Explain how real machines can never recover the full amount of energy they were supplied with?.
Sounds like efficiency to me. In general, efficiency is defined as the work output acquired from some work input.
#w#is work. #q_H#is the heat flow into the engine from the hot reservoir. #q_C#is the heat flow from the engine into the cold reservoir. #T_H#is the temperature of the hot reservoir. #T_C#is the temperature of the cold reservoir.
The ideal machine can achieve
As I derived above, a cyclic process has a change in entropy of
If we separate the cyclic process into two processes, then
#q_H/T_H + q_C/T_C = 0#
Also from the above, I calculated that
#color(blue)(e) = |w|/q_H = (q_H + q_C)/q_H = color(blue)(1 + q_C/q_H)#.
Or, using the first relation:
#color(blue)(e) = |w|/q_H = (q_H + q_C)/q_H = color(blue)(1 - T_C/T_H)#.
So, we could even say that since all absolute temperatures are positive (i.e. when in
Therefore, the efficiency can never be more than