## Important equations in An Introduction to Thermal Physics

A reviewer has pointed out that the most important equations in the text should be highlighted. Of course, different readers will have different opinions on which are the most important equations. Here, however, is my personal list. Please don't accept this list uncritically; instead, think about whether you believe each of these equations is really important, and consider whether other equations should be included on your personal list.

• 1.5 (ideal gas law)
• 1.23 (equipartition theorem)
• 1.24 (first law of thermodynamics)
• 1.29 (compression work)
• 1.44 (heat capacity at constant volume)
• 1.50 (definition of latent heat)
• 1.51 (definition of enthalpy)
• 1.56 (heat capacity at constant pressure)
• 1.60 (Fourier heat conduction law)
• 1.69 (shear stress in terms of viscosity)
• 1.70 (Fick's law of diffusion)

• 2.6 (multiplicity of a two-state system)
• 2.9 (multiplicity of an Einstein solid)
• 2.14 (Stirling's approximation)
• 2.16 (Stirling's less accurate approximation for ln N!)
• 2.40 (multiplicity of a monatomic ideal gas)
• 2.45 (definition of entropy)
• 2.49 (entropy of an ideal gas)

• 3.5 (theoretical definition of temperature)
• 3.17 (change in entropy in terms of heat)
• 3.32 (magnetization of an ideal two-state paramagnet)
• 3.39 (pressure in terms of entropy)
• 3.46 (thermodynamic identity)
• 3.48 (change in entropy in terms of heat)
• 3.55 (definition of chemical potential)
• 3.58 (generalized thermodynamic identity)

• 4.1 (definition of efficiency)
• 4.5 (efficiency limit for an engine)
• 4.6 (definition of COP)
• 4.9 (limit on COP)

• 5.2 (Helmholtz free energy)
• 5.3 (Gibbs free energy)
• 5.5 (change in F related to work)
• 5.8 (change in G related to other work)
• 5.18 (thermodynamic identity for H)
• 5.20 (thermodynamic identity for F)
• 5.23 (thermodynamic identity for G)
• 5.35 (G related directly to chemical potential)
• 5.37 (generalization of 5.35 to mixtures)
• 5.40 (chemical potential of an ideal gas in terms of pressure)
• 5.46 (Clausius-Clapeyron relation, in terms of entropy)
• 5.47 (Clausius-Clapeyron relation, in terms of latent heat)
• 5.49 (van der Waals equation of state)
• 5.61 (Gibbs free energy of an ideal mixture)
• 5.69 (chemical potential of solvent in a dilute solution)
• 5.72 (chemical potential of a dilute solute)
• 5.78 (van't Hoff's formula for osmotic pressure)
• 5.86 (Raoult's law for the vapor pressure of a dilute solution)
• 5.90 (shift in boiling temperature of a dilute solution)
• 5.102 (equilibrium condition in terms of chemical potentials)
• 5.107 (equilibrium constant defined)
• 5.108 (law of mass action)
• 5.130 (Saha equation for ionization of hydrogen)

• 6.8 (Boltzmann distribution)
• 6.10 (partition function)
• 6.18 (general formula for average values)
• 6.25 (average energy in terms of Z)
• 6.40 (equipartition theorem)
• 6.50 (Maxwell speed distribution)
• 6.56 (relation between F and Z)
• 6.69 (total Z for a collection of noninteracting, distinguishable systems)
• 6.70 (total Z for a collection of noninteracting, indistinguishable particles)
• 6.80 (quantum length)
• 6.82 (translational partition function of a single molecule)
• 6.83 (quantum volume)
• 6.85 (partition function of an ideal gas)
• 6.93 (chemical potential of an ideal gas)

• 7.6 (Gibbs distribution)
• 7.7 (grand partition function)
• 7.23 (Fermi-Dirac distribution)
• 7.28 (Bose-Einstein distribution)
• 7.33 (Fermi energy defined)
• 7.39 (Fermi energy of a free electron gas)
• 7.47 (total energy of a free electron gas)
• 7.51 (density of states of a free electron gas)
• 7.72 (Planck distribution)
• 7.84 (Planck spectrum)
• 7.86 (total energy of a photon gas)
• 7.97 (Stefan's law of blackbody radiation)
• 7.112 (total energy of phonons in Debye approximation)
• 7.126 (condensation temperature of a free Bose gas)

• 8.7 (configuration integral for a nonideal gas)
• 8.23 (cluster expansion for the configuration integral)
• 8.38 (interaction energy in the Ising model)
• 8.39 (partition function of the Ising model)
• 8.43 (Ising model partition function in one dimension)
• 8.50 (implicit solution of Ising model in mean field approximation)

• A.2 (Einstein relation for photon energy)
• A.3 (de Broglie relation for particle wavelength)
• A.7 (uncertainty principle)
• A.11 (energies of a particle in a box)
• A.17 (energies of a quantum harmonic oscillator)
• A.19 (energies of a hydrogen atom)
• A.21 (rotational energies of a diatomic molecule)
• A.23 (energies of a dipole in a magnetic field)

• B.6 (integral of a Gaussian)
• B.8 (integral of a Gaussian times x^2)
• B.12 (definition of the gamma function)
• B.14 (recursion relation for the gamma function)
• B.16 (Stirling's approximation)
• B.18 (Stirling's approximation for ln n!)
• B.25 (surface "area" of a hypersphere)
• B.36 (quantum statistics integrals in terms of the zeta function)
Goodness, that's a long list--more than a hundred important equations! Although these "important" equations occur on average only once every four pages, you may also like to see a shorter list of the really important equations. Here, then, are the ones that I consider to be absolutely central; you may wish to write them on the inside cover for reference. Again, think carefully about whether your personal list should be different from mine.
• 1.5 (ideal gas law)
• 1.23 (equipartition theorem)
• 1.24 (first law of thermodynamics)
• 1.51 (definition of enthalpy)
• 2.45 (definition of entropy)
• 3.58 (generalized thermodynamic identity)
• 5.2 (Helmholtz free energy)
• 5.3 (Gibbs free energy)
• 6.8 (Boltzmann distribution)
• 6.10 (partition function)
• 7.6 (Gibbs distribution)
• 7.7 (grand partition function)
• 7.23 (Fermi-Dirac distribution)
• 7.28 (Bose-Einstein distribution)