# Thermodynamics

## Table of Contents

- Notation
- Equations / Laws
- Zeroth Law
- First Law of Thermodynamics
- Maxwell-Boltzmann distribution
- Compressibility
- Bulk modulus
- Heat capacity
- Thermal expansivity
- Bernoulli Equation
- Ideal gas law
- Work (reversible process)
- TODO Boltzmann Distribution
- Adiabatic expansion
- Clapeyron equation
- Van der Waals equation
- Lennard-Jones potential
- Ionic bonding
- Surface energy
- Capillary rise
- Viscosity
- Reynolds number
- Braggs Law
- Miller indices
- Entropy
- Internal energy
- Gibbs function
- Enthalpy
- Helmholtz function
- Second Law of Thermodynamics
- Maxwell's relations

- Definitions

## Notation

- is
**heat / thermal energy** - is
**internal energy**

## Equations / Laws

### Zeroth Law

If each of two systems is in thermal equilibrium with a third system they are in thermal equilibrium with each other.

### First Law of Thermodynamics

### Maxwell-Boltzmann distribution

### Compressibility

### Bulk modulus

### Heat capacity

### Thermal expansivity

### Bernoulli Equation

### Ideal gas law

### Work (reversible process)

#### "Derivation"

which gives us

#### Differential

where work is defined as the work done ON the system *by its surroundings* .

### TODO Boltzmann Distribution

In the case where we have *no* degeneracy (or you simply care about the probability of specific state corresponding to some energy $ε$_{i} instead of the probability of the energy itself, i.e. don't want to included all possible states which can take on the this energy):

#### Notation

- is the expected number of particles in the energy level indexed by
- total number of particles
- the energy of the ith energy level
- is the
*degeneracy*of the ith energy level with energy (this is always an integer), which corresponds to the number of quantum mechanical observables which can take on an energy of

#### Derivation

In our deduction we're going to consider a microcanonical ensemble.

To begin with, we ignore the problem of degeneracy; i.e. we assume there is only one way to put particles into the energy level .

The number of possible ways to "bin" the particles in the different energies:

where each of the factor on the first line represents the number of possible ways to partition all particles into groups of , then the rest of the particles partitioned into bins of size , and so on. I.e. we end up counting the number of possible ways to partition all particles into the different "energy-bins" with particles in the ith "energy-bin".

Because we're "exhausting" the number of particles, i.e. we keep "binning" until we have no particles left (each particle MUST be assigned a bin), then , and we can write

Which is just the **multinomial coefficient** , the number of ways to arrange objects into bins, ignoring order / permutations.

Now we want to take into account the possible degeneracy degree of each energy level. The corresponding to the energy can then be arranged in ways within the energy-level (since for each particle with energy can go in either of the "sub-bins"). Thus,

BUT when doing this we're treating the particles as *distinguishable* , i.e. the order of arranging into the "sub-bins" matter, which leads to a "invalid" entropy (not *extensive* , which means that the entropy is not proportional to the amount of substance, which it really ought to be). This is called the Gibbs paradox. This leads to the Bose-Einstein expression for :

NOT FINISHED YET DUE TO REALIZING THIS NOT FITTING AS WELL INTO AN ANKI CARD AS I HAD HOPED.

### Adiabatic expansion

### Clapeyron equation

Which is basically saying that when we're moving between two phases, the change in pressure wrt. temperature is equal to the ratio between the change in **entropy** and **volume**.

#### TODO Derivation

### Van der Waals equation

- Weak force
- Isotropic
- Noble gases and neutral molecules
- Potential given by Lennard-Jones potential

#### TODO Derivation

### Lennard-Jones potential

where the first term is the *repulsion* and the second term is the *Van der Waals* forces.

### Ionic bonding

- assumes single and complete ionisation, which is why we can consider the pairwise potential as the
*Coloumb force*(1st term) together with a*repulsive*force (2nd term) - attraction via electron exchanges to produces filled orbitals results in charged particles that attract

### Surface energy

where is the *seperation energy* and is the change in *nearest neighbours* .

#### TODO Derivation

### Capillary rise

### Viscosity

**Viscosity** characterizes the sheer forces that exist in a moving fluid.

For a **gas** we have

### Reynolds number

### Braggs Law

### Miller indices

Index system tused to identify equally-spaced parallel planes intersecting lattice points.

It works as follows:

If the plane *intercepts* at:

- along the x-axis
- along the y-axis
- along the z-axis

the the lattice has **Miller indices**

### Entropy

Or with a fixed energy, for a system in a particular *macrostate* , defined by the number of associated *micro-states* :

### Internal energy

This is true regardless of whether or not the process is reversible!

### Gibbs function

#### The function

Is a **state-function**.

and

#### Criterion for spontanous process

and in *equilibrium*

### Enthalpy

Is a **state-function**.

and

### Helmholtz function

and

### Second Law of Thermodynamics

for any process, in total. That is,

### Maxwell's relations

## Definitions

### Temperature

The temperature of a system is a property that determines whether or not that system would be in thermal equilibrium with other systems.

### Equilibrium state

An equilibrium state is one in which all the bulk physical properties do not change with time and are uniform throughout the system

### Young's Modulus

**Young's modulus** is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression.

### Systems

#### Closed system

Cannot exchange matter with its surroundings, but may exchange energy.

#### Isolated system

No exchange of any material or energy with surroundings.

### Reversible process

A process where every step for the system *and* its surroundings can be reversed. A reversible process involves a series of equilibrium states.

Path-independent integral over wrt. for an ideal gas (this is true for any *state variables* )

### Ensembles

#### Microcanonical ensemble

A **microcanonical ensemble** is a mechanical ensemble which represents the *possible states* of the particles in a system where the *total energy of the system is exactly known* .

#### Grand canonical ensemble

A **grand canonical ensemble** is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that is being maintained in thermodynamic equilibrium (thermal and chemical) with a reservoir.

#### Canonical ensemble

A **canonical ensemble** is the statistical ensemble that is used to represent the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchage energy with the heat bath, so that the states of the system will differ in total energy.

### Adiabatic process

A **adiabatic** process is a *reversible* and *adiathermal* (thermally isolated) process.

### Bonding

#### Covalent bonding

Strong and directional based bonding based on **electron-sharing** .

#### Metallic bonding

Attraction of like species via sharing of **free** electrons.

It's similar to *covalent* , but the electrons are completely delocalized and free to move.

#### Hydrogen bonding

Hydrogen atom itself is shared between atoms.

### Isotropic

A process is called **isotropic** if .