The is a key concept in thermodynamics, setting limits on and efficiency in cyclic processes. It states that the integral of heat transfer divided by temperature is always less than or equal to zero for any cycle.
This inequality has major implications for real-world systems like engines and refrigerators. It shows that no heat engine can be 100% efficient and no refrigerator can operate without external work, placing fundamental limits on their performance.
The Clausius Inequality
Clausius inequality and significance
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Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency | Physics View original
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Applications of Thermodynamics: Heat Pumps and Refrigerators | Physics View original
Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency | Physics View original
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Applications of Thermodynamics: Heat Pumps and Refrigerators | Physics View original
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States for a system undergoing a cyclic process, the integral of [T](https://www.fiveableKeyTerm:t)[dQ](https://www.fiveableKeyTerm:dq) (heat transfer divided by absolute temperature) is always less than or equal to zero ∮TdQ≤0
Fundamental statement of the establishes direction of heat transfer and limits on efficiency of heat engines (steam turbines) and refrigerators (air conditioners)
Valid for any cyclic process, both reversible () and irreversible ()
Heat transfer direction in cycles
Determines the direction of net heat transfer in a cyclic process
For a reversible cyclic process, the integral of TdQ equals zero ∮TdQ=0
For an irreversible cyclic process, the integral of TdQ is always less than zero ∮TdQ<0
Indicates heat flows from a high-temperature reservoir (combustion chamber) to a low-temperature reservoir (condenser) in a cyclic process
Clausius inequality vs second law
Mathematical statement of the second law of thermodynamics reinforces idea that heat cannot spontaneously flow from a cold reservoir to a hot reservoir without external work being done on the system
Implies no heat engine can be 100% efficient, as some heat must always be rejected to a low-temperature reservoir (atmosphere)
Similarly, no refrigerator can operate without external work input, as heat cannot spontaneously flow from a cold reservoir (inside fridge) to a hot reservoir (room temperature)
Implications for thermal efficiency
Sets an upper limit on the efficiency of heat engines
The efficiency of a heat engine is always less than the efficiency of a operating between the same two reservoirs
Carnot efficiency: ηCarnot=1−THTL, where TL and TH are the absolute temperatures of the low and high-temperature reservoirs
Sets a lower limit on the (COP) of refrigerators
The COP of a refrigerator is always less than the COP of a operating between the same two reservoirs
Carnot refrigerator COP: COPCarnot=TH−TLTL
Demonstrates no real heat engine (internal combustion engine) or refrigerator (heat pump) can achieve the efficiency or COP of a Carnot device, which operates under ideal, reversible conditions