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.