P-V Diagram for the Carnot engine
We can derive Carnot's result if we know a little more about the cycle in a Carnot engine. It involves four steps:
| Step | Description | Heat | DS |
|---|
| 1 | Isothermal Expansion
at Th | Qh |
| DS1 |
= + |
| |Qh|
|  |
| Th
|
|
|
| 2 | Adiabatic Expansion
to Tc | 0 | 0
|
| 3 | Isothermal Compression
at Tc | Qc |
| DS3 |
= - |
| |Qc|
| |
| Tc
|
|
|
| 4 | Adiabatic Compression
to Th | 0 | 0
|
| Full Cycle | - | |Qh| - |Qc| | 0
|
After completing a cycle the system returns to its original state - the original pressure , volume, temperature, and entropy. If the entropy is the same the total DS for the cycle is zero.
| DS1 + DS3 =
| | |Qh|
| |
| Th
|
|
- |
| |Qc|
| |
| Tc
|
|
= 0 |
| Therefore:
| | |Qh|
| |
| Th
|
|
= |
| |Qc|
| |
| Tc
|
|
This is Carnot's result.