Table of Contents
- Introduction
- What Is Thermodynamics?
- The Four Laws of Thermodynamics
- Internal Energy and the First Law
- Work, Heat, and State Functions
- The Concept of Entropy
- Entropy and Reversible vs Irreversible Processes
- The Second Law and the Arrow of Time
- Entropy in Statistical Mechanics
- Thermodynamic Potentials
- Maxwell’s Relations
- Real-World Applications
- Conclusion
1. Introduction
Thermodynamics is the science of energy transformations and the relationships between heat, work, and internal energy. It is both a phenomenological and fundamental science, underpinning engines, phase changes, chemical reactions, and even black holes.
At the heart of thermodynamics lies entropy — a concept that bridges physics, probability, and the arrow of time.
2. What Is Thermodynamics?
Thermodynamics studies the macroscopic properties of matter — temperature, pressure, volume, and energy — without needing microscopic detail.
Key questions include:
- How does energy flow?
- What processes are possible or forbidden?
- Why do systems evolve toward equilibrium?
3. The Four Laws of Thermodynamics
Zeroth Law (Thermal Equilibrium):
If \( A = B \) and \( B = C \) are in thermal equilibrium, then \( A = C \).
This allows the definition of temperature as a measurable, transitive quantity.
First Law (Energy Conservation):
\[
\Delta U = Q – W
\]
Change in internal energy equals heat added minus work done by the system.
Second Law (Entropy Increase):
In any spontaneous process:
\[
\Delta S_{\text{universe}} \geq 0
\]
Entropy tends to increase, leading to irreversibility.
Third Law (Zero Entropy at Absolute Zero):
As \( T \rightarrow 0 \), \( S \rightarrow 0 \) for a perfect crystal.
4. Internal Energy and the First Law
Internal energy \( U \) is the total microscopic energy of a system (kinetic + potential of atoms).
The first law states:
\[
dU = \delta Q – \delta W
\]
Where:
- \(delta Q\): infinitesimal heat input
- \(delta W\): infinitesimal work output
5. Work, Heat, and State Functions
- Work and heat are path-dependent (not state functions)
- Internal energy, entropy, and volume are state functions
For example, in a quasistatic expansion:
\[
\delta W = P dV
\]
The total work done depends on the path taken in P-VP\text{-}VP-V space.
6. The Concept of Entropy
Entropy SSS quantifies disorder, multiplicity of microstates, or unavailable energy.
Clausius defined:
\[
dS = \frac{\delta Q_{\text{rev}}}{T}
\]
For a reversible process.
Entropy is a state function — its change depends only on initial and final states, not the path.
7. Entropy and Reversible vs Irreversible Processes
For Reversible processes :
\[
\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}
\]
For Irreversible (real) processes:
\[
\Delta S > \int \frac{\delta Q}{T}
\]
Entropy increases in spontaneous processes — never decreases for isolated systems.
8. The Second Law and the Arrow of Time
The second law introduces time asymmetry — systems evolve from less probable to more probable states.
Entropy gives a direction to time. Systems evolve toward higher entropy states:
- Ice melts
- Gases expand
- Energy dissipates
These processes are irreversible, even though microscopic laws are time-symmetric.
9. Entropy in Statistical Mechanics
Boltzmann connected entropy to microstates:
\[
S = k_B \ln \Omega
\]
Where:
- \(Omega\): number of microstates corresponding to a macrostate
- \(k_B\): Boltzmann constant
This formulation bridges classical thermodynamics with statistical physics.
10. Thermodynamic Potentials
Useful for systems with constraints (e.g., constant pressure, temperature).
- Internal energy: \( U = U(S, V) \)
- Enthalpy: \( H = U + PV \) (constant pressure processes)
- Helmholtz Free Energy: \( F = U – TS \) (constant temperature and volume)
- Gibbs Free Energy: \( G = H – TS \) (constant pressure and temperature)
Each potential is minimized under different constraints, in other words systems minimize appropriate potentials in equilibrium.
11. Maxwell’s Relations
Derived from thermodynamic potentials via Legendre transformations:
Example:
\[
\left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V
\]
These provide useful identities between thermodynamic variables.
There are four such relations. They relate otherwise hard-to-measure quantities using measurable ones.
12. Real-World Applications
Thermodynamics and entropy are central in:
- Heat engines and refrigerators (Carnot cycle)
- Chemical reactions (Gibbs free energy and spontaneity)
- Black hole entropy and thermodynamics
- Biological processes (entropy production, ATP hydrolysis)
- Information theory (Shannon entropy is thermodynamic entropy analog)
13. Conclusion
Thermodynamics provides deep insights into the behavior of matter and energy. Entropy, once viewed as abstract, is now seen as the fundamental driver of irreversible processes and the bridge between order and chaos.
Understanding entropy is not only key to mastering thermodynamics, but also to appreciating the deep structure of the universe — from steam engines to black holes to information theory.
