Zeroth law and Mathematics

Jun 2018
so i have a question.

zeroth law of thermodynamics states that if two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other.

in mathematics its defined that if a=b and b=c then a=c. its same as the zeroth law.

which was discovered first ? i mean which was defied first ?

if the mathematics rule was defined first then its applicable to everything then why zeroth law is developed individually ? why it has a special attention in thermodynamics ?

and what is this logic for developing / discovering zeroth law because its already stated in terms of mathematics.
Oct 2017
why it has a special attention in thermodynamics ?
The mathematical law is just a statement that things are the same. It states that if a = 1, we can infer that c = 1 directly because we know that a = b and b = c. It's a statement about identity or value semantics.

The zeroth law of thermodynamics is a little bit different because it involves more than just value semantics. It's also a law of thermodynamics that is often taken for granted because we are educated with the knowledge that all forms of heat are basically the same; heat is heat, so individual heat flows across the boundaries of control volumes can be summed and there exists net heat flow. There was a time in the past when the nature of heat was not as well known and the behaviour of thermal systems was not so well understood... can a 100 J transfer of heat from A to B really 'cancel out' another 100 J heat transfer from B to A? What about 'carriers' of heat?

So... what is so special about it? It formalises the fact that if many components of a system in thermal equilibrium with their adjacent components, it is valid to say that components can be in thermal equilibrium with each other, even if they are not adjacent. This is important for situations where we have components and sub-components and makes the mathematics convenient.

Here's an example. Consider a toastie that we want to grill which has two slices of bread and a filling. The two pieces of bread and filling are initially in thermal equilibrium at room temperature. We place the toastie into a sandwich grill, which puts a constant heat flux q into each of the bread layers. We then wait a little while until the toastie is in thermal equilibrium because the losses to the grill equal the flux from the grill. Can we just calculate the net heat flux into the sandwich and treat the sandwich as having a single temperature or do we need to model the individual layers of the toastie? Thanks to the zeroth law, both options are viable, depending on the level of detail we want to go to. If the zeroth law wasn't true, we couldn't do the former and we would always be forced to model all sub-components of a system because we couldn't know whether any of the layers are in thermal equilibrium until we calculate them and verify the equilibrium.
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