Carnot Engine — Revision Notes

⚡ 30-Second Revision

Carnot Engine: — Ideal, theoretical heat engine with maximum possible efficiency.
Carnot Cycle: — Four reversible processes: Isothermal expansion ($T_H$), Adiabatic expansion ($T_H \to T_C$), Isothermal compression ($T_C$), Adiabatic compression ($T_C \to T_H$).
Efficiency Formula: — $\eta = 1 - \frac{T_C}{T_H}$ (Temperatures in Kelvin).
Heat-Temperature Relation: — $\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$.
Work Done: — $W = Q_H - Q_C$.
Carnot's Theorems: — No engine can be more efficient than a reversible one between same $T_H, T_C$. All reversible engines between same $T_H, T_C$ have same efficiency.
Key Point: — Efficiency is independent of working substance. Cannot be 100% unless $T_C = 0,\text{K}$.

2-Minute Revision

The Carnot engine is a theoretical construct, an ideal heat engine that operates on the Carnot cycle, which comprises four perfectly reversible processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.

It works between a high-temperature reservoir ($T_H$) and a low-temperature reservoir ($T_C$), absorbing heat $Q_H$ and rejecting $Q_C$, doing net work $W = Q_H - Q_C$. Its thermal efficiency is given by $\eta = 1 - \frac{T_C}{T_H}$, where $T_H$ and $T_C$ must be in Kelvin.

This formula shows that efficiency depends only on the absolute temperatures of the reservoirs and is independent of the working substance. Carnot's theorems state that no engine can be more efficient than a reversible engine operating between the same two temperatures, and all reversible engines operating between the same two temperatures have the same efficiency.

The Carnot engine sets the maximum possible efficiency, a fundamental limit for all real heat engines, which always have lower efficiencies due to irreversibilities.

5-Minute Revision

The Carnot engine is the most efficient theoretical heat engine, operating on the Carnot cycle, which is a sequence of four reversible processes. These processes are: 1. Isothermal expansion at $T_H$ (heat $Q_H$ absorbed, work done).

2. Adiabatic expansion (temperature drops from $T_H$ to $T_C$, work done). 3. Isothermal compression at $T_C$ (heat $Q_C$ rejected, work done on gas). 4. Adiabatic compression (temperature rises from $T_C$ to $T_H$, work done on gas).

The net work done by the engine is $W = Q_H - Q_C$.

The thermal efficiency of a Carnot engine is given by $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$. Crucially, for a Carnot engine, the ratio of heat exchanged is equal to the ratio of absolute temperatures: $\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$. Substituting this into the efficiency formula yields the famous Carnot efficiency: $\eta = 1 - \frac{T_C}{T_H}$. Remember, $T_H$ and $T_C$ must always be in Kelvin.

Example: A Carnot engine operates between $400,\text{K}$ and $200,\text{K}$. Its efficiency is $\eta = 1 - \frac{200}{400} = 1 - 0.5 = 0.5$, or $50\%$. If it absorbs $1000,\text{J}$ from the hot reservoir, the work done is $W = \eta Q_H = 0.5 \times 1000,\text{J} = 500,\text{J}$. The heat rejected is $Q_C = Q_H - W = 1000,\text{J} - 500,\text{J} = 500,\text{J}$.

Carnot's theorems are vital: no engine can be more efficient than a reversible one between the same temperatures, and all reversible engines between the same temperatures have the same efficiency, independent of the working substance. This engine is a theoretical limit; real engines always have lower efficiency due to irreversibilities and cannot achieve 100% efficiency as $T_C = 0,\text{K}$ is unattainable.

Prelims Revision Notes

The Carnot engine is an idealized heat engine, crucial for understanding the limits of energy conversion. Its operation is based on the Carnot cycle, a sequence of four reversible processes: two isothermal and two adiabatic.

1
Isothermal Expansion (A \to B): — Working substance absorbs heat $Q_H$ from hot reservoir at constant temperature $T_H$. Work is done by the gas. $\Delta U = 0$ for ideal gas.
2
Adiabatic Expansion (B \to C): — Working substance expands, doing work, without heat exchange. Temperature drops from $T_H$ to $T_C$. Internal energy decreases.
3
Isothermal Compression (C \to D): — Working substance rejects heat $Q_C$ to cold reservoir at constant temperature $T_C$. Work is done on the gas. $\Delta U = 0$.
4
Adiabatic Compression (D \to A): — Working substance is compressed without heat exchange. Temperature rises from $T_C$ to $T_H$. Internal energy increases.

Key Formulas:

Thermal Efficiency: — $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$.
Carnot Efficiency: — $\eta_{Carnot} = 1 - \frac{T_C}{T_H}$. (Always use absolute temperatures in Kelvin! $T_K = T_C + 273.15$).
Heat-Temperature Relation (for Carnot engine): — $\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$. This implies $Q_C = Q_H \frac{T_C}{T_H}$.
Net Work Done: — $W = Q_H - Q_C$.

Carnot's Theorems:

1
No engine operating between two given temperatures can be more efficient than a reversible engine (Carnot engine) operating between the same two temperatures.
2
All reversible engines operating between the same two temperatures have the same efficiency, irrespective of the nature of the working substance.

Important Points for NEET:

Carnot engine is theoretical; real engines are always less efficient.
Efficiency is always less than 100% (unless $T_C = 0,\text{K}$, which is impossible).
Efficiency depends ONLY on $T_H$ and $T_C$, not on the working substance.
To increase efficiency, increase $T_H$ or decrease $T_C$.
Be careful with temperature units (Celsius vs. Kelvin).

Vyyuha Quick Recall

To remember the Carnot Cycle processes: In All Ideal Applications, Expansion Comes Easily Compressed.

Isothermal Expansion (A to B) Adiabatic Expansion (B to C) Isothermal Compression (C to D) Adiabatic Compression (D to A)