Zeroth law of thermodynamic

Statement:- If two system is in thermal equilibrium¹ with a third system then the are in thermal equilibrium with each other.

Explanation:- Let tow system A and B are in thermal equilibrium with a third system C i.e.
the all the  thermodynamic parameter of system A and B are exactly same
with the thermodynamic parameters of system C. then according to the Zeroth law of thermodynamic the systems A and B ere in thermal equilibrium with each other i.e. all the  thermodynamic parameter of system A and B are exactly same.

Concept of Temperature:

The temperature of a system is a property that determine whether or not the system is in thermal equilibrium with other system. This property can be represented by a function of the states co-ordinates of the system.

Let a system A with state co-ordinates X_A Y_A be in thermal equilibrium with a system C with state co-ordinate X_C , Y_C . Then the functional from of this thermal equilibrium condition may be represented by

              f_AC (X_A , Y_A ;X_C , Y_C )=0          ……………………(1)

Similarly the thermal equilibrium between the system B with X_B , Y_B and the system C is represented by

              f_BC (X_B , Y_B ; X_C , Y_C )=0          ……………………………..(2)

Where f_BC may be different from f_AC.

Now, from (1)          Y_C = g_AC (X_A , Y_A, X_C )

     &  from (2)           Y_C = g_BC (X_B , Y_B, X_C)

Where g_AC and g_BC are new function.

           g_AC (X_A , Y_A, X_C ) = g_BC (X_B , Y_B, X_C)         .……………………………(3)

According to Zeroth law of thermodynamic, the thermal equilibrium between A & C and the thermal equilibrium between B & C imply thermal equilibrium between A and B. Hence

             f_AB (X_A , Y_A ; X_B , Y_B) = 0       ………………………………(4)

The equition (4) must agree with the equition (3) because they represented the same equlilibrim situation. This is possible if the co-ordinates X drops out from equition (3). Then we get

              h_A (X_A , Y_A) = h_B(X_B , Y_B)    where h_A and h_B ere new functions.

Applying the same arrangement a second time with A & C in equilibrium with B, we get

               h_A (X_A , Y_A) = h_C (X_C , Y_C )

Thus when the three system are in thermal equilibrium, we get ,

               h_A (X_A , Y_A) = h_B(X_B , Y_B) = h_C (X_C , Y_C ) = θ (say)

Then θ the common value of the function is called the temperature thermodynamically common to all the systems.

[NOTES: - For a hydrostatic system the thermodynamic parameter are P, V & T. thus if system A & B are hydrostatic system then the equition of states of the system is given by   P’V’ = RT’  and  P’’V’’ = RT’’ . Now if system A & B are in equilibrium with each other then

                                                            P’V’ = P”V”                (think about it )

                                                            P’V’ – P’’V’’ =0

                                                            f (P’, V’, P’’, V”  ) = 0     

  which represented the equilibrium condition between system A & B . 

 1.   Here by the  thermodynamic equilibrium term we mainly interested in thermal equilibrium condition. ]        

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