The Thermodynamic Efficiency of a Heat Engine (and the concept of Entropy)

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The Thermodynamic Efficiency of a Heat Engine (and the concept of Entropy)
It was shown here that a heat engine can never be 100% efficient.  
The thermodynamic efficiency of an engine is a measure of how much work it does compared with how much energy it takes from the energy source (from here on we will just say "efficiency".)  
   
When considering the efficiency of heat engines, we often represent the engine symbolically as shown below.  
We will not be concerned with the (probably quite complicated) mechanical details of the engine itself.  
For this reason, it is represented simply by the circle.  
We will just consider the energy flows into and out of the engine.  
   
We assume that the engine is working in a cycle, repeating the same actions over and over.  
It could be, for example, a car engine, taking in fuel, burning it, ejecting burnt fuel, taking in more fuel etc.  
   
   
 
During each cycle of operation of the engine:
ΔQH is the energy taken from the hot source
w is the work done by the engine
ΔQC is the energy given to the cold sink (the wasted energy)
 
 
 
 
   
The efficiency, η of a heat engine is defined as follows  
 
This fraction is usually multiplied by 100 to give a % so, if the net work done is only equal to half of the energy taken from the source, the engine has an efficiency of 50%.   
   
From the principle of conservation of energy, we have   
 
therefore  
 
   
Experiments show that increases as the difference between TH and TC increases.  
It can be shown that the theoretical maximum efficiency of a heat engine is given by    
 
It is worth emphasizing here that this gives the theoretical maximum efficiency.  
Practical engines like car engines have efficiencies of around 20% (for petrol) and 38% (for Diesel).    
These are much lower than you would get from applying the above formula.  
   
Comparing the last two equations, we can say that for an engine operating at maximum efficiency  
 
and rearranging this gives   
 
In other words, for an engine operating at the theoretical maximum efficiency, the quantity ΔQ/T for the source will be equal in magnitude to the same quantity for the sink.   
   
Realizing the importance of this quantity, Rudolf Clausius gave it a name: he said this ratio represents the change in entropy of the body.  
   
So, in general, we have the following definition of entropy, S  
 
where ΔQ represents the quantity of energy entering or leaving the body and T represents the absolute (or Kelvin or thermodynamic) temperature at which the energy transfer takes place.  
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