# How can $U$ have an arbitrary zero point and still be extensive?

You have taken the puzzle incorrectly backwards in a sense. The first sentence categorically says that the math that must apply to work with internal energy must apply exactly as below:

$$\Delta U_1 = U_B - U_A = \Delta U_2 = (U_B + U_o) - (U_A + U_o)$$

In this mathematical language, $$U_o$$ is one of two things. Either it is an energy that is added to both systems or it is an offset due to the measuring device.

Let's take the first case. Before we add any energy, combine the two systems as $$U_{To} = U_A + U_B$$. Now, add $$U_o$$ to this combined system to obtain $$U_{T1} = U_A + U_B + U_o$$. Now, let's split the new system. In order to remain true, we cannot MULTIPLY or ADD ANY OTHER energy during the process that we split the system. We must also split $$U_o$$ in the same way that we split back to $$U_A$$ and $$U_B$$. This results in

$$U_{T1} = U_{To} + U_o = (U_A + f_A U_o) + (U_B + f_B U_o)$$

In this, $$f_A$$ and $$f_B$$ are the proportions that go to A and B respectively. Because $$f_A + f_B = 1$$, we cannot make any combination of systems where $$f_A = 1, f_B = 1$$ simultaneously.

You might think about this with voltages (V) or electrical potentials (eV). You are trying to argue that adding 5 V to a 12 V source is the same as adding 5 V each to an 8 V and a 4 V source, then adding the two together. In other words, you are arguing that this should be true:

$$12 + 5 = (8 + 5) + (4 + 5)$$

You are stating that you can split one voltage source that is offset by 5 V into two voltage sources in parallel that sum to the original voltage and that are each individually offset by 5 V also with both offsets adding in parallel. You are doing so because you have heard that current flow does not depend on the absolute voltage of an object but only on the difference in voltage between two objects. So, why not just increase the voltage on all three states by a fixed amount (5 V)? As you might now see, this approach reverses the required math in exactly the wrong way.

By analogy, heat flow and work (flow) between two systems do not depend on the absolute internal energy of one system or another. Those energy flows depend only on the difference in internal energy between the two systems.

You can also make an analogy with ladders. You have two ladders that are 8 m and 4 m high respectively as measured from the first floor of the building. When you drop a ball from either, the results will give two different end point kinetic energies. The difference is a height of 4 m (in kinetic energy). Now, stack the ladders while you are on the first floor. You get 8 m + 4 m = 12 m. Now take the ladders to the second floor to drop a ball all the way back to the first floor level. Stand them separately. The difference is still 4 m (in kinetic energy). What you are trying to argue is that, when you take the ladders to the second floor of the building and stack them, you should be able to stack them in this way:

(8 m ladder + floor) + (4 m ladder + floor) = (12 m ladder + floor)

Of course, this is nonsense. We do NOT add the floor twice when we stack the ladders. But this is how you argue from a (mis)-interpretation of Callen.

You might instead be asking about the physical significance of a system truly at $$U$$ versus one truly at $$U + Uo$$ versus one truly at $$U$$ but measured by a device with a set offset of $$Uo$$ to give an apparent value of $$U + Uo$$. The latter case is what is meant by the second sentence in the first quote. Taking a particular state as having an arbitrary zero value is setting the zero point offset of a system according to our measuring tool. Although we change the offset of the measuring device that we use, we do not change the absolute energy content of the system. Setting a system to have a zero arbitrary energy is absolutely not the same as adding or removing energy to make the energy of that system zero. In conclusion, the seeming contradiction that $$U + Uo \neq U$$ measured with a device having an offset $$Uo$$ is solved by realizing that we cannot use two different measuring devices to compare the true state of any two systems.

Finally, one other way that the contradiction can be avoided is when we focus the first statement explicitly this way.

For any process undertaken on a single system, only differences of energy between the ending point and the starting point, rather than absolute values of the energy in either the starting or the ending point, have physical significance, either at the atomic level or in macroscopic systems.

Your step to combine two different systems violates the principles behind these unstated phrases. While this is an odd way out, it is still possibly a valid one depending on the larger context around the first statement.

In conclusion, the contradiction is to believe that shifting a zero point of a system is the same as adding/removing energy from a system. These are not the same. Callen is certifying that we can shift the zero point of our measuring frame because differences between systems are the only physically meaningful quantity. He absolutely is not certifying that we can arbitrarily add/remove a specific amount of energy to two different systems, combine them, and expect the summation is only shifted by that same amount of energy. As shown by the examples given here, the belief would lead to a gross, inverted violation of first principles based entirely on the mathematics alone. The counter that such a belief is categorically false is dutifully shown across a host of other physical phenomena where differences rather than absolutes drive the macroscopic or fundamental physical processes.

