# Derivation of the Ideal Gas Law

Nov 11, 2007 - gas law. Undergraduate and graduate physical chemistry and physics textbooks, for ... Boyle's law states that the volume of fixed amoun...
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Derivation of the Ideal Gas Law Alexander Laugier IdPCES - Centre d’affaires PATTON, 6, rue Franz Heller, bât. A, F35700 Rennes, France

József Garai* Department of Earth Sciences, Florida International University, University Park, Miami, FL 33199; *[email protected]

In the gas phase the relationship among the thermodynamic variables, pressure ( p), volume (V ), and temperature (T ), for a one mole gas is given in the form This expression originally was proposed by Horstmann (1). The more general form of eq 1 is (2)

pV = n RT

where n is the number of moles. Equation 2 is called the ideal gas law. Undergraduate and graduate physical chemistry and physics textbooks, for example, Petrucci et al. (2), Atkins (3), McMurry and Fay (4), describe the behavior of an ideal gas by introducing Boyle’s and Charles’s laws along with Avogadro’s principle, and they state that combining these laws gives the ideal gas law. This statement gives the impression that a simple substitution should give the desired result, which is far from the truth. The derivation of the ideal gas law given by Cage (5), Bosch et al. (6), and Levine (7) requires rigorous mathematical treatment of the gas laws. In this article we present a simple solution that does not require the knowledge of higher mathematics, although the multivariable nature of the gas laws is emphasized. An alternative derivation, which combines the phenomenological laws with the use of advanced mathematics, is given in the Supplemental Material.W The Gas Laws Boyle’s law states that the volume of fixed amount of gas at constant temperature is inversely proportional to the pressure: 1 p

or

pV = const

n ,T

(3)

n,T

Charles and Gay-Lussac found for a fixed amount of gas under constant pressure that the volume varies linearly with the temperature: V ∝ T

n, p

or V = const T

n, p

(4)

A similar relationship has been established for the pressure and the temperature at constant volume: p ∝ T

n,V

or p = const T

n,V

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p,T

or V = const n

p,T

(6)

The constants in these empirical laws, eqs 3–6, depend on the variables that are held constant in the experiments. Derivation of the Ideal Gas Law Boyle’s law, Charles’s law, and Avogadro’s principle can be rewritten as p V = f (n, T )

(7)

V = T g (n, p )

(8)

p = T ᐉ(n, V )

(9)

V = n h ( p, T )

(10)

where f, g, ᐉ, and h are arbitrary functions of the mole number and the temperature, the mole number and the pressure, the mole number and the volume, and the pressure and the temperature, respectively. Equations 7 to 10 are the general mathematical forms of the macroscopic laws of Boyle, Charles, and Avogadro. Combining eq 7 with eqs 9 and 10 gives the following general expression for the ideal gas law:

pV = f (n, T ) = nT ᐉ (n,V ) h ( p, T )

(11)

According to Avogadro’s principle, the volume is proportional to the mole number at constant pressure and temperature leading to the expression V ∝ n

p,T

pV = const n = n h ( p,T ) ᐉ(n,V ) (12) T p ,T

Since eq 12 is used at constant pressure and temperature it implies that h ( p,T ) ᐉ (n,V ) = const

p ,T

⇒ ᐉ (n,V ) = const (13)

Equation 11 can then be written as

pV = const nT h ( p,T )

(14)

The constant in eq 14 can be removed by redefining the function h as

(5)

Avogadro suggested that the number of molecules or atoms occupying a volume under the same conditions of pressure

1832

V ∝ n

(1)

pV = RT

V ∝

and temperature is independent from the chemical identity of the gas:

const h → h

(15)

pV = nT h ( p , T )

(16)

So eq 14 becomes

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Charles’s law states that the pressure is proportional to temperature at constant mole number and constant volume. Applying this law to eq 16 gives

p ∝ T

n,V

pV = const T = T h ( p,T ) n

n ,V

(17)

Resulting from eq 17 h ( p,T ) = const

n,V

⇒ h ( p,T ) = const

(18)

Acknowledgment The authors like to thank to Jeffrey Joens for reading the manuscript and making comments. W

Supplemental Material

An alternative derivation that combines the phenomenological laws with the use of advanced mathematics is available in this issue of JCE Online. Literature Cited

Thus, eq 16 becomes (19)

pV = const nT

Identifying the constant as universal gas constant, R, recovers the ideal gas law, eq 2: (20)

pV = nRT Conclusion

The derivation of the ideal gas law should not be left to the students. Either it should be provided in the textbooks or a citation should be given; otherwise students get the impression that a simple substitution should give the desired result. One example of such derivation emphasizing the multivariable nature of the gas laws is presented.

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1. Horstmann, A. Ann. Phys. 1873, 170, 192. 2. Petrucci, R. H.; Harwood, W. S.; Herring, G. E.; Madura, J. General Chemistry: Principles and Modern Application, 9th ed.; Prentice-Hall, Inc.: Upper Saddle River, NJ, 2006. 3. Atkins, P. Physical Chemistry, 8th ed.; W. H. Freeman and Company: New York, 2006. 4. McMurry, J.; Fay, R. C. Chemistry, 4th ed; Prentice-Hall, Inc.: Upper Saddle River, NJ, 2004. 5. Cage, F. W., Jr. J. Chem. Educ. 1973, 50, 692. 6. Bosch, W. L.; Crawford, C. M.; Gensler, W. J.; Haim, A.; Levine, I. N.; Linde, P. F.; Salzsieder, J. C.; Silberszye, W.; Viehland, L. A.; Waser, J. J. Chem. Educ. 1980, 57, 201– 202. 7. Levine, S. J. Chem. Educ. 1985, 62, 399.

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