P-V Diagram for the Carnot engine
Step 1: Isothermal expansion at Th:
Step 1: Adiabatic expansion to Tc:
Step 1: Adiabatic compression to Th:
Step 4: Isothermal compression at Tc:
| Step | Description | Heat | ΔS |
|---|
| 1 | Isothermal Expansion
at Th | Qh |
| ΔS1 |
= + |
| |Qh|
|  |
| Th
|
|
|
| 2 | Adiabatic Expansion
to Tc | 0 | 0
|
| 3 | Isothermal Compression
at Tc | Qc |
| ΔS3 |
= - |
| |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, then ΔS for the cycle is zero.
| ΔS1 + ΔS3 =
| | |Qh|
| |
| Th
|
|
- |
| |Qc|
| |
| Tc
|
|
= 0 |
| Therefore:
| | |Qh|
| |
| Th
|
|
= |
| |Qc|
| |
| Tc
|
|
| Comparing this to our previous result: η
| = 1 - |
| |Qc|
| |
| |Qh|
|
|
| we obtain ηc
| = 1 - |
| Tc
|  |
| Th
|
|