Thermodynamics

Table of Contents

Notation

  • thermo_88c6e1c008509ea8de35316286b272c4107bed01.png is heat / thermal energy
  • thermo_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png 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

thermo_cf30c9d74b603ec843ed512465ec88bbbb2b4662.png

Maxwell-Boltzmann distribution

thermo_b0e5259d394d3c47e203ec91fa1c1722ec3b4abe.png

Compressibility

thermo_b4df7456ae306305cde4756c5422b444877d99b0.png

Bulk modulus

thermo_49411d49635b6daba6648d4f74a22fe21caf230f.png

Heat capacity

thermo_a4aff969d8d624f8a2e67fe5f8c62b108e57ae97.png

Thermal expansivity

thermo_00fa5a2092fd6e08953956492da63ec140010619.png

Bernoulli Equation

thermo_90f044cff27da7af250a2c81dd5714942f0738eb.png

Ideal gas law

thermo_f97de35db1615616c1bc7a59a0ffeca77fdede3a.png

Work (reversible process)

"Derivation"

thermodynamics_work_reversible_fig.png

thermo_e2e81ab2cb1925493969fcaeda9d2f61abf8028e.png

which gives us

thermo_9b06823c10dd6f2826c53886adecfe557ae7445e.png

Differential

thermo_60990579c7f7df5039bada8172b1e41c8e28a1a8.png

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

TODO Boltzmann Distribution

thermo_d9cbd7fb884cfd6ec0958decd1535f7224ba5809.png

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 thermo_72103cbefc02cdcd1cdaffca00f896bdf855b8e4.png itself, i.e. don't want to included all possible states which can take on the this energy):

thermo_82bc41263115ad8705b5d55e7849725deb49a5a7.png

Notation

  • thermo_55f5662fd23a57bfce820288cf77c6e6e3f5464d.png is the expected number of particles in the energy level indexed by thermo_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png
  • thermo_e10e2b430f95617381cdd6d6b52aed29fb971dff.png total number of particles
  • thermo_72103cbefc02cdcd1cdaffca00f896bdf855b8e4.png the energy of the ith energy level
  • thermo_5b75907baddc1f20479fae69727ff4819121a7ed.png is the degeneracy of the ith energy level with energy thermo_ec92215b33c48a62ab467c2d83ab5ffac34eead8.png (this is always an integer), which corresponds to the number of quantum mechanical observables which can take on an energy of thermo_ec92215b33c48a62ab467c2d83ab5ffac34eead8.png

Derivation

Source

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 thermo_d5c68033ab14423ef7e55781f5e969f5f114e984.png particles into the energy level thermo_72103cbefc02cdcd1cdaffca00f896bdf855b8e4.png.

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

thermo_1f26d0d63fc94d8579b515d66954f1e611aba0f2.png

where each of the factor on the first line represents the number of possible ways to partition all thermo_e10e2b430f95617381cdd6d6b52aed29fb971dff.png particles into groups of thermo_3ef591d1a6b4a1809496ae120e8d985d204abc60.png, then the rest of the particles partitioned into bins of size thermo_d552ce98eee30679c545a46f61e8ca745914d0a8.png, and so on. I.e. we end up counting the number of possible ways to partition all thermo_e10e2b430f95617381cdd6d6b52aed29fb971dff.png particles into the different "energy-bins" with thermo_d5c68033ab14423ef7e55781f5e969f5f114e984.png 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 thermo_ca18b2f1994310535527443f28a711c812a73b88.png, and we can write

thermo_9328c972bd2ca849d623decaf236f8890121ca28.png

Which is just the multinomial coefficient , the number of ways to arrange thermo_e10e2b430f95617381cdd6d6b52aed29fb971dff.png objects into thermo_094b02afce734f4ce51933d0093ef3d2da9f8123.png bins, ignoring order / permutations.

