How to understand the zeroth law through these specific equations?

[zenterix]

Homework Statement

I've studied the zeroth law in other books and am just reviewing it in a new book.

Relevant Equations

This new book shows some math to explain the zeroth law.

I've seen this math also in a lecture once. It seems very vague to me.

Here is the relevant part of the book

$F_1$ and $F_2$ are introduced as seen above. There is, as far as I can tell, no previous mention of them.

Why does $F_1(P_A,V_A,P_B,V_B)=0$ signify thermal equilibrium?

$P_A$ and $V_A$ determine $T_A$ and the same for the $B$ thermodynamic variables.

Thus, it seems that this equation indicates that given, say, $P_A$ and $V_A$, ie $T_A$, then there are specific pairs of $P_B$ and $V_B$ that make the equation true.

The book where this equation appears does not define thermal equilibrium.

In Zemansky and Dittman we have the following definition

Thermal Equilibrium: state achieved by two (or more) systems, characterized by restricted values of the coordinates of the systems, after they have been in communication with each other through a diathermic wall.

This means that systems $A$ and $B$ can exchange energy (I think this means just heat).

Again in Zemansky and Dittman

Experiment shows that there exists a whole set of states of a system A, any one of which is in thermal equilibrium with some specific state of system B, and all of which are in equilibrium with each other.

All such states lie on a curve in $xy$-plane called an **isotherm:** locus of all points representing states in which a system is in thermal equilibrium with one state of another system.

The specific state of system B is also on an isotherm, and the isotherm of system A and the isotherm of system B in question are called corresponding isotherms.

Suppose we keep $V_B$ constant and vary $P_B$ and $T_B$. Then we are moving among different isotherms.

In the equation for $F_1$ in the screenshot above, it seems that this type of change in $B$ should not be allowed.

But I cannot see what it is that prevents it from being allowed with just that equation.

Next, the book isolates $P_B$ in the two initial equations and reaches

$$f_1(P_A,V_A,V_B)=f_2(P_C,V_C,V_B)$$

and says this means $A$ and $C$ are in equilibrium. Why?

 

[jedishrfu]

The wikipedia article on the Zeroth Law seems pretty good and might give you clues on how to understand the text above:

https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics

They have a section on ideal gases and the PV/n=RT formula where P is the pressure, V is the volume, and n is the number of moles. R represents a constant, and T represents the temperature.

So, using a thermometer to measure the temperature of one system A, getting a reading of 100 Celsius when system A and the thermometer reach thermal equilibrium, and then measuring a second system B with the same thermometer and getting 100 Celsius, one can conclude that system A and system B are in thermal equilibrium by the Zeroth Law.

The thermometer is the means to determine the thermal equilibrium.

 

[Lord Jestocost]

From the book „A History of Thermodynamics – The Doctrine of Energy and Entropy“ by Ingo Müller (Springer-Verlag Berlin Heidelberg 2007); in chapter 1 “Temperature” one reads:

For the early researchers there was no need to define temperature. They knew, or thought they knew, what temperature was when they stuck their thermometer into well-water, or into the armpit of a healthy man. They were unaware of the implicit assumption, – or considered it unimportant, or self-evident – that the temperature of the thermometric substance, gas or mercury, or alcohol, was equal to the temperature of the measured object.

This in fact is the defining property of temperature: That the temperature field is continuous at the surface of the thermometer; hence temperature is measurable. Axiomatists call this the zeroth law of thermodynamics because, by the time when they recognized the need for a definition of temperature, the first and second laws were already firmly labelled.

 

[zenterix]

I understand the zeroth law, I just don't understand how these specific mathematical equations are expressing it.