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: 
ΔQ_{H} is the
energy taken from the hot source 
w is the work done by the engine 
ΔQ_{C} 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 T_{H} and T_{C} 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. 
