How Do Carnot's Efficiency Formulas for Heat Engines Align?

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SUMMARY

Carnot's efficiency formulas for heat engines are defined by two key equations: 1 - T2/T1 and 1 - Q2/Q1. The first formula indicates that maximum efficiency decreases as the temperature of the cold source (T2) increases relative to the hot source (T1). The second formula, which relates efficiency to heat exchanged (Q), can cause confusion as it suggests that higher temperatures result in less heat exchange with the cold source. However, both formulas are consistent when considering the relationship between pressure, volume, and temperature in gas laws, where P x V is proportional to absolute temperature (T).

PREREQUISITES
  • Understanding of Carnot cycle principles
  • Familiarity with thermodynamic efficiency equations
  • Knowledge of gas laws and their implications
  • Basic grasp of pressure-volume work in thermodynamics
NEXT STEPS
  • Study the derivation of Carnot's efficiency formulas
  • Explore the relationship between pressure, volume, and temperature in ideal gases
  • Investigate real-world applications of Carnot's theorem in heat engines
  • Learn about the limitations of Carnot efficiency in practical scenarios
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Students and professionals in thermodynamics, mechanical engineers, and anyone involved in the design or analysis of heat engines will benefit from this discussion.

Tulio Cesar
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My question is: according to Carnot cycle, the maximum efficiency of a heat engine is given by 1 - T2/T1, where T2 is the temperature of the cold source and T1 the temperature of the hot source. So, accordingly, as higher T2 is for a same T1, lowest is the efficiency of the engine. But, the other formula for efficiency (1 - Q2/Q1), gets me confuse because as bigger temperature is, minus heat is exchanged with the cold source, and as lower temperature is, more heat is exchanged with de hot source. I don't know if I missed any point but these two formulas seem contradictory for me.
 
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The analysis of heat engines centres around the expansion of gases. It is possible to show that the work (energy) involved in compressing or expanding gases is give by P x V ,( area under P~ V graph or ∫P.dV ).
From the gas laws it is possible to show that P x V ∝ T.
This means that heat energy is proportional to Absolute temp (T) and therefore the expressions Q in your equation can be replaced by T
 

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