# Maxwell construction

Maxwell construction
The black curve is an isotherm in the pressure-volume phase diagram of a model for a real gas that can undergo a phase transition to a liquid. The oscillating middle part of it is in reality replaced by a horizontal line. The two monotonically decreasing parts that are removed describe metastable states (overheated liquid, undercooled gas), while the rising part in the middle is absolutely unstable. The height of the horizontal line is such that the two shaded areas are equal.

In thermodynamic equilibrium, a necessary condition for stability is that pressure P does not increase with volume V. This basic consistency requirement -- and similar ones for other conjugate pairs of variables -- are sometimes violated in analytic models for first order phase transitions. The most famous case is the Van der Waals equation for real gases, see Fig.~1 where a typical isotherm is drawn (black curve). The Maxwell construction is a way of correcting this deficiency. The decreasing right hand part of the curve in Fig.~1 describes a diluted gas, while its left part describes a liquid. The intermediate (rising) part of the curve in Fig.~1 would be correct, if these two parts were to be joined smoothly -- meaning in particular that the system would remain also in this region spatially uniform with a well defined density. But this is not what happens. If the volume of a vessel containing a fixed amount of liquid is expanded at constant temperature, there comes a point where some of the liquid boils and the system consists of two well separated phases. While this two-phase coexistence holds as the volume continues to increase, the pressure remains constant. It decreases again, after all liquid is evaporated and the gas expands. Thus the sinusoidal part of the isotherm is replaced by a horizontal line (red line in Fig.~1). According to the Maxwell construction (or "equal area rule"), the height of the horizontal line is such that the two green areas in Fig.~1 are equal. The Maxwell construction is derived from the condition that the Gibbs free energies of the gas and the liquid must be equal when they coexist. Essentially the same applies to any other thermodynamic system, where P and V are replaced by a different pair of conjugate variables, e.g. magnetic field and magnetization or chemical potential and number of particles.

## References

• Maxwell, J. C. (1875). "On the Dynamical Evidence of the Molecular Constitution of Bodies.". Nature: 357–359. doi:10.1351/pac200173081349.
• Reichl, L. E. (2009). A Modern Course in Statistical Physics (3rd ed.). New York, NY USA: Wiley-VCH. ISBN 978-527-40782-8.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Maxwell Air Force Base — Part of Air Education and Training Command (AETC) Located near: Montgomery, Alabama …   Wikipedia

• Maxwell-Kirby House — U.S. National Register of Historic Places Location: 8671 Northshore Dr Knoxville, Tennessee …   Wikipedia

• Maxwell Fernie — (25 April 1910 – 22 May 1999) was a New Zealand organist, teacher and conductor. He was an authority on Gregorian chant, sixteenth century polyphony, organ construction and tonal design. Contents 1 Life 2 Awards 3 The Maxwell Fernie Organ …   Wikipedia

• Maxwell Street — 1330 South Location: Chicago Maxwell Street is an east west street in Chicago, Illinois that intersects with Halsted Street just south of Roosevelt Road. It runs at 1330 South in the numbering system running from 500 West to 1126 West.[1] The… …   Wikipedia

• Maxwell Lock & Dam — Location East Millsboro, Pennsylvania Coordinates …   Wikipedia

• Maxwell Fry — and his wife Jane Drew, in 1984, at Lartington Hall on the occasion of a dinner to celebrate his 85th birthday Edwin Maxwell Fry, CBE, RA, FRIBA, FRTPI, known as Maxwell Fry (2 August 1899 – 3 September 1987), was an English modernist architect… …   Wikipedia

• Maxwell Hunter — Maxwell White Hunter II (March 11, 1922 – November 10, 2001) was a prominent American aerospace engineer. He worked on the design of the Douglas B 42 and B 43 bombers, the Honest John, Nike Ajax, and Nike Zeus missiles, the Thor IRBM, and on… …   Wikipedia

• Construction automobile aux États-Unis — Usines d assemblage d automobiles aux États Unis en 2004 : Cercle : entreprise américaine Carré : entreprise étrangère Triangle : Coentreprise La construction automobile aux États Unis, traditionnellement dominée par les Big… …   Wikipédia en Français

• Maxwell, James Clerk — born June 13, 1831, Edinburgh, Scot. died Nov. 5, 1879, Cambridge, Cambridgeshire, Eng. Scottish physicist. He published his first scientific paper at age 14, entered the University of Edinburgh at 16, and graduated from Cambridge University. He… …   Universalium

• Maxwell's thermodynamic surface — Historic photograph of Maxwell’s plaster model (taken by James Pickands II, and published in 1942).[1] Maxwell’s thermodynamic surface is an 1874 sculpture[2] made by Scottish physicist …   Wikipedia