B.4.3 Thermodynamic processes and heat engines (HL only)

The Carnot Cycle and Heat Engine Efficiency

Carnot cycle

Definition

Carnot cycle

The Carnot cycle represents an idealized model of a heat engine that achieves the maximum possible efficiency for given reservoir temperatures.

The Carnot cycle consists of four reversible processes:

Isothermal Expansion (Process 1-2)
- Gas expands at constant temperature $T_{h}$
- Absorbs heat $Q_{h}$ from hot reservoir
- Work is done by the gas
Adiabatic Expansion (Process 2-3)
- No heat exchange with surroundings
- Temperature drops from $T_{h}$ to $T_{c}$
- Work is done by the gas
Isothermal Compression (Process 3-4)
- Gas compressed at constant temperature $T_{c}$
- Releases heat $Q_{c}$ to cold reservoir
- Work is done on the gas
Adiabatic Compression (Process 4-1)
- No heat exchange with surroundings
- Temperature increases from $T_{c}$ to $T_{h}$
- Work is done on the gas

The Adiabatic Equation for a Monatomic Ideal Gas

During an adiabatic process, the relationship between pressure and volume for a monatomic ideal gas is given by:

$P V^{\frac{5}{3}} = constant$

This equation arises from combining the ideal gas law with the first law of thermodynamics under adiabatic conditions.

Example

Suppose a gas undergoes adiabatic expansion from an initial state $P_{1}$ , $V_{1}$ to a final state $P_{2}$ , $V_{2}$ .

The adiabatic equation ensures:

$P_{1} V_{1}^{\frac{5}{3}} = P_{2} V_{2}^{\frac{5}{3}}$

Heat Engine Cycles

A heat engine is a device that converts thermal energy into mechanical work by operating in a cyclic process.

Efficiency of a Heat Engine

Definition

Efficiency

The efficiency ( $η$ ) of a heat engine measures how effectively it converts input energy into useful work.

It is defined as:

$η = \frac{Useful work}{Input energy} = \frac{W}{Q_{in}}$

Example

If a heat engine receives 500 J of energy from a hot reservoir and performs 200 J of work, its efficiency is:

$η = \frac{200 J}{500 J} = 0.4 or 40 %$

Carnot Efficiency

Definition

Carnot efficiency

The Carnot efficiency represents the maximum possible efficiency of a heat engine operating between two temperatures, $T_{h}$ (hot reservoir) and $T_{c}$ (cold reservoir).

It is given by:

$η_{Carnot} = 1 - \frac{T_{c}}{T_{h}}$

Example

For an engine operating between a hot reservoir at 600 K and a cold reservoir at 300 K, the Carnot efficiency is:

$η_{Carnot} = 1 - \frac{300}{600} = 0.5 or 50 %$

Note

The Carnot efficiency is an ideal limit.
No real engine can exceed this efficiency due to practical limitations such as friction and heat loss.
This is because it assumes perfectly reversible processes with no energy losses due to friction, turbulence, or heat dissipation.
In reality, all real engines experience irreversibilities, making the Carnot cycle an upper limit on efficiency rather than an achievable process.

Real Heat Engines and Their Efficiency

In real heat engines, such as the Otto and Diesel engines, the cycle approximates but does not replicate the Carnot cycle.
These engines have lower efficiencies due to:

Mechanical energy losses
Non-reversible processes
Non-ideal working conditions, such as imperfect insulation

Comparing Thermodynamic Processes

Process	Constant Quantity	Key Equation	Work Done
Isovolumetric	Volume ( $V$ )	$Q = Δ U$	$W = 0$
Isobaric	Pressure ( $P$ )	$Q = Δ U + P Δ V$	$W = P Δ V$
Isothermal	Temperature ( $T$ )	$Q = W$	$W = n R T \ln \frac{V_{2}}{V_{1}}$
Adiabatic	No heat exchange ( $Q = 0$ )	$P V^{γ} = constant$	$W = \frac{P_{1} V_{1} - P_{2} V_{2}}{γ - 1}$

Reflection

Self review

What is the main difference between an isothermal and an adiabatic process?
How is the efficiency of a heat engine calculated?
Why can no real engine exceed the Carnot efficiency?

Theory of Knowledge

How do the limitations imposed by the second law of thermodynamics influence the development of sustainable energy technologies?
Can we ever achieve a perfectly efficient engine?

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B.4.3 Thermodynamic processes and heat engines (HL only)

The Carnot Cycle and Heat Engine Efficiency

Carnot cycle

The Adiabatic Equation for a Monatomic Ideal Gas

Heat Engine Cycles

Efficiency of a Heat Engine

Carnot Efficiency

Real Heat Engines and Their Efficiency

Comparing Thermodynamic Processes

Reflection

You've read 2/2 free chapters this week.

Thermodynamic Processes and Heat Engines