# Ideal gas law

 Zazzle.com ideal gas law print for T-shirts and mugs. [4]
In gas laws, the ideal gas law is an equation of state describing the behavior of a body of ideal or perfect gas, defined by the expression:

$~ PV = nRT ~$

where P is the pressure of the body of gas, V the volume, n as the number of mols, R the gas constant, and T the temperature.

History
The first verbalized statement of the ideal gas law seems to have been made in parts in 1738 by Dutch-born Swiss mathematician Daniel Bernoulli.

In 1834, French engineer Emile Clapyron was defining the gas law as such:

$~ Pv = R(267 + t) ~$

which he says is Mariotte’s law (PV = k, at constant temperature) combined with that of Gay-Lussac's law (P = kT, at constant volume). [5]

In 1850, German physicist Rudolf Clausius, in his "On the Moving Force of Heat", was defining the gas law as such:

$pv = R(a + t) \,$

which he says is "the combined laws of Mariotte and Gay-Lussac" and in which he derives a value for the constant a of:

$a = 273 \,$

with t the temperature in degrees centigrade. In reference to this gas law, Clausius introduces the notion of dependent variables and independent variables, to the effect that in a three variable equation, i.e. p, v, and t, one can treat any one of these three magnitudes, which then is called the "dependent variable", as a function of the other two, which are then called the "independent variables".

Classical version | Molar version | Avogadro's law
The first statement of the modern version of the gas law, with the particle count measure 'n' in the formula, seems to have been done in the 1893 Theoretical Chemistry from the Standpoint of Avogadro’s Rule and Thermodynamics by German chemist Walther Nernst, in which he integrated of Avogadro's rule (1811) into modern chemical thermodynamics, to the result that he states that if n g-mol of different gases are under the pressure P, filling a volume V, at an absolute temperature T, then in the sense of Dalton's law: [1][8]

$~ PV = nRT ~$

In 1897, German physicist Max Planck was also using the same modern-version formula in his chapter on "Molecular Weight". [2] This expression became commonplace with its use in the classic 1923 textbook Thermodynamics and the Free Energy of Chemical Substances by American physical chemists Gilbert Lewis and Merle Randall. [3]

Statistical version | Molecular version | Boltzmann's constant
If n, the number of g-moles, gram moles, or moles, of gas, is set equal to: [9]

where N is the actual number of molecules and NA is Avogadro's number, and one defines the commonly used ratio of the gas constant R to Avogadro's number to be equal to a new so-called constant of nature:

$k_{B} = \frac{R}{N_{\rm A}}\,$

where the subscript "B" is in honor of Austrian physicist Ludwig Boltzmann, via substitution, the classical "molar-based" ideal gas law can also be written in the alternative so-called "molecular" or statistical mechanical form:

or

with the subscript "B" (for Boltzmann) assumed, where kB is called the Boltzmann constant, a constant that was first calculated in the 1900 work of German physicist Max Planck.

Other
In 1936, Italian-born American physicist Enrico Fermi was defining what he called the "equation of state for the ideal gas or perfect gas" as follows: [6]

$pV = \frac{m}{M} RT \,$

where m is grams of a gas, and M is molecular weight, and expression which Fermi says includes Boyle's law, Gay-Lussac's law, and Avogadro's law. [7] If, according to Fermi, the number of grams of the gas equals the molecular weight of the gas, then:

$m = M \,$

which he says is the condition for what is called the "gram-molecule" amount of a gas. Hence, the gram molecule ideal gas law becomes:

$pV = RT \,$

Of note, the inconsistency with which authors upper or lower case letters for pressure volume and temperature is a bit puzzling, as there seems to be some type of unwritten rule practiced, albeit inconsistently.

Social ideal gas law

References
1. Nernst, Walther. (1893). Theoretical Chemistry: from the Standpoint of Avogadro’s Rule and Thermodynamics (PV=nRT, pgs. 32, 41). MacMillan and Co.
2. Planck, Max. (1897). Treatise on Thermodynamics (ch. 2: Molecular Weight, pgs. 23-33
). Longmans, Green and Co.
3. Lewis, Gilbert N. and Randall, Merle. (1923). Thermodynamics and the Free Energy of Chemical Substances (pg. 63). McGraw-Hill Book Co., Inc.

4. Ideal gas law (T-shirt) – Zazzle.com.
5. Clapeyron, Émile. (1834). “Memoir on the Motive Power of Heat”, Journal de l’Ecole Polytechnique. XIV, 153 (and Poggendorff's Annalender Physick, LIX, [1843] 446, 566).
6. Fermi, Enrico. (1936). Thermodynamics (pg. 8). Prentice Hall.