Question

Indicate whether each statement is true or false. (a) The entropy of the universe increases for any spontaneous process. (b) The entropy change of the system is equal and opposite that of the surroundings for any irreversible process. (c) The entropy of the system must increase in any spontaneous process. (a) The entropy change for an isothermal process depends on both the absolute temperature and the amount of heat reversibly transferred.

Short answer

(a) True. The entropy of the universe increases for any spontaneous process according to the second law of thermodynamics. (b) False. The entropy change of the system is equal and opposite that of the surroundings for reversible processes, not irreversible processes. (c) False. The entropy of the system doesn't have to increase in any spontaneous process, as long as the entropy of the surroundings increases by an equal or greater amount. (a) in the last sentence: True. The entropy change for an isothermal process depends on both the absolute temperature and the amount of heat reversibly transferred, given by the formula \(\Delta S = \frac{q_{rev}}{T}\).

Step-by-step solution

Statement (a)

Statement (a) states that "The entropy of the universe increases for any spontaneous process." This statement is true. According to the second law of thermodynamics, for any spontaneous process occurring in an isolated system (which can be considered as the universe), the total entropy of the universe always increases or remains constant. Entropy, in general, is a measure of the disorder or randomness of a system.

Statement (b)

Statement (b) states that "The entropy change of the system is equal and opposite that of the surroundings for any irreversible process." This statement is false. This is actually true for reversible processes. For irreversible processes, while the entropy change of the system is indeed opposite to that of the surroundings, the magnitude may not be equal, as there are additional entropy changes due to the irreversibilities in the process.

Statement (c)

Statement (c) states that "The entropy of the system must increase in any spontaneous process." This statement is false. Although the entropy of the universe always increases or stays constant during a spontaneous process, entropy of the system does not necessarily have to increase. It could decrease as long as the entropy of the surroundings increases by an equal or greater amount.

Statement (a) in the last sentence

Statement (a) in the last sentence states "The entropy change for an isothermal process depends on both the absolute temperature and the amount of heat reversibly transferred." This statement is true. For an isothermal process, the entropy change is given by the formula: \(\Delta S = \frac{q_{rev}}{T}\) where \(\Delta S\) is the change in entropy, \(q_{rev}\) is the heat transferred reversibly, and \(T\) is the absolute temperature. So, the entropy change depends on both the temperature and the heat transferred.

Key concepts

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle which predicts the direction of spontaneous processes and their feasibility. It states that the total entropy of an isolated system can never decrease over time. In simpler terms, this law speaks to the tendency for energy differences within a system to level out and for systems to become more disordered over time.

How does this relate to real-life phenomena? Think of a cup of hot coffee placed in a room. Without any interference, the coffee cools down while the room warms up slightly until both reach the same temperature. Here, entropy increases as the energy distribution becomes more uniform and disordered.

Understanding Entropy: Entropy is often associated with the amount of disorder within a system. When a system evolves spontaneously, it often moves from a state of order to a state that's more chaotic and random. For instance, a block of ice in a warm room will melt, increasing entropy as the structured arrangement of water molecules in the ice becomes the more disordered liquid water.

However, this law doesn't merely concern itself with disorder. It fundamentally limits the efficiency of energy conversions and power cycles - since some energy will always dissipate as heat, increasing the entropy of the surroundings.

Reversible and Irreversible Processes

In thermodynamics, processes are categorized as reversible or irreversible based on their ability to return a system to its original state without leaving any trace in the surroundings. Reversible processes are idealistic and theoretical constructs where the system and its surroundings can be returned to their exact initial states after an infinitesimally slow change. They mark the upper limit of efficiency for any process, but in reality, all natural processes are irreversible.

Ideal vs. Reality: A real-world process is irreversible, meaning once it occurs, you cannot revert the system and surroundings back to their original state without some change, such as increased entropy. For example, when you mix cream into coffee, you cannot unmix them to their original separate states.

Irreversible processes, including spontaneous processes like the melting of an ice cube or the mixing of substances, all contribute to the increase of the universe's overall entropy. Energy lost as heat in friction or dissipated sound waves in a collision are examples of factors that make a process irreversible.

Isothermal Process Entropy Change

During an isothermal process, a system's temperature remains constant. 'Isothermal' literally means 'equal temperature'. In such a process, the entropy change is particularly simple to calculate because it only depends on the heat transferred reversibly and the absolute temperature as shown in the formula:
\[ \Delta S = \frac{q_{rev}}{T} \]
where \(\Delta S\) represents the change in entropy, \(q_{rev}\) is the heat transferred in a reversible process, and \(T\) is the absolute temperature of the system.

Real-World Implications: Imagine a gas that expands in a cylinder with a moveable piston. If the expansion happens slowly enough, allowing for heat transfer with the surroundings to maintain a constant temperature, that is isothermal expansion. The gas molecules spread out, occupying more space, which reflects an increase in disorder or entropy. If heat is added to the system during this process, the entropy change of the system is positive, indicating a gain in randomness. This concept is paramount in understanding phenomena such as engine cycles and refrigeration, where control of temperatures and heat transfer leads to work output or cooling effects.