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The Van 't Hoff equation relates the change in the equilibrium constant, K eq, of a chemical reaction to the change in temperature, T, given the standard enthalpy change, Δ r H ⊖, for the process. The subscript r {\displaystyle r} means "reaction" and the superscript ⊖ {\displaystyle \ominus } means "standard".
The Boltzmann constant (k B or k) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. [2] It occurs in the definitions of the kelvin (K) and the gas constant , in Planck's law of black-body radiation and Boltzmann's entropy formula , and is used in ...
for reactions in solution and unimolecular gas reactions A = (ek B T/h) exp(ΔS ‡ /R), while for bimolecular gas reactions A = (e 2 k B T/h) (RT/p) exp(ΔS ‡ /R). In these equations e is the base of natural logarithms, h is the Planck constant, k B is the Boltzmann constant and T the absolute temperature. R′ is the ideal gas constant. The ...
= , where k B is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability. d S = δ Q T {\displaystyle dS={\frac {\delta Q}{T}}} , for reversible processes only
Boltzmann's entropy formula—carved on his gravestone. [1]In statistical mechanics, Boltzmann's entropy formula (also known as the Boltzmann–Planck equation, not to be confused with the more general Boltzmann equation, which is a partial differential equation) is a probability equation relating the entropy, also written as , of an ideal gas to the multiplicity (commonly denoted as or ), the ...
where ΔU is the change in internal energy, K is the equilibrium constant of the reaction, R is the universal gas constant, and T is thermodynamic temperature. Based on experimental work, in 1889, Svante Arrhenius proposed a similar expression for the rate constant of a reaction, given as follows:
Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times the Boltzmann constant k B = 1.38 × 10 −23 J K −1. The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero provided that its ground state is unique, because ln(1) = 0.
The gas constant occurs in the ideal gas law: = = where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. R specific is the mass-specific gas constant. The gas constant is expressed in the same unit as molar heat.