Question

Explain how real machines can never recover the full amount of energy they were supplied with?.

Answer 1

Sounds like efficiency to me. In general, efficiency is defined as the work output acquired from some work input.

where:

• `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 `100%` efficiency, and the work it performs is `100%` reversible. Real machines, however, can never quite achieve `100%` efficiency, and the work they do is nearly `100%` reversible.

As I derived above, a cyclic process has a change in entropy of `DeltaS = 0`, and if heat flow `q` is reversible, then `DeltaS = q_"rev"/T`.

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)`.

`q_C` flows out of the engine, and thus, `q_C <= 0`. When `q_C = 0`, the efficiency is `100%`.

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 `"K"`), as they are in this equation, `0 <= e <= 1`. When `T_C = T_H`, `e = 0`, and when `T_C = "0 K"` (which has yet to be accomplished!), or `T_H` is absurdly large, `e ~~ 1`.

Therefore, the efficiency can never be more than `100%`, and for real machines, the efficiency is effectively never exactly `100%`.