Thermodynamics is a perfect science any small inconsistency and the whole structure is destroyed. It therefore pays to be very clear and explicit about each little detail that is to be brought forth to understand anything. For now, we concentrate on what we mean by equilibrium and slightly formalize what we mean by it.
First we curve out a universe for ourselves which consists of everything we care to know about, we call this the system. Anything that is not contained in it is called the environment or the surrounding. The two spheres of course do not live un-related by are separated by a boundary or a wall.
For a closed system the boundary does not allow for any interaction(no matter exchange) while an open system does allow for matter exchange.
ASIDE: The same kind of taxonomy appears also in quantum mechanics. There for a closed system we have nice unitary evolution wonderfully described by the Schrödinger’s equation and for open systems we have non-unitary evolutions in which classical behaviour emerges. It is often said sloppily that quantum is for the small and classical is for the big, it is more accurate to say quantum is for closed systems and classical is for open systems.
Let us consider a closed system. From experience we know that after a while system reaches a state when no changes occur. In particular, pressure becomes uniform. This state can be labelled by two independent variables (P,V). The amazing thing is that knowing P,V and the masses in the system fixes all other bulk properties that one might what to know about.
We therefore arrive at what we mean by and equilibrium state: this is one in which all bulk physical properties of the system are uniform throughout the system and do not change with time.
I have chosen the pair of variables P,V but in fact we can choose two any two independent variables e.g for a wire you might choose the tension and the length as your variables. These pairs are called thermodynamic variables or co-ordinates.
It turns out there are functions which take on unique values as a function of these thermodynamic variables at different equilibrium states. These are called state functions. The concept of a state function is very important because of a very important property namely, the values it takes at different equilibrium states do not depend on history of the system. First of all this is how we shall arrive at state functions but this property is important because in the thermal dynamics we often run into irreversible processes that connect two equilibrium states and we rarely ever have nice neat formulae for these processes but that does not matter because of state functions. All we need to do is describe reversible processes connecting these two equilibrium states (which is nice because reversible processes are easy to describe) and we can understand the physics. This is how we shall arrive at the concept of entropy.
We know we can bring two systems together so that they interact thermally and after a while there will be no further changes to take place in pressure and volume. When no further changes take place we say these two systems are in thermal equilibrium. A transitive property applies to states in thermal equilibrium i.e If A is thermal equilibrium with B and B is in thermal equilibrium with C then A is in thermal equilibrium with C. We can of course apply this argument ad infinitum to describe a whole series of systems that are in thermal equilibrium. This is in essence the zeroth law of thermodynamics.
The amazing thing is that as a consequence of the zeroth law we can prove that all systems in thermal equilibrium share one property, we call this the temperature. The proof which is borrowed from Zemansky's book "Heat and Thermodynamics" will follow momentarily.
We first note a subtle distinction often skirted over namely the difference between thermal equilibrium and thermodynamic equilibrium. Thermal equilibrium does not guarantee thermodynamic equilibrium. In order to have thermodynamic equilibrium we must have mechanical equilibrium i.e all forces are balanced and we must have chemical equilibrium i.e no chemical reactions should be taking place between the two systems.
Now at thermal equilibrium it must be that P,V and T are not independent but are related by some functional equation f(P,V,T)=0. This is called the equation of state. For an ideal gas we have the famous PV -NKT=0.
We now present a proof that given the zeroth law there exits one property that all systems in thermal equilibrium with one another share. This of course we know a head of time is temperature.
Suppose we have to systems A and C with thermal dynamic variables being X and Y for A and X'' and Y'' for C. We know there is and equation of state \begin{equation}
f_{AC} (X,Y,X'',Y'') =0 \hspace{10mm} \text{eq.1}
\end{equation}.
Also let us say there is another system B with variables X' and Y' which is in equilibrium with C so that there is another equation of state
\begin{equation}
f_{BC} (X',Y',X'',Y'') =0 \hspace{10mm} \text{eq.2}
\end{equation}
Now we can solve for Y'' in both equations and set the resulting equations equal to each other
\begin{equation}
g_{AC}(X,Y,X'') = g_{BC}(X',Y',X'') \hspace{10mm} \text{eq.3}
\end{equation}
The zeroth law guarantees that A is in equilibrium with B so that
\begin{equation}
f_{AB} (X,Y,X',Y') =0 \hspace{10mm} \text{eq.4}
\end{equation}
but in particular eq.4 and eq.3 must be in fact equal (remember these are state functions and take on unique values at equilibrium states). This means that X'' is an extraneous variable and can be removed so that at thermal equilibrium we have
\begin{equation}
g_{A}(X,Y) = g_{B}(X',Y') = g_{C}(X'',Y'') = t \hspace{10mm} \text{eq.3}
\end{equation}.
So we have that there exists a function which parametrizes these sets of co-ordinates and these functions are all equal at thermal equilibrium. This common value is empirically known as the temperature.
We now see the importance of the zeroth law it guarantees the existence of temperature. I often wondered about the importance of the zeroth law. It is mentioned in the first few pages of thermodynamic textbooks and never makes any appearance as one moves forward. Well, now we know it allows us to talk about thermal equilibrium and once that has happened it is forgotten.
