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    Why does the zeroth law mean that objects have a measurable physical quantity that determines the direction of energy transfer of thermal energy, form the the basis for the definition of temperature, form the basis for comparing temperatures and is important for the mathematical formulation of laws about the effect of changing temperature? 😄

    I know that's a ton to ask but I'm not sure how a law that essentially says that if two objects are in contact with another object and are at the same temperature as the object then all objects will be in thermal equilibrium can do all of those things 🙂
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    Instead of directly responding to your questions, I think it might be beneficial for you to consider something as simple as a thermometer.

    Imagine you have a volume of water kept at a constant 50 degrees Celsius, we can reference this as object A. Now we introduce a thermometer, which consists of a glass wall (object B) housing another volume of liquid (object C).

    As the zeroth law of thermodynamics states, if object A is in thermal equilibrium with object B (the water and glass), and object B is in thermal eqm. with object C (the liquid), then object A and object B should also be in thermal eqm.

    Given we know certain properties about the liquid within the thermometer (relating to its thermal expansion), do you see how when thermal equilibrium is reached, and we have some relative indication of the difference in this 'quantity', how the zeroth law played a crucial part?
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    Sorry but I have no clue! Plus that just raised another question of why the measured temperature change is accurate when you place a thermometer into a solution then the solution reacts - won't the thermometer decrease the actual temperature of the solution and if so why isn't that decrease negligible? 🙂
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    I now understand everything but how the law forms the basis for the definition of temperature and that last one 😄
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