Now we want to take into account the possible degeneracy degree thermo_03d96c52df5ea6a510607e2260e0745d11e4bd85.png of each energy level. The thermo_d5c68033ab14423ef7e55781f5e969f5f114e984.png corresponding to the energy thermo_72103cbefc02cdcd1cdaffca00f896bdf855b8e4.png can then be arranged in thermo_5d8959b98729a7c8e75b3d1a6f6723a6af6e1dbb.png ways within the energy-level (since for each particle with energy thermo_72103cbefc02cdcd1cdaffca00f896bdf855b8e4.png can go in either of the thermo_5b75907baddc1f20479fae69727ff4819121a7ed.png "sub-bins"). Thus,

thermo_beced9ead536abb10162b4429427dee8fc41adc3.png

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 thermo_fe7108f38f2db55db9cb7eee3f63077437a2e510.png :

thermo_eab4831dd7acd5373702d6f6cf2f7a40c6a6f28a.png

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

Adiabatic expansion

thermo_8ab5ddb0a4b32d2947c73e60200d1a5eff44280e.png

Clapeyron equation

thermo_9cf61b2d41a133bb4ed055c6ff1f60021be6a472.png

Which is basically saying that when we're moving between two phases, the change in pressure thermo_fefe9e556d399665a26a37824ec578cbffb0cabe.png wrt. temperature thermo_b3731d37cd447bd6c31809075a6be43b3d0b04ec.png is equal to the ratio between the change in entropy and volume.

TODO Derivation

Van der Waals equation

thermo_10c3e3e26f3c5fe199beb50dddadcab55e0e7d11.png

TODO Derivation

Lennard-Jones potential

thermo_9afa60e466513d3b9804b4f1726d85fff8560a00.png

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

Ionic bonding

thermo_14b565863303b6035d4be8e5e174d0786b6c7e0d.png

  • 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

thermo_4382ac7175546cc78c6d0a8c418788473fb782b4.png

where thermo_03d96c52df5ea6a510607e2260e0745d11e4bd85.png is the seperation energy and thermo_24e22d2561a069baa75e64a19861206f150f93df.png 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

thermo_eeb9bba802f9d6b4cef6208c573a51e2b4844cf9.png

Reynolds number

thermo_3495eaf0ad84a83b73e9d504172141e858940f70.png

Braggs Law

thermo_c4b152ab02e2948ba1383c4937304405acd96f3e.png

Miller indices

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

It works as follows:

If the plane intercepts at:

  • thermo_60019aa69125e72d256a67bd7e024b740025e239.png along the x-axis
  • thermo_97f451b249a3bd189b2ba0db19b8768f01538f76.png along the y-axis
  • thermo_bdd2ebd9b14864d01d3bf1ac29e2b25223a27615.png along the z-axis

the the lattice has Miller indices thermo_b629674635a032b42f56641b087cabffcdc27e83.png

Entropy

thermo_53b976ebb2b0e7cfe897389c2eb1e1b74cd52d78.png

Or with a fixed energy, for a system in a particular macrostate , defined by the number of associated micro-states thermo_cdfd3a815ed27b71796fe6289ce253c6c19a2803.png :

thermo_9d66ec34e4608bbd93f9d16e2fb1578cb8765085.png

Internal energy

thermo_2445eb67fb4849bd0bf227a23b90a7acf39ca152.png

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

Gibbs function

The function

Is a state-function.

thermo_85fa62276e76894eb70e1ffa3f9701ef6df93858.png

and

thermo_a6170eb3b0f843f8f5e2aefa04ed742cf5da70ee.png

Criterion for spontanous process

thermo_9b6aaefa4e5a92515e82ee667e65ce2a62927ccd.png

and in equilibrium

thermo_ba9c2a3bff1fc1b2db7796592b17ca52f92c68ee.png

Enthalpy

Is a state-function.

thermo_2f6baca07660cd88c89869b616052936bf43cd47.png

and

thermo_01aae6b8221e87e08c1939705b0e9b57681abdad.png

Helmholtz function

thermo_aa0b55c98a9bca57a676998ab3911ae0842c4b4d.png

and

thermo_672a0bd791bab45fcc4d6fa756deb8d8cf3e78f9.png

Second Law of Thermodynamics

thermo_ec27a990f5c12924426a19e44ce27dddbeca1adf.png

for any process, in total. That is,

thermo_e8ff8be21d3ad5f9e0e6927cc6c6ba6ab72a3812.png

Maxwell's relations

thermo_e15408da7b8a415de57bca07ff766d9a89100b62.png

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 thermo_fd9fe5003b5dfa76cd57dccf213781e787a553d0.png wrt. thermo_79dc7b66c784a82e557ef3df794e82b67a292b17.png 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 thermo_a374079c5d50e82206b8c0c71faeb27cbc866e78.png.