First we curve out a universe for ourselves which consists of everything we care to know about, we call this the system. Anything that is not contained in it is called the environment or the surrounding. The two spheres of course do not live un-related by are separated by a boundary or a wall.
For a closed system the boundary does not allow for any interaction(no matter exchange) while an open system does allow for matter exchange.
ASIDE: The same kind of taxonomy appears also in quantum mechanics. There for a closed system we have nice unitary evolution wonderfully described by the Schrödinger’s equation and for open systems we have non-unitary evolutions in which classical behaviour emerges. It is often said sloppily that quantum is for the small and classical is for the big, it is more accurate to say quantum is for closed systems and classical is for open systems.
Let us consider a closed system. From experience we know that after a while system reaches a state when no changes occur. In particular, pressure becomes uniform. This state can be labelled by two independent variables (P,V). The amazing thing is that knowing P,V and the masses in the system fixes all other bulk properties that one might what to know about.
We therefore arrive at what we mean by and equilibrium state: this is one in which all bulk physical properties of the system are uniform throughout the system and do not change with time.
I have chosen the pair of variables P,V but in fact we can choose two any two independent variables e.g for a wire you might choose the tension and the length as your variables. These pairs are called thermodynamic variables or co-ordinates.
It turns out there are functions which take on unique values as a function of these thermodynamic variables at different equilibrium states. These are called state functions. The concept of a state function is very important because of a very important property namely, the values it takes at different equilibrium states do not depend on history of the system. First of all this is how we shall arrive at state functions but this property is important because in the thermal dynamics we often run into irreversible processes that connect two equilibrium states and we rarely ever have nice neat formulae for these processes but that does not matter because of state functions. All we need to do is describe reversible processes connecting these two equilibrium states (which is nice because reversible processes are easy to describe) and we can understand the physics. This is how we shall arrive at the concept of entropy.
We know we can bring two systems together so that they interact thermally and after a while there will be no further changes to take place in pressure and volume. When no further changes take place we say these two systems are in thermal equilibrium. A transitive property applies to states in thermal equilibrium i.e If A is thermal equilibrium with B and B is in thermal equilibrium with C then A is in thermal equilibrium with C. We can of course apply this argument ad infinitum to describe a whole series of systems that are in thermal equilibrium. This is in essence the zeroth law of thermodynamics.
The amazing thing is that as a consequence of the zeroth law we can prove that all systems in thermal equilibrium share one property, we call this the temperature. The proof which is borrowed from Zemansky's book "Heat and Thermodynamics" will follow momentarily.
We first note a subtle distinction often skirted over namely the difference between thermal equilibrium and thermodynamic equilibrium. Thermal equilibrium does not guarantee thermodynamic equilibrium. In order to have thermodynamic equilibrium we must have mechanical equilibrium i.e all forces are balanced and we must have chemical equilibrium i.e no chemical reactions should be taking place between the two systems.
Now at thermal equilibrium it must be that P,V and T are not independent but are related by some functional equation f(P,V,T)=0. This is called the equation of state. For an ideal gas we have the famous PV -NKT=0.
We now present a proof that given the zeroth law there exits one property that all systems in thermal equilibrium with one another share. This of course we know a head of time is temperature.
Suppose we have to systems A and C with thermal dynamic variables being X and Y for A and X'' and Y'' for C. We know there is and equation of state \begin{equation}
f_{AC} (X,Y,X'',Y'') =0 \hspace{10mm} \text{eq.1}
\end{equation}.
Also let us say there is another system B with variables X' and Y' which is in equilibrium with C so that there is another equation of state
\begin{equation}
f_{BC} (X',Y',X'',Y'') =0 \hspace{10mm} \text{eq.2}
\end{equation}
Now we can solve for Y'' in both equations and set the resulting equations equal to each other
\begin{equation}
g_{AC}(X,Y,X'') = g_{BC}(X',Y',X'') \hspace{10mm} \text{eq.3}
\end{equation}
The zeroth law guarantees that A is in equilibrium with B so that
\begin{equation}
f_{AB} (X,Y,X',Y') =0 \hspace{10mm} \text{eq.4}
\end{equation}
but in particular eq.4 and eq.3 must be in fact equal (remember these are state functions and take on unique values at equilibrium states). This means that X'' is an extraneous variable and can be removed so that at thermal equilibrium we have
\begin{equation}
g_{A}(X,Y) = g_{B}(X',Y') = g_{C}(X'',Y'') = t \hspace{10mm} \text{eq.3}
\end{equation}.
So we have that there exists a function which parametrizes these sets of co-ordinates and these functions are all equal at thermal equilibrium. This common value is empirically known as the temperature.
We now see the importance of the zeroth law it guarantees the existence of temperature. I often wondered about the importance of the zeroth law. It is mentioned in the first few pages of thermodynamic textbooks and never makes any appearance as one moves forward. Well, now we know it allows us to talk about thermal equilibrium and once that has happened it is forgotten.
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