)) measures how heat exchange (\(q\)) affects a system's level of disorder at a given temperature (\(T\)). When heat is introduced, entropy typically increases, signaling more disorder.

The formula \(\Delta S = \frac{q}{T}\) provides a straightforward way to compute entropy change. It indicates that the change in entropy is directly proportional to the heat added or removed and inversely proportional to the temperature. Essentially, for a constant amount of heat, changes in temperature significantly impact how disorder is affected within a system. This establishes an important balance between energy flow and thermal states in understanding entropy behavior.

Thermodynamics

Thermodynamics is the branch of physics that studies the interplay between heat, energy, and work within physical systems. It’s a critical framework, crucial for understanding how different processes are driven by energy changes. At its core, thermodynamics embodies several principles often called laws, which dictate how systems interact energetically.

In the context of entropy, thermodynamics introduces the idea that every energy transfer increases the total entropy of an isolated system. This is highlighted by the Second Law of Thermodynamics, stating that the natural progression of any system is towards greater disorder. Thus, in any heat transaction, like adding heat to surroundings, there’s an inherent increase in disorder or entropy.

Heat Exchange

Heat exchange is a process in which energy in the form of heat is transferred between a system and its surroundings. It plays a critical role in how entropy is modified, exemplified by \(\Delta S = \frac{q}{T}\). This relation underscores how the flow of heat into or out of a system affects its thermodynamic state.

When a system absorbs heat, its temperature may rise, causing an increase in entropy, signifying greater randomness. Conversely, when heat is removed, the system's entropy can decrease, indicating a more ordered state. The concepts of heat exchange are pivotal in engineering applications like energy systems and environmental science, showing how energy efficiencies are realized through entropy management.

Impact of Temperature on Entropy

Temperature is a critical factor in how entropy changes across various systems. According to the relation \(\Delta S = \frac{q}{T}\), as the temperature increases, the change in entropy for a given amount of heat becomes smaller. This occurs because when temperatures are higher, systems are already more thermally active, hence, additional heat leads to relatively smaller changes in their disorder.

On the other hand, introducing heat at lower temperatures can cause a substantial increase in entropy since systems tend to be more "organized" at lower energy states. This aspect of entropy and temperature explains why colder climates often exhibit noticeable changes in entropy with energy exchanges, whereas warmer conditions reflect subtler shifts. Understanding the impact of temperature elucidates how energy distributions affect systems, guiding predictions about material behaviors and energetic dynamics in diverse scenarios.