Statistical physics is a branch of physics that seeks to explain the macroscopic properties of matter in terms of the microscopic behaviour of its constituent particles. At the heart of this fascinating field lies the study of thermodynamic functions, which play a crucial role in connecting the microscopic world of particles to the macroscopic world described by classical thermodynamics.
Background:
Thermodynamics, a classical branch of physics, deals with the macroscopic properties of matter and energy. It relies on a set of variables called state functions, including internal energy (U), temperature (T), pressure (P), and volume (V). However, classical thermodynamics doesn’t provide insights into the underlying microscopic processes responsible for these macroscopic properties.
Statistical Physics:
Statistical physics, on the other hand, delves into the microscopic details of a system by considering the statistical behaviour of its particles. It aims to bridge the gap between the microscopic and macroscopic worlds, providing a deeper understanding of thermodynamic phenomena.
Ensemble Theory:
A fundamental concept in statistical physics is the ensemble theory, which involves considering all possible states a system can occupy. The most commonly used ensembles are the micro-canonical, canonical, and grand canonical ensembles. These ensembles provide a statistical framework for calculating the probabilities of a system existing in various states.
Partition Function:
The cornerstone of statistical physics is the partition function (Z), a function that encapsulates the statistical information about a system. It serves as a bridge between microscopic and macroscopic properties, connecting the probabilities of different states to thermodynamic quantities. The partition function is essential for calculating various thermodynamic functions.
There are four fundamental thermodynamic functions:
- Internal energy U,
- Enthalpy H,
- Helmholtz free energy F, and
- Gibbs free energy G.
1. INTERNAL ENERGY (U)
Energy of a system is its capacity to do work. If work is done on the system, its energy will increase. If work is done by the system, its energy will decrease. Total energy of a system consists of the external energy & the internal energy. The total energy of a system is the sum of kinetic and gravitational potential energy, and this Total energy is the sum of all different types of energies a body can have.
The change in the internal energy of a system is the sum of the heat transferred and the work done.
U = Q − W
ENTROPY:
Entropy is the measure of randomness or disorder in a system. More the disorder, more is entropy, less the disorder, less is entropy. It is also the measure of the unavailable energy for work. If you increase temperature, you increase entropy. More energy put into a system excites the molecules and the amount of random activity. As a gas expands in a system, entropy increases. When a liquid becomes a gas, its entropy increases.
PROPERTIES OF ENTROPY :
- It is a thermodynamic function.
- It is a state function. It depends on the state of the system and not the path that is followed.
- It is represented by S but in the standard state, it is represented by S°.
- Its SI unit is J/Kmol.
- Its CGS unit is cal/Kmol.
- Entropy is an extensive property which means that it scales with the size or extent of a system.
- The greater disorder will be seen in an isolated system, hence entropy also increases.
- In chemical reactions, if reactants break into more number of products, entropy also increases.
- A system at higher temperatures has greater randomness than a system at a lower temperature. From these examples, it is clear that entropy increases with a decrease in regularity.
- Entropy order: gas > liquid > solids.
2. Enthalpy(H) :
Enthalpy, the sum of internal energy and the product of the pressure and volume of a thermodynamic system. Enthalpy is an energy-like property or state function. It has the dimensions of energy (and is thus measured in units of Joules or ergs), and its value is determined entirely by the temperature, pressure, and composition of the system and not by its history.
In symbols, the enthalpy, H, equals the sum of the internal energy, U, and the product of the pressure, P, and volume, V, of the system :
H = U + PV
3. Helmholtz Free Energy (F) :
The Helmholtz free energy is a thermodynamic potential defined in terms of the partition function. It is particularly useful for systems at constant temperature and volume. The relationship between the Helmholtz free energy and other thermodynamic quantities allows scientists to make predictions about a system’s behaviour.
The Helmholtz energy is defined as
F = U − TS
This is the free energy for gas phase reactions in isolated systems kept at constant volume.
Eg. Combustion of fuel in Bomb Calorimeter is carried out at Constant Volume. Hence the heat of the reaction q is a direct measure of the free energy change, i.e.
q = ∆E
If U is the internal energy of the system, PV is the product of pressure & volume, and TS is the product of temperature & entropy, then, F = U – TS Thus, under constant volume, the free energy of the system is called Helmholtz free energy F.
4. Gibbs Free Energy (G):
The Gibbs free energy is another key thermodynamic potential, particularly relevant for systems at constant temperature and pressure. It incorporates information about the system’s entropy and enthalpy, providing insights into spontaneity and equilibrium conditions.
G = H − TS
or
G = U + PV − TS
Where, U is internal energy (SI unit: Joule)
P is pressure (SI unit: Pascal)
V is volume (SI unit: m3)
T is temperature (SI unit: Kelvin)
S is entropy (SI unit: Joule/Kelvin)
H is the enthalpy (SI unit: Joule)
Applications:
Statistical physics has widespread applications, from understanding the behaviour of gases and liquids to explaining phase transitions and the properties of complex systems. It provides a foundation for diverse fields, including chemistry, materials science, and astrophysics.
Quantum Statistical Mechanics:
In quantum statistical mechanics, statistical physics merges with quantum mechanics to describe systems at the atomic and subatomic levels. The principles of quantum statistics and the density matrix become essential tools in understanding the behaviour of particles obeying quantum principles.
Conclusion:
The study of thermodynamic functions in statistical physics enriches our comprehension of the physical world by linking the microscopic behaviour of particles to the macroscopic observables described by classical thermodynamics. This interdisciplinary approach has paved the way for numerous technological advancements and continues to be a fertile ground for research, uncovering the intricacies of matter and energy on both the smallest and largest scales.
