ThermoD Carnot Cycle

### **3. Net Work Output**

The net work done by the Carnot engine:

\[

W_{\text{net}} = Q_{\text{hot}} - Q_{\text{cold}} = \eta \cdot Q_{\text{hot}}

\]

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### **4. Adiabatic Process Relations**

For adiabatic steps (no heat exchange), the following hold:

- **Temperature-Volume Relation**:

\[

T V^{\gamma - 1} = \text{constant} \quad \Rightarrow \quad T_{\text{hot}} V_2^{\gamma - 1} = T_{\text{cold}} V_3^{\gamma - 1}

\]

- **Pressure-Volume Relation**:

\[

P V^\gamma = \text{constant}

\]

where \( \gamma = \frac{C_p}{C_v} \) (heat capacity ratio).

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### **5. Coefficient of Performance (COP)**

- **For a Carnot Refrigerator**:

\[

\text{COP}_{\text{refrigerator}} = \frac{Q_{\text{cold}}}{W_{\text{net}}} = \frac{T_{\text{cold}}}{T_{\text{hot}} - T_{\text{cold}}}

\]

- **For a Carnot Heat Pump**:

\[

\text{COP}_{\text{heat pump}} = \frac{Q_{\text{hot}}}{W_{\text{net}}} = \frac{T_{\text{hot}}}{T_{\text{hot}} - T_{\text{cold}}}

\]

---

### **Key Relationships**

- Ratio of heats:

\[

\frac{Q_{\text{cold}}}{Q_{\text{hot}}} = \frac{T_{\text{cold}}}{T_{\text{hot}}}

\]

- Volume ratios in adiabatic steps:

\[

\frac{V_3}{V_2} = \left(\frac{T_{\text{hot}}}{T_{\text{cold}}}\right)^{\frac{1}{\gamma - 1}}, \quad \frac{V_4}{V_1} = \left(\frac{T_{\text{hot}}}{T_{\text{cold}}}\right)^{\frac{1}{\gamma - 1}}

\]

---

### **Summary**

- The Carnot cycle consists of **two isothermal** and **two adiabatic** processes.

- Efficiency depends **only** on the reservoir temperatures.

- Use \( \eta = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \) for maximum efficiency calculations.

- Heat transfers and work are linked via the logarithmic volume ratios and adiabatic relations.

These formulas are foundational for analyzing Carnot engines, refrigerators, and heat pumps in thermodynamics.