The problem is that shifting the zero-point of the energy does not always translate to the same shifts for each sub system as well as the entire combined system.

To make this explicit with a simple example, let's say you have two point masses of $$m_1$$ and $$m_2$$ in a uniform gravitational field. Then $$U_1=m_1gh_1$$ and $$U_2=m_2gh_2$$. Now let's change the zero-point. Then $$U_1'=m_1g(h_1-h_0)$$ and $$U_2'=m_2g(h_2-h_0)$$. Notice how the energy for mass 1 changed by $$m_1gh_0$$ and the energy for mass 2 changed by $$m_2gh_0$$, and the total energy changed by $$(m_1+m_2)gh_0$$. This is consistent with considering shifting the zero-point for the entire system.

Therefore, the transformation you describe is not correct in general, and this is where the issue lies. Changing the zero-point does not mean adding the same energy amount to all energies. It means changing where $$U=0$$, and so the corresponding energies then change accordingly by shifting appropriate amounts based on parameters of each system.

This statement you quote

Only differences of energy, rather than absolute values of the energy, have physical significance, either at the atomic level or in macroscopic systems.

refers to differences in energy across space which determine the forces. These differences are not affected by shifting the zero-point, since the shift effects all points in space equally.

Reading carefully what Callen writes in the same chapter, clarifies the source of troubles. I assume that Callen's book can be considered a reputable source. It is true that

Only differences of energy, rather than absolute values of the energy, have physical significance, either at the atomic level or in macroscopic systems.

However, what is the energy he was speaking about? Let's confine the discussion to macroscopic systems. Macroscopic (internal) energy is defined by Callen, in the first chapter of his book, as the adiabatic work necessary to connect two equilibrium states, accordingly to a well established tradition in thermodynamics. Formally, one could write $$U(X_B) = U(X_A) + \int_{X_A}^{X_B}{\rm d}xW_{ad}(x)$$ i.e. the internal energy of the generic state $$B$$ is defined as the internal energy of the reference state $$A$$ (which is arbitrary) plus the adiabatic work along the transformation. It is the definition of adiabatic work which ensures that the integral does not depend on the path, but only on the initial ad final state.

But, let's look more carefully how each state is described. One independent variable should be the value of the variable the work $$W_{ad}$$ depends on. In this example it is $$x$$. In addition to this varible, a thermodynamic system is characterized by other variables, which are kept fixed in the adiabatic process. In particular, among the variables, some represent the size of the system. This is the case of the number of moles or the number of molecules ($$N$$).

Rewriting the previous formula with the explicit dependence on $$N$$, we have: $$U(X_B,N) = U(X_A,N) + \int_{X_A}^{X_B}{\rm d}xW_{ad}(x,N).$$ Therefore, it is true that only differences of energy have physical significance, because the difference between two generic states with the same number of molecules, $$U(X_B',N)-U(X_B,N)$$ does not depend on the value of the reference energy $$U(X_A,N)$$. However, the explicit argument $$N$$ shows that it is not possible to arrive to the same conclusion if the two states correspond to a different number of molecules: in that case, the difference $$U(X_B',N')-U(X_B,N)$$ depends explicitly on the difference $$U(X_A,N')-U(X_A,N)$$.

Indeed, this point was clear to Callen since, at the end of section 1-7 (I am referring to the second edition of his book) he stated explicitly:

...the methods of mechanics permit us to measure the energy difference of any two states with equal mole number.

Two paragraphs after this sentence, he added a recipe for relating the energies of states with a different number of moles: the energy of a compound system is defined as the sum of the original subsystems.

This implies that once one has arbitrarily chosen the zero of the energy of each isolated subsystem, it is not allowed to vary the zero of energy of the compound system. More formally, if energy of a subsystem $$1$$ is defined as $$U(X_{1B},N_1) = U(X_{1A},N_1) + \int_{X_{1A}}^{X_{1B}}{\rm d}xW_{ad}(x,N_1)$$ and similarly for sybsystem $$2$$, the energy of the compound system is $$U(X_{1B},N_1) +U(X_{2B},N_2)= U(X_{1A},N_1) + U(X_{2A},N_2) + \int_{X_{1A}}^{X_{1B}}{\rm d}xW_{ad}(x,N_1) +\int_{X_{2A}}^{X_{2B}}{\rm d}xW_{ad}(x,N_2)$$ and the energy of the reference state of the compound system $$U(X_{1A},N_1) + U(X_{2A},N_2)$$ cannot be changed at will. It has to stick to the (arbitrary) choice of the energy of the reference states of the subsystems. Said in another way, it is not allowed to vary independently the zero of energy of each subsystem and that of the compound system.