Question

Describe the equilibrium condition in terms of the entropy changes of a system and its surroundings. What does this description say about the entropy change of the universe?

Short answer

At equilibrium, the total entropy change of the system and its surroundings is zero. This implies the entropy change of the universe is also zero.

Step-by-step solution

Identify the Concept of Equilibrium

In thermodynamics, equilibrium is a state where the system and its surroundings do not experience any further net change in properties such as pressure, temperature, and volume. In this state, there will be no net flow of energy or matter.

Define Entropy

Entropy is a measure of the disorder or randomness in a system. It is a key concept in the second law of thermodynamics, which states that in an isolated system, the total entropy will never decrease over time.

State the Entropy Change Condition for Equilibrium

At equilibrium, the total entropy change \(\triangle S_{\text{total}}\) of the system and its surroundings is zero. This can be expressed as:\[\triangle S_{\text{total}} = \triangle S_{\text{system}} + \triangle S_{\text{surroundings}} = 0\]

Interpretation of the Entropy Condition

Since \(\triangle S_{\text{total}}\) equals zero at equilibrium, it implies that any positive entropy change in the system is exactly balanced by a negative entropy change in the surroundings, and vice versa. This is an essential condition for equilibrium.

Entropy Change of the Universe

The universe is considered as the combination of the system and its surroundings. Thus, at equilibrium for any spontaneous process, the sum of the entropy changes of the system and surroundings is zero. This implies that the entropy change of the universe is zero at equilibrium. In mathematical terms: \[\triangle S_{\text{universe}} = \triangle S_{\text{system}} + \triangle S_{\text{surroundings}} = 0\]

Key concepts

Thermodynamic Equilibrium

In thermodynamics, equilibrium represents a state where a system and its surroundings reach a balance, causing no further net change in properties such as pressure, temperature, and volume. This means that energy and matter no longer flow between the system and its surroundings. When a system is in thermodynamic equilibrium, it has reached a stable condition. For instance, if you have a cup of hot tea in a room, it will eventually cool down until it reaches room temperature. At that point, the tea and the room are in thermodynamic equilibrium. No heat flows between them anymore. This balance is crucial in understanding how systems behave and interact.

Second Law of Thermodynamics

The second law of thermodynamics states that in an isolated system, the total entropy will never decrease over time. Entropy is a measure of disorder or randomness. For the system to reach thermodynamic equilibrium, it must maximize its entropy inside an isolated environment. For example, gas molecules in a box will spread out evenly over time due to increased entropy, reaching a steady state where they are uniformly distributed. This state of maximum entropy aligns with the concept of equilibrium. The second law also implies that natural processes have a direction: they tend to move towards greater disorder or higher entropy. This is why heat flows from hot to cold and why systems move toward equilibrium.

Entropy Change

Entropy change, denoted as \(\triangle S\), measures the difference in entropy between the initial and final states of a process. At equilibrium, the total entropy change of the system and its surroundings is zero:\[\triangle S_{\text{total}} = \triangle S_{\text{system}} + \triangle S_{\text{surroundings}} = 0\]This means that for any change occurring in the system, there is an equal and opposite change in the surroundings to maintain equilibrium. Additionally, considering the universe as the combination of the system and its surroundings, the universe's entropy change at equilibrium is also zero:\[\triangle S_{\text{universe}} = \triangle S_{\text{system}} + \triangle S_{\text{surroundings}} = 0\]This principle helps us understand that any increase in disorder in one part of an isolated system must be balanced by a decrease elsewhere, keeping the overall change neutral.