Zeroth law of thermodynamics

The "zeroth law of thermodynamics" is a generalized statement about thermal equilibrium between bodies in contact. It is the result of the definition and properties of temperaturecite book |last=Reif |first=F. |title= Fundamentals of Statistical and Thermal Physics |year=1965 |publisher=McGraw-Hill |location=New York |isbn=07-051800-9 |pages=102 |chapter=Chapter 3: Statistical Thermodynamics] . A common enunciation 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

The term "zeroth law" was coined by Ralph H. FowlerFact|date=June 2008. In many ways, the law is more fundamental than any of the othersdn. 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

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 $delta\; U$ to the temperature of the "i"th system $T\_i$ and the entropy fluctuation in the "i"th system $delta\; S\_i$ by, :$delta\; U=sum\_i^NT\_idelta\; S\_i$. The adiabatic isolation of the system from the remaining universe requires that the total sum of the entropy fluctuations vanishes,:$sum\_i^Ndelta\; S\_i=0$,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 $delta\; S\_i$. In order for a non-trivial solution to exist,:$delta\; S\_i\; e\; 0$,the determinant of the matrix formed by $T\_i-T\_j$ must vanish for all choices of $N$. However, according to Jacobi's 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 $PV/N\; =\; ext\{const\}$ defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that $PV/N\; =\; RT$ where R is some constant. These systems can now be used as a thermometer to calibrate other systems.

References

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