The zeroth law of thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. In everyday language, if A feels the same as C, and B feels the same as C, then A and B feel the same. Nothing in the statement tells us how “hot” or “cold” any of them are, it only guarantees that the word “same” is transitive. That tiny logical point is enough to let us write a number on a thermometer and believe it.
To see why, we can take three metal blocks with energies , , . Bring A and C into contact, after a while nothing changes. Repeat with B and C, again nothing will change. The zeroth law says A and B can be now placed side by side and still nothing will happen. Whatever variable settled down in the C contacts must therefore be a shared, measurable property. We call that variable temperature and give it the symbol . is simply the label that lets us predict “no net energy will move”.
Equilibrium : A condition in which all acting influences are canceled by others, resulting in a stable, balanced, or unchanging system.
Once we have the label we can draw the first tiny piece of mathematics which is, if and , then .
That one line syllogism is the entire zeroth law. From it follows every thermometer ever built, mercury expands until its T matches the room’s T, a thermistor’s resistance drifts until its T equals the coffee’s T. The device needs no knowledge of what it is measuring; it only needs to reach the same T.
First Law of thermodynamics enters when we ask what happens if the temperatures are not equal. Suppose a sealed box contains a small battery and a resistor. While the battery runs, the resistor warms and its energy rises by . Somewhere else the battery’s stored energy falls by the same amount. Write that as
Extend the idea to any system: if heat Q flows in and work W is done on the system, the internal energy changes by
when the system does work on the outside (a piston moves out) W is negative; when heat leaks out Q is negative. The bookkeeping always balances to zero, so the total energy of the universe remains the same no exceptions have ever been found.
We could already notice two laws fit together, the zeroth law gives us reliable thermometer, so we can decide when Q will flow (when T’s differ). The first law tells how much energy that Q carries and where it ends up.