PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Statistical mechanics and thermodynamics → Classical thermodynamics ↓


Carnot cycle


The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Leonard Sadi Carnot in 1824. It establishes an ideal model for heat engines, forming an essential basis in classical thermodynamics. The beauty of the Carnot cycle lies in its reversibility, which makes it the most efficient cycle for a heat engine operating between two heat reservoirs. The efficiency of actual heat engines is often compared to the theoretical maximum efficiency of the Carnot cycle.

To better understand the Carnot cycle, one must have a good understanding of some fundamental concepts of thermodynamics: thermodynamic systems, heat engines, and the laws of thermodynamics.

Thermodynamic systems and heat engines

A thermodynamic system is a quantity or region of matter in space that is selected for analysis. The system is separated from its surroundings by a boundary, with which it can exchange energy and matter.

A heat engine is a type of thermodynamic system that performs work in a cycle. It takes in energy as heat and gives out some energy as heat while operating in the cycle and does work. The Carnot cycle is an ideal heat engine cycle that provides the maximum possible efficiency that any heat engine can achieve while operating between two reservoirs.

Laws of thermodynamics

The Carnot cycle respects the laws of thermodynamics, specifically the first and second laws:

  • First Law: The energy of a closed system is conserved. Mathematically, ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.
  • Second Law: The entropy of any isolated system always increases. This law states that energy conversions are not perfectly efficient and some energy will be lost as heat.

Carnot cycle process

The Carnot cycle involves four reversible processes: two isothermal (constant temperature) and two adiabatic (no heat exchange). Let's explore each process:

1. Isothermal expansion (Process 1-2)

In this stage, the gas in the engine is put in contact with a high temperature heat reservoir at temperature TH. Since the process is isothermal, the temperature of the gas remains constant. The gas absorbs heat QH from the reservoir and expands, doing work W1-2 on the surroundings. The given formula represents the amount of work done:

        W1-2 = Qh = nRTh ln(V2/V1)

where n is the number of moles, R is the universal gas constant, and V1 and V2 are the initial and final volumes of the gas, respectively.

qh th 1 2

2. Adiabatic expansion (Process 2-3)

Here, the system is perfectly insulated; there is no heat exchange with the surrounding environment. The gas continues to expand, doing work W2-3 on the environment. Since the process is adiabatic, the temperature of the gas decreases to a lower temperature TL

        THV2γ-1 = TLV3γ-1

Where γ (gamma) is the adiabatic index, the ratio of specific heats (Cp/Cv).

th TL

3. Isothermal compression (Process 3-4)

The gas is then placed in contact with a low temperature heat reservoir at TL and isothermally compressed. It releases heat QL to the surroundings and loses work W3-4 in compressing the gas.

        W3-4 = Ql = nRTl ln(V4/V3)

ql TL 3 4

4. Adiabatic compression (Process 4-1)

Finally, the gas is adiabatically compressed back to its original state, regaining the original temperature TH. Again, since this process is adiabatic, no heat is exchanged with the surroundings, and all the work done is converted into internal energy, raising the temperature.

        TLV4γ-1 = THV1γ-1

TL th

Efficiency of the Carnot cycle

The efficiency η of a heat engine is defined as the ratio of the work done W to the heat input QH:

        η = W / QH

For a Carnot engine, the work done is the difference between the heat absorbed from the high temperature reservoir and the heat expelled into the low temperature reservoir:

        w = qh - ql

Combining both equations we get the efficiency of the Carnot cycle:

        η = 1 − (QL/QH)

Using the fact that for a reversible process, the heat exchanged is proportional to the temperature:

        QL/QH = TL/TH

Substitution gives the Carnot efficiency:

        η = 1 − (Tl/Th)

This shows that the maximum efficiency depends only on the temperature of the heat reservoir and is always less than 1. No real engine operating between two heat reservoirs can be more efficient than a Carnot engine.

Limitations and real-world applications

Although the Carnot cycle sets the upper limit of thermodynamic efficiency, it is an idealized concept. Real engines cannot achieve truly reversible processes due to friction, heat losses, and other inefficiencies, making the Carnot cycle unattainable in practice. However, it serves as a valuable benchmark for evaluating the performance of real heat engines.

In practical applications, engineers attempt to design engines that get as close as possible to Carnot efficiency by minimizing irreversibilities. Understanding the principles of the Carnot cycle also helps in optimizing refrigeration cycles and improving energy efficiency in various technologies.

Conclusion

The Carnot cycle remains a cornerstone of thermodynamics and statistical mechanics. By providing a theoretical framework for maximizing the efficiency of heat engines, it illuminates important understandings of heat, work, and energy transformations. Even though real-world constraints prevent achieving Carnot's ideal, its principles guide innovations that increase thermal efficiency and energy conservation.


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Carnot cycle

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