Page 124 - Essentials of physical chemistry

86 Essentials of Physical Chemistry

that, we want to evaluate the efficiency, E eff , of a Carnot heat engine in terms of how much work
is done on the environment relative to how much energy is input in the first step, so we need to
use Sw and in the previous analysis of the Carnot steps, we have given w for each step:

V 2 V 4
RT h ln C V (T l T h ) þ RT l ln C V (T h T l )
( Sw) V 1 V 3
:

E eff ¼
q I V 2
RT h ln
V 1
1

V 4 V 1 V 2
Once again, we use the key relationship from the adiabatic steps as ¼ ¼ and
V 3 V 2 V 1
substituting that relationship into the ln ( ) terms reverses the sign of one, which allows a lot of
cancelation. This results in a very useful and simple formula for the efficiency of a Carnot engine:
T h T l
:
E eff ¼
T h
This tells us that the efficiency of the hypothetical Carnot engine only depends on the high
temperature of the input energy and the low temperature. Let us think about this as (T h T l )
representing the heat actually used in the conversion to work, but that the heat released at T l is
wasted. The most important result is that if the (P, V) graph is drawn for any real heat engine, the
cycle can be overlaid with a grid of isotherms and adiabats to show that within each tiny calculus
Þ
differential Carnot cycle all the ds ¼ 0 terms cancel internally, and the sum around the outer real
engine PV graph also satisfies the same conditions and leads to the same formula.

EFFICIENCY OF REAL HEAT ENGINES
It is important to note that the efficiency formula uses absolute temperatures in 8K. On planet earth,
ambient temperatures limit the exit temperature T l of most heat engines to roughly 2738K or higher
temperatures. Thus, most heat engines can harness heat energy from hot sources to do work but
waste a lot of that heat. A little thought shows that it ought to be possible to select a gas that could
operate as a cyclic heat-transport medium between the common temperatures of ice water and steam
as in a ‘‘steam engine,’’ noting that the gas need not be steam but could be some gas trapped in a
cyclic system operating between 273.158K and 373.158K. For that semirealistic situation, the
efficiency would be

373:15 273:15
ffi 0:268 ’ 27%:
373:15
E eff ¼
Early in the development of the steam engine, it was realized that if actual water steam is super-
heated as a gas well above the boiling point of water to say 8008K and regular liquid is used as a
coolant, the engine could be operated between about 8008K and 3738K to improve the efficiency to
roughly

800 373
¼ 0:534 ffi 53:4%:
E eff ¼
800
This is actually very good for a heat engine or any engine for that matter. Compare that to an internal
combustion engine at the extreme operating temperatures. Thus, if we assume ideal conditions of
about 23008K for combustion of gasoline and a maximum temperature of the exhaust manifold and