The Zeroth Law

As has been mentioned already, everyone is familiar with such elementary notions as ‘A is warmer than B’, ‘B may gain heat from A’, and the qualitative notion of the ‘flow of heat’. Also, everyone knows that, when the flow of heat between two systems has ceased, those systems are said to be in thermal equilibrium. The above might all be termed ‘facts of experience ’and, in many arguments which lead to the formulation of the laws of thermal physics, such ‘facts of experience’ play an important role.

Now, if the influence of, for example, electric or magnetic fields is absent, it is a fact of experience that the properties of a stationary fluid are determined completely by just two properties - the pressure p and volume of the containing vessel V. A system defined by only two properties is termed a two-coordinate system. Such systems occur widely but the formalism may be generalised easily to cope with multi-coordinate systems.

Consider two isolated systems consisting of fluids with coordinates p1, V1 and p2, V2. If brought into thermal contact and left for a long time, the properties of these two systems will change so that a state of thermal equilibrium is achieved. This means that all components making up the system are allowed to interact thermally until, after a long time, no further changes are observed in the bulk properties of the system. Generally, heat will be exchanged and work done in attaining this final situation. Eventually, the two come to thermal equilibrium such that their thermodynamic coordinates assume the values p1’,V1’ and p2’,V2’. However, it is a fact of experience that the four coordinates cannot be totally independent if the two systems are in thermal equilibrium. Hence, there must be some relation linking these four quantities:

p1V1p2V2=0.

This equation enables one quantity to be found in terms of the other three.

So far, use of the word ‘temperature’ has been avoided but a suitable definition may be found by using another ‘fact of experience’ which is so central to the subject that it appears as a law of thermodynamics - the Zeroth Law. This states that

Systems 1 and 3 being in thermal equilibrium means a relation of the form

p1V1p3V3=0

holds. This may be expressed alternatively as

3=fp1V1V3.

Similarly, systems 2 and 3 being in thermal equilibrium implies p3 = g(p2,V2, V3).

Hence,

p1V1V3=gp2V2V3.

However, according to the Zeroth Law, systems 1 and 2 must be in thermal equilibrium also and so, a relation of the form

p1V1p2V2=0

must hold.

This implies that (2.1) must be of a form such that the dependence on V3 on either side cancels; that is, for example,

p1V1V3=ϕ3p1V1ξV3+ηV3

and

p2V2V3=ϕ2p2V2ξV3+ηV3.

Hence, if the dependence on V3 is cancelled out, it is seen that, in thermal equilibrium

1p1V1=ϕ2p2V2=ϕ3p3V3=t=constant.

This is the logical consequence of the Zeroth Law - there exists a function of p and V which may well vary in form from one system to another but which takes a constant value for all systems in thermal equilibrium with one another. Different equilibrium states will be characterised by different constants. This constant , which characterises the equilibrium, is termed a function of state - that is, it is a quantity which assumes a definite value for a particular equilibrium state - and is called the empirical temperature t. (Here it might be noted that the word empirical means based on observation and experiment, not on theory.)

An equation of state relating p and V to this empirical temperature emerges also

pV=t.

All the combinations of p and V which correspond to a fixed value of the empirical temperature t may be found from experiment. Of these three quantities, any two are sufficient to define the equilibrium state completely. Lines of constant t plotted on a pressure - volume, (p-V), diagram are called isotherms.

At this stage, the empirical temperature looks nothing like what is normally called temperature. To place everything on a firm experimental foundation, a thermometric scale must be chosen. Once that is fixed for one system, it will be fixed for all others since all systems must have the same value of the empirical temperature when they are in thermal equilibrium.

Now consider a device which, in attaining thermal equilibrium with a physical system, disturbs that system to a negligible extent. Also, suppose that all its thermodynamic variables except one are constrained to practically fixed values. When this device is allowed to reach thermal equilibrium with a system, ideally its variable characteristic is a strictly increasing or strictly decreasing function of the empirical temperature. Hence, the device may be calibrated to read temperature directly and is a thermometer. If a gas is used as the medium, the thermometer may be of the constant volume or constant pressure variety.

Note that the notions of ‘hot’ and ‘cold’ may be associated with high and low temperatures respectively - but the reverse may be true equally well. As yet, it is not required to restrict the choice of an empirical temperature and so, both possibilities must be allowed.

Exercises A

Familiarity with partial differentiation is crucial for many thermodynamic manipulations. Therefore, this first set of examples is intended to help with revision of the techniques of partial differentiation which are so important in the present context. The main basic results required are derived in the appendix.

(1) Given u = x 2 + 2x - 1 and x = t2 - 1, find du/dt by

(a) substituting t for x,

(b) using the chain rule.

(2) Given u(x,y) = (x + 1)2 - 3xy2 + 4y, find

(a) u(2,-1), (b) u(1/x, xly).

(3) Find all the first partial derivatives of

(a) u(x,y) = tan(x/y)

(b) f(r,θ) = r2sin2θ + r3

(c) u(r,s,t) = r3 + s2t + (t - 1)(r - 3)

(d) f(p,q) = exp(p2logq).

(4) The relation between the pressure p, temperature t and volume V of a certain amount of hydrogen gas may be expressed over a limited range by the equation

(V-B)=At

where B is independent of p but is a function of t ; A is a constant. Find (∂V / ∂t) p and (∂V / ∂p) t and express dV as a function of t and p.

(5) If z = f(x,y), prove the important result

x∂yz∂y∂zx∂z∂xy=−1.

This result should be known; it will prove to be of use on innumerable occasions. [Here the symbol outside each bracket indicates that that variable is the one to be held constant in the differentiation. This particular element of notation is peculiar to physics and to thermodynamics in particular. It does have the merit of making it quite clear which variable is being held constant.]

(6) An increment

Z=L(x,y)dx+M(x,y)dy

is an exact differential if there exists a function Z(x,y) such that

(x,y)=(∂Z/∂x)y;M(x,y)=(∂Z/∂y)x.

A necessary and sufficient condition for d'Z to be exact is

∂L/∂y)x=(∂M/∂x)y.

The volume of a figure is given by V = arbh3‒b, where a and b are constants.
Find (∂V /∂h)r and (∂V /∂r)h. Give an...