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Consider this - A function of state is any physical quantity that has a well-defined value for each equilibrium state of the system. (Blundell and Blundell - Thermal Physics)

My question is - Are equilibrium states not defined by the function of state variables themselves?

That is,function(s) of state defines a equilibrium state and function(s) of state are themselves characterised by having a unique/well-defined value at an equilibrium point.

Is this not a circularity?

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Whenever i say that the pressure of a gas is 2 atm then it is known that at the time i say this, the gas is in equilibrium. Well what i want to say that we define a state of any system when it is in equilibrium and by equilibrium i mean thermal equilibrium which means every knid of equilibrium like mechanical, chemical, etc. Now to define a state we use thermodynamic terms like temperature, pressure and density.

Now if we look at the state variable like internal heat, entropy, etc we will find that they are depending on these thermodynamic terms which i mentioned above. For example take the example of internal heat. The first law of thermodynamic says that the cyclic integral of heat is equal to the cyclic integral of the work done from which on further solving we get the expression of $dU=\delta Q-\delta W$. so we can see that $dU$ is depending on the temperature, pressure, etc.

Conclusion:


(1) A state can only be defined at a thermodynamic equilibrum.
(2) A property (temperature, pressure, etc) can be defined as any quantity that depend on the state of the system and conversely the state is specified or described by the property.[Fundamentals of thermodynamics, Sixth Edition].
(3) The state variables are dependent on these properties.
Hence logically from the above three statements we can say that the state variables are defined at the thermodynamic equilibrium or conversely thermodynamic equilibrium state is defined by the state variables

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