Zeroth law of thermodynamics

This MedLibrary.org supplementary page on Zeroth law of thermodynamics is provided directly from the open source Wikipedia as a service to our readers. Please see the note below on authorship of this content, as well as the Wikipedia usage guidelines. To search for other content from our encyclopedia supplement, please use the form below:

Related Sponsors

BM Pharmacy Generic pharmaceuticals at unbeatable prices. Insomnia, men's health, hair health, pain control. bmpharmacy.com

Online Pharmacy Carisoprodol, tadalafil, sildenafil, vardenafil and more... xlpharmacy.com

MedBasket.com Discreet. Inexpensive. Fast shipping. Great selection. What more can we say? medbasket.com

Frugalmed 2000 reasons to shop with us first. frugalmed.com

Ads by Tiva

For the Zeroth Law used in Isaac Asimov's Robot Series, see Laws of Robotics.

Laws of thermodynamics

Zeroth Law

v • d • e

1 History
2 Thermal equilibrium
3 Thermal equilibrium between many systems
4 Temperature and the zeroth law
5 Notes
6 References

The zeroth law of thermodynamics is a generalized statement about the thermodynamic equilibrium between bodies in contact. It is the result of the definition and properties of temperature.¹

A common phrasing of the zeroth law of thermodynamics is:

If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other.

The law can be expressed in mathematical form as a simple transitive relation between the temperature T of the bodies A, B, and C:

$\mathrm{if}~T(A)=T(B)$

$\mathrm{if}~T(B)=T(C)$

$\mathrm{then}~T(A)=T(C).$

History

Please help improve this section by expanding it. Further information might be found on the talk page. (June 2008)

The term zeroth law was coined by Ralph H. Fowler^{citation needed}. In many ways, the law is more fundamental than any of the others. However, the need to state it explicitly as a law was not perceived until the first third of the 20th century, long after the first three laws were already widely in use and named as such, hence the zero numbering.

Thermal equilibrium

Main article: Thermal equilibrium

A system is said to be in thermal equilibrium when its temperature is stable (i.e does not change over time). Mathematically, this could be expressed as:

$\frac{dT}{dt}=0.$

Thermal equilibrium between many systems

Many systems are said to be in equilibrium if the small exchanges (due to Brownian motion, for example) between them do not lead to a net change in the total energy summed over all systems. A simple example illustrates why the zeroth law is necessary to complete the equilibrium description.

Consider $N$ systems in adiabatic isolation from the rest of the universe (i.e no heat exchange is possible outside of these N systems), all of which have a constant volume and composition, who can only exchange heat with one another.

The combined first and second laws relate the fluctuations in total energy $δ U$ to the temperature of the i th system $T i$ and the entropy fluctuation in the i th system $δ S i$ by,

$\delta U=\sum_i^NT_i\delta S_i$ .

The adiabatic isolation of the system from the remaining universe requires that the total sum of the entropy fluctuations vanishes,

$\sum_i^N\delta S_i=0$ ,

that is, entropy can only be exchanged between the $N$ systems. This constraint can be used to re-arrange the expression for the total energy fluctuation to give,

$\delta U=\sum_{i}^N(T_i-T_j)\delta S_i$ ,

where $T j$ is the temperature of any system $j$ we may choose to single out among the $N$ systems. Finally, equilibrium requires the total fluctuation in energy to vanish, so we arrive at,

$\sum_{i}^N(T_i-T_j)\delta S_i=0$ ,

which can be thought of as the vanishing of the product of an anti-symmetric matrix $T i - T j$ and a vector of entropy fluctuations $δ S i$ . In order for a non-trivial solution to exist,

$\delta S_i\ne 0$ ,

the determinant of the matrix formed by $T i - T j$ must vanish for all choices of $N$ . However, according to Jacobi's four-square theorem, the determinant of a $N$ x $N$ anti-symmetric matrix is always zero if $N$ is odd, although for $N$ even we find that all of the entries must vanish, $T i - T j = 0$ , in order to obtain a vanishing determinant, and hence $T i = T j$ at equilibrium. This non-intuitive result means that an odd number of systems are always in equilibrium regardless of their temperatures and entropy fluctuations, while equality of temperatures is only required between an even number of systems to achieve equilibrium in the presence of entropy fluctuations.

The zeroth law solves this odd vs. even paradox, because it can readily be used to reduce an odd-numbered system to an even number by considering any three of the $N$ systems and eliminating one by application of its principle, and hence reduce the problem to even $N$ which subsequently leads to the same equilibrium condition that we expect in every case, i.e., $T i = T j$ . The same result applies to fluctuations in any extensive quantity, such as volume (yielding the equal pressure condition), or fluctuations in mass (leading to equality of chemical potentials), and therefore the zeroth law carries implications for a great deal more than temperature alone. In general, we see that the zeroth law breaks a certain kind of anti-symmetry that exists in the first and second laws.

Temperature and the zeroth law

It is often claimed, for instance by Max Planck in his influential textbook on thermodynamics, that this law proves that we can define a temperature function or more informally, that we can 'construct a thermometer'. Whether this is true is a subject in the philosophy of thermal and statistical physics.

In the space of thermodynamic parameters, zones of constant temperature will form a surface, which provides a natural order of nearby surfaces. It is then simple to construct a global temperature function that provides a continuous ordering of states. Note that the dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters (thus, for an ideal gas described with 3 thermodynamic parameter P, V and n, they are 2D surfaces). The temperature so defined may indeed not look like the Celsius temperature scale, but it is a temperature function.

For example, if two systems of ideal gas are in equilibrium, then P₁V₁/N₁ = P₂V₂/N₂ where P_i is the pressure in the i^th system, V_i is the volume, and N_i is the 'amount' (in moles, or simply number of atoms) of gas.

The surface $P V / N = const$ defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that $P V / N = R T$ where R is some constant. These systems can now be used as a thermometer to calibrate other systems.

Notes

^ Reif, F. (1965). "Chapter 3: Statistical Thermodynamics". Fundamentals of Statistical and Thermal Physics. New York: McGraw-Hill. pp.102. ISBN 07-051800-9.

References

Jos Uffink, J. van Dis, S. Muijs; Grondslagen van de Thermische en Statistische Fysica; Utrecht University

See also Four Laws That Drive the Universe by Peter Atkins (Professor of Chemistry at Oxford University

Wikipedia content modification information:

This page was last modified on 26 November 2008, at 18:09.

Wikipedia Authorship and Review

Wikipedia content provided here is not reviewed directly by MedLibrary.org. Wikipedia content is authored by an open community of volunteers and is not produced by or in any way affiliated with MedLibrary.org.

Wikipedia Usage Guidelines

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article on "Zeroth law of thermodynamics".

The URL for this specific entry is:

http://medlibrary.org/medwiki/Zeroth_law_of_thermodynamics

All Wikipedia text is available under the terms of the GNU Free Documentation License. (See Copyrights for details). Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc.

Welcome to the MedLibrary.org Wikipedia Supplement on Zeroth law of thermodynamics -- Please Click to Return to Front Page