The Clausius inequality is a fundamental concept in thermodynamics that relates to the second law of thermodynamics. It establishes a criterion for the direction of heat transfer between two bodies or systems. The inequality is named after Rudolf Clausius, a prominent physicist and thermodynamicist.
The Clausius inequality states that for a cyclic process involving a heat engine or a refrigeration cycle, the integral of the heat transfer (δQ) divided by the temperature (T) over the entire cycle is always less than or equal to zero:
∮ (δQ / T) ≤ 0
In this inequality, the symbol "∮" represents the integration over the entire cycle.
The Clausius inequality can be understood in the context of heat flow. It states that heat cannot spontaneously flow from a colder body to a hotter body without the input of external work. In other words, in a closed system, heat naturally flows from a region of higher temperature to a region of lower temperature, and not the other way around. The inequality quantifies this concept by relating the heat transfer to the temperature at which it occurs.
Mathematically, the inequality can be derived from the Kelvin-Planck statement of the second law of thermodynamics, which states that no heat engine can operate in a cycle while transferring heat solely from a colder reservoir to a hotter reservoir without any other effects.
The Clausius inequality is a powerful tool for analyzing and understanding the behavior of heat engines, refrigeration systems, and processes involving heat transfer. It provides a quantitative criterion for the direction of heat flow and helps establish the irreversibility of certain processes.