The Clausius inequality is a fundamental principle in thermodynamics that establishes a relationship between the change in entropy of a system and the amount of heat exchanged with its surroundings. It is a mathematical expression of the Second Law of Thermodynamics and is closely related to the concept of entropy.
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The Clausius inequality states that the change in entropy of a system plus the change in entropy of the surroundings is greater than or equal to zero for any real process.
The equality holds true only for reversible processes, while the inequality holds true for all other (irreversible) processes.
The Clausius inequality can be expressed mathematically as: $\Delta S_{system} + \Delta S_{surroundings} \geq 0$, where $\Delta S_{system}$ is the change in entropy of the system and $\Delta S_{surroundings}$ is the change in entropy of the surroundings.
The Clausius inequality is a powerful tool in understanding the directionality of heat transfer and the spontaneity of processes in thermodynamics.
The Clausius inequality is a fundamental principle that underlies the concept of the arrow of time and the irreversibility of many natural processes.
Review Questions
Explain the relationship between the Clausius inequality and the Second Law of Thermodynamics.
The Clausius inequality is a mathematical expression of the Second Law of Thermodynamics. It states that the total change in entropy of a system and its surroundings is always greater than or equal to zero for any real process. This means that entropy can only increase or remain constant, and never decrease, in an isolated system. The Clausius inequality reflects the fundamental principle that heat cannot spontaneously flow from a colder object to a hotter object without the input of work, which is a key tenet of the Second Law of Thermodynamics.
Describe the conditions under which the Clausius inequality becomes an equality.
The Clausius inequality, $\Delta S_{system} + \Delta S_{surroundings} \geq 0$, becomes an equality only for reversible processes. A reversible process is an idealized process in which the system and its surroundings can be returned to their initial states without leaving any trace on the surroundings. In other words, a reversible process is one that can be run in both the forward and reverse directions without any net changes to the system or its surroundings. For a reversible process, the change in entropy of the system plus the change in entropy of the surroundings is exactly zero, resulting in the Clausius equality: $\Delta S_{system} + \Delta S_{surroundings} = 0$.
Explain how the Clausius inequality can be used to determine the spontaneity of a process.
The Clausius inequality can be used to determine the spontaneity of a process. If the total change in entropy of the system and its surroundings is positive ($\Delta S_{system} + \Delta S_{surroundings} > 0$), then the process is spontaneous and will occur naturally. Conversely, if the total change in entropy is negative ($\Delta S_{system} + \Delta S_{surroundings} < 0$), then the process is non-spontaneous and will not occur naturally without the input of work. The Clausius inequality, therefore, provides a criterion for determining the spontaneity of a process based on the changes in entropy of the system and its surroundings.
The Second Law of Thermodynamics states that the entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.
A reversible process is an idealized process in which the system and its surroundings can be returned to their initial states without leaving any trace on the surroundings.