CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 7, Exercise 2 Solution
==========================================================
Applying the Zeroth Law Without Direct Contact

PROBLEM
-------
Three metal blocks, A, B, and C, are tested with a thermometer. Block
A and Block C both read exactly 25 degrees Celsius on the
thermometer. Using the Zeroth Law, what can you conclude about the
thermal relationship between Block A and Block C directly, without
ever placing them in contact with each other?

SOLUTION
--------
The thermometer itself is the "third system" referred to in the
Zeroth Law. When the thermometer is placed in contact with Block A
and reaches thermal equilibrium with it (shown by a steady reading of
25 degrees Celsius), the thermometer and Block A are in thermal
equilibrium with each other. The same is separately true for the
thermometer and Block C, which also brings the thermometer to
equilibrium at 25 degrees Celsius.

The Zeroth Law states that if two systems (here, Block A and Block C)
are each separately in thermal equilibrium with a third system (the
thermometer), then Block A and Block C must be in thermal equilibrium
with each other as well.

ANSWER: Block A and Block C are in thermal equilibrium with each
other - meaning if they were placed in direct contact, no net heat
would flow between them - even though they were never actually
brought into contact with one another. This can be concluded purely
from each one separately reaching equilibrium with the same
thermometer.

----
WHY THIS WORKS AS AN ANSWER

This is exactly the real-world justification for why a thermometer
works as a measuring instrument at all: it lets you compare the
temperatures of two objects indirectly, through a shared third object,
rather than needing to physically place every pair of objects you
want to compare into direct contact with each other. Without the
Zeroth Law's guarantee, a matching thermometer reading on two
different objects would tell you nothing reliable about how those two
objects would behave if you actually put them together.