Work Out Efficiency from Kelvin Temperatures Only
The Carnot engine is the best possible engine between two temperatures, and its efficiency depends on those two temperatures alone. Heat-engine questions are therefore short: convert to kelvin, find the efficiency, and split the heat into work and rejected heat.
Carnot Engine and the Second Law is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
Efficiency and the heat balance
A reservoir at 627 °C (900 K) and a sink at 27 °C (300 K) give an efficiency of 2/3, so 3 x 10⁶ cal drawn from the source yields 2 x 10⁶ cal of work. Between 300 K and 600 K the efficiency is 1/2, so 800 J of work per cycle needs 1600 J from the source. Between the freezing and boiling points of water the efficiency is 1 - 273/373, about 27%.
Changing the temperatures
Questions that say the efficiency changes when a temperature is shifted give two equations in Th and Tc. If a Carnot engine has efficiency 1/2 and, with the sink lowered by 100 K, efficiency 2/3, then Tc = Th/2 and (Tc - 100) = Th/3, so Th/6 = 100 and Th = 600 K.
Always solve for the unknown temperature first, then recompute the efficiency.
Second law and real engines
The second law (Kelvin-Planck form) says that no cyclic engine can convert heat completely into work; some heat must be rejected to a colder body. This limit exists even for a perfect frictionless engine, so it is not caused by friction. Work can be converted completely into heat, but not the other way around in a cycle, and no engine between the same two temperatures can be more efficient than a Carnot engine. For a given P-V cycle of an ideal gas, heat is extracted from the source only on the legs where heat is absorbed, and the efficiency is the net work divided by that heat.
OlympIQ exam lens
- ✓Efficiency = 1 - Tc/Th with temperatures in kelvin
- ✓W = efficiency x Qh and Qc = Qh - W
- ✓For temperature-change questions, write two efficiency equations and solve for Th or Tc first
- ✓The second law forbids complete conversion of heat to work in a cycle; friction is not the reason
- ✓For a P-V cycle, efficiency = net work / heat absorbed on the heating legs only
Do not put Celsius temperatures into the Carnot formula. A 27 °C sink and a 627 °C source give 1 - 27/627 if you forget to convert, which is wrong; the correct efficiency uses 300 K and 900 K